{"id":5160,"date":"2014-12-11T02:29:23","date_gmt":"2014-12-11T02:29:23","guid":{"rendered":"https:\/\/courses.candelalearning.com\/colphysics\/?post_type=chapter&#038;p=5160"},"modified":"2016-11-03T18:47:06","modified_gmt":"2016-11-03T18:47:06","slug":"23-12-rlc-series-ac-circuits","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-physics\/chapter\/23-12-rlc-series-ac-circuits\/","title":{"raw":"RLC Series AC Circuits","rendered":"RLC Series AC Circuits"},"content":{"raw":"<div>\r\n<div>\r\n<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<div>\r\n<div>\r\n<ul>\r\n\t<li>Calculate the impedance, phase angle, resonant frequency, power, power factor, voltage, and\/or current in a RLC series circuit.<\/li>\r\n\t<li>Draw the circuit diagram for an RLC series circuit.<\/li>\r\n\t<li>Explain the significance of the resonant frequency.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div title=\"Impedance\">\r\n<div>\r\n<div>\r\n<div>\r\n<h2>Impedance<\/h2>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWhen alone in an AC circuit, inductors, capacitors, and resistors all impede current. How do they behave when all three occur together? Interestingly, their individual resistances in ohms do not simply add. Because inductors and capacitors behave in opposite ways, they partially to totally cancel each other\u2019s effect. Figure 1\u00a0shows an <em>RLC <\/em>series circuit with an AC voltage source, the behavior of which is the subject of this section. The crux of the analysis of an <em>RLC<\/em> circuit is the frequency dependence of <em>X<\/em><sub><em>L<\/em><\/sub> and <em>X<\/em><sub><em>C<\/em><\/sub>, and the effect they have on the phase of voltage versus current (established in the preceding section). These give rise to the frequency dependence of the circuit, with important \u201cresonance\u201d features that are the basis of many applications, such as radio tuners.\r\n<div title=\"Figure 23.49.\">\r\n<div>\r\n<div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"275\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20110957\/Figure_24_12_01a.jpg\" alt=\"The figure describes an R LC series circuit. It shows a resistor R connected in series with an inductor L, connected to a capacitor C in series to an A C source V. The voltage of the A C source is given by V equals V zero sine two pi f t. The voltage across R is V R, across L is V L and across C is V C.\" width=\"275\" height=\"417\" \/> Figure 1. An RLC series circuit with an AC voltage source.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe combined effect of resistance <em>R<\/em>, inductive reactance <em>X<\/em><sub><em>L<\/em><\/sub>, and capacitive reactance <em>X<\/em><sub><em>C<\/em><\/sub> is defined to be <em> impedance<\/em>, an AC analogue to resistance in a DC circuit. Current, voltage, and impedance in an <em>RLC<\/em> circuit are related by an AC version of Ohm\u2019s law:\r\n<p style=\"text-align: center;\">[latex]{I}_{0}=\\frac{{V}_{0}}{Z}\\text{ or }{I}_{\\text{rms}}=\\frac{{V}_{\\text{rms}}}{Z}\\\\[\/latex].<\/p>\r\nHere <em>I<\/em><sub>0<\/sub> is the peak current, <em>V<\/em><sub>0<\/sub> the peak source voltage, and <em>Z<\/em> is the impedance of the circuit. The units of impedance are ohms, and its effect on the circuit is as you might expect: the greater the impedance, the smaller the current. To get an expression for <em>Z<\/em> in terms of <em>R<\/em> , <em>X<\/em><sub><em>L<\/em><\/sub>, and <em>X<\/em><sub><em>C<\/em><\/sub>, we will now examine how the voltages across the various components are related to the source voltage. Those voltages are labeled <em>V<\/em><sub><em>R<\/em><\/sub>, <em>V<\/em><sub><em>L<\/em><\/sub>, and <em>V<\/em><sub><em>C<\/em><\/sub> in Figure 1. Conservation of charge requires current to be the same in each part of the circuit at all times, so that we can say the currents in <em>R<\/em>, <em>L<\/em>, and <em>C<\/em> are equal and in phase. But we know from the preceding section that the voltage across the inductor <em>V<\/em><sub><em>L<\/em><\/sub> leads the current by one-fourth of a cycle, the voltage across the capacitor <em>V<\/em><sub><em>C<\/em><\/sub> follows the current by one-fourth of a cycle, and the voltage across the resistor <em>V<\/em><sub><em>R<\/em><\/sub> is exactly in phase with the current. Figure 2\u00a0shows these relationships in one graph, as well as showing the total voltage around the circuit <em>V\u00a0<\/em>=\u00a0<em>V<\/em><sub><em>R\u00a0<\/em><\/sub>+\u00a0<em>V<\/em><sub><em>L\u00a0<\/em><\/sub>+\u00a0<em>V<\/em><sub><em>C<\/em><\/sub>, where all four voltages are the instantaneous values. According to Kirchhoff\u2019s loop rule, the total voltage around the circuit <em>V<\/em> is also the voltage of the source. You can see from Figure 2\u00a0that while <em>V<\/em><sub><em>R<\/em><\/sub> is in phase with the current, <em>V<\/em><sub><em>L<\/em><\/sub> leads by 90\u00ba, and <em>V<\/em><sub><em>C<\/em><\/sub> follows by 90\u00ba. Thus <em>V<\/em><sub><em>L<\/em><\/sub> and <em>V<\/em><sub><em>C<\/em><\/sub> are 180\u00ba out of phase (crest to trough) and tend to cancel, although not completely unless they have the same magnitude. Since the peak voltages are not aligned (not in phase), the peak voltage <em>V<\/em><sub>0<\/sub> of the source does <em>not<\/em> equal the sum of the peak voltages across <em>R<\/em>, <em>L<\/em>, and <em>C<\/em>. The actual relationship is\r\n<p style=\"text-align: center;\">[latex]{V}_{0}=\\sqrt{{{{V}_{0R}}^{2}+\\left({V}_{0L}-{V}_{0C}\\right)^{2}}}\\\\[\/latex],<\/p>\r\nwhere <em>V<\/em><sub>0<em>R<\/em><\/sub>, <em>V<\/em><sub>0<em>L<\/em><\/sub>, and <em>V<\/em><sub>0<em>C<\/em><\/sub> are the peak voltages across <em>R<\/em>, <em>L<\/em>, and <em>C<\/em>, respectively. Now, using Ohm\u2019s law and definitions from <a title=\"23.11. Reactance, Inductive and Capacitive\" href=\".\/chapter\/23-11-reactance-inductive-and-capacitive\/\" target=\"_blank\">Reactance, Inductive and Capacitive<\/a>, we substitute <em>V<\/em><sub>0\u00a0<\/sub>=\u00a0<em>I<\/em><sub>0<\/sub><em>Z<\/em> into the above, as well as <em>V<\/em><sub>0<em>R\u00a0<\/em><\/sub>=\u00a0<em>I<\/em><sub>0<\/sub><em>R<\/em>, <em>V<\/em><sub>0<em>L\u00a0<\/em><\/sub>=\u00a0<em>I<\/em><sub>0<\/sub><em>X<\/em><sub><em>L<\/em><\/sub>, and <em>V<\/em><sub>0<em>C\u00a0<\/em><\/sub>=\u00a0<em>I<\/em><sub>0<\/sub><em>X<\/em><sub><em>C<\/em><\/sub>, yielding\r\n<p style=\"text-align: center;\">[latex]{I}_{0}Z=\\sqrt{{{I}_{0}}^{2}{R}^{2}+\\left({I}_{0}{X}_{L}-{I}_{0}{X}_{C}\\right)^{2}}={I}_{0}\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\[\/latex]<\/p>\r\n<em>I<\/em><sub>0<\/sub> cancels to yield an expression for <em>Z<\/em>:\r\n<p style=\"text-align: center;\">[latex]Z=\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\[\/latex],<\/p>\r\nwhich is the impedance of an <em>RLC<\/em> series AC circuit. For circuits without a resistor, take <em>R<\/em> = 0; for those without an inductor, take <em>X<\/em><sub><em>L\u00a0<\/em><\/sub>= 0; and for those without a capacitor, take <em>X<\/em><sub><em>C\u00a0<\/em><\/sub>= 0.\r\n<div title=\"Figure 23.50.\">\r\n<div>\r\n<div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"300\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20111001\/Figure_24_12_02a.jpg\" alt=\"The figure shows graphs showing the relationships of the voltages in an RLC circuit to the current. It has five graphs on the left and two graphs on the right. The first graph on the right is for current I versus time t. Current is plotted along Y axis and time is along X axis. The curve is a smooth progressive sine wave. The second graph is on the right is for voltage V R versus time t. Voltage V R is plotted along Y axis and time is along X axis. The curve is a smooth progressive sine wave. The third graph is on the right is for voltage V L versus time t. Voltage V L is plotted along Y axis and time is along X axis. The curve is a smooth progressive cosine wave. The fourth graph is on the right is for voltage V C versus time t. Voltage V C is plotted along Y axis and time t is along X axis. The curve is a smooth progressive cosine wave starting from negative Y axis. The fifth graph shows the voltage V verses time t for the R L C circuit. Voltage V is plotted along Y axis and time t is along X axis. The curve is a smooth progressive sine wave starting from a point near to origin on negative X axis. The first and the fifth graphs are again shown on the right and their amplitudes and phases compared. The current graph is shown to have a lesser amplitude.\" width=\"300\" height=\"476\" \/> Figure 2. This graph shows the relationships of the voltages in an <em>RLC<\/em> circuit to the current. The voltages across the circuit elements add to equal the voltage of the source, which is seen to be out of phase with the current.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div title=\"Example 23.12. Calculating Impedance and Current\">\r\n<div class=\"textbox examples\">\r\n<h3>Example 1. Calculating Impedance and Current<\/h3>\r\n<div>\r\n\r\nAn <em>RLC <\/em>series circuit has a 40.0 \u03a9 resistor, a 3.00 mH inductor, and a 5.00 \u03bcF capacitor. (a) Find the circuit\u2019s impedance at 60.0 Hz and 10.0 kHz, noting that these frequencies and the values for <em>L<\/em> and <em>C<\/em> are the same as in Example 1 and Example 2 from <a href=\".\/chapter\/23-11-reactance-inductive-and-capacitive\/\" target=\"_blank\">Reactance, Inductive, and Capacitive<\/a>.\u00a0(b) If the voltage source has <em>V<\/em><sub>rms\u00a0<\/sub>= 120 V, what is <em>I<\/em><sub>rms<\/sub> at each frequency?\r\n<h4><strong>Strategy<\/strong><\/h4>\r\nFor each frequency, we use [latex]Z=\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\[\/latex]\u00a0to find the impedance and then Ohm\u2019s law to find current. We can take advantage of the results of the previous two examples rather than calculate the reactances again.\r\n<h4><strong>Solution for (a)<\/strong><\/h4>\r\nAt 60.0 Hz, the values of the reactances were found in Example 1 from <a href=\".\/chapter\/23-11-reactance-inductive-and-capacitive\/\" target=\"_blank\">Reactance, Inductive, and Capacitive<\/a>\u00a0to be\u00a0<span id=\"MathJax-Element-7174-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-113330\" class=\"math\"><span id=\"MathJax-Span-113331\" class=\"mrow\"><span id=\"MathJax-Span-113332\" class=\"semantics\"><span id=\"MathJax-Span-113333\" class=\"mrow\"><span id=\"MathJax-Span-113334\" class=\"mrow\"><span id=\"MathJax-Span-113335\" class=\"mrow\"><span id=\"MathJax-Span-113336\" class=\"mrow\"><span id=\"MathJax-Span-113337\" class=\"mrow\"><em><span id=\"MathJax-Span-113338\" class=\"msub\"><span id=\"MathJax-Span-113339\" class=\"mi\">X<\/span><sub><span id=\"MathJax-Span-113340\" class=\"mrow\"><span id=\"MathJax-Span-113341\" class=\"mi\">L\u00a0<\/span><\/span><\/sub><\/span><\/em><span id=\"MathJax-Span-113342\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-113343\" class=\"mn\">1<\/span><\/span><span id=\"MathJax-Span-113344\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-113345\" class=\"mtext\">13\u00a0<\/span><span id=\"MathJax-Span-113346\" class=\"mspace\"><\/span><span id=\"MathJax-Span-113347\" class=\"mo\">\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0and in Example 2 from <a href=\".\/chapter\/23-11-reactance-inductive-and-capacitive\/\" target=\"_blank\">Reactance, Inductive, and Capacitive<\/a>\u00a0to be\u00a0<span id=\"MathJax-Element-7175-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-113349\" class=\"math\"><span id=\"MathJax-Span-113350\" class=\"mrow\"><span id=\"MathJax-Span-113351\" class=\"semantics\"><span id=\"MathJax-Span-113352\" class=\"mrow\"><span id=\"MathJax-Span-113353\" class=\"mrow\"><span id=\"MathJax-Span-113354\" class=\"mrow\"><span id=\"MathJax-Span-113355\" class=\"mrow\"><span id=\"MathJax-Span-113356\" class=\"mrow\"><em><span id=\"MathJax-Span-113357\" class=\"msub\"><span id=\"MathJax-Span-113358\" class=\"mi\">X<\/span><span id=\"MathJax-Span-113359\" class=\"mrow\"><span id=\"MathJax-Span-113360\" class=\"mi\"><sub>C<\/sub>\u00a0<\/span><\/span><\/span><\/em><span id=\"MathJax-Span-113361\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-113362\" class=\"mtext\">531\u00a0<\/span><\/span><span id=\"MathJax-Span-113363\" class=\"mspace\"><\/span><span id=\"MathJax-Span-113364\" class=\"mo\">\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>. Entering these and the given 40.0 \u03a9 for resistance into [latex]Z=\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\[\/latex]\u00a0yields\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}Z &amp; =&amp; \\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\ &amp; =&amp; \\sqrt{\\left(40.0\\text{ }\\Omega\\right)^{2}+\\left(1.13\\text{ }\\Omega -531\\text{ }\\Omega\\right)^{2}}\\\\ &amp; =&amp; 531\\text{ }\\Omega\\text{ at }60.0\\text{ Hz}\\end{array}\\\\[\/latex]<\/p>\r\nSimilarly, at 10.0 kHz, <em>X<sub>L\u00a0<\/sub><\/em>= 188 \u03a9\u00a0and\u00a0<em>X<sub>C\u00a0<\/sub><\/em>= 3.18 \u03a9, so that\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}Z &amp; =&amp; \\sqrt{\\left(40.0\\text{ }\\Omega\\right)^{2}+\\left(188\\text{ }\\Omega -3.18\\text{ }\\Omega\\right)^{2}}\\\\ &amp; =&amp; 190\\text{ }\\Omega\\text{ at }10.0\\text{ kHz}\\end{array}\\\\[\/latex]<\/p>\r\n\r\n<h4><strong>Discussion for (a)<\/strong><\/h4>\r\nIn both cases, the result is nearly the same as the largest value, and the impedance is definitely not the sum of the individual values. It is clear that <em>X<\/em><sub><em>L<\/em><\/sub> dominates at high frequency and <em>X<\/em><sub><em>C<\/em><\/sub> dominates at low frequency.\r\n<h4><strong>Solution for (b)<\/strong><\/h4>\r\nThe current <em>I<\/em><sub>rms<\/sub> can be found using the AC version of Ohm\u2019s law in the equation <em>I<\/em><sub>rms<\/sub>=<em>V<\/em><sub>rms<\/sub>\/<em>Z<\/em>:\r\n<p style=\"text-align: center;\">[latex]{I}_{\\text{rms}}=\\frac{{V}_{\\text{rms}}}{Z}=\\frac{120\\text{V}}{531\\text{ }\\Omega}=0.226\\text{ A}\\\\[\/latex] at 60.0 Hz<\/p>\r\nFinally, at 10.0 kHz, we find\r\n<p style=\"text-align: center;\">[latex]{I}_{\\text{rms}}=\\frac{{V}_{\\text{rms}}}{Z}=\\frac{120\\text{ V}}{190\\text{ }\\Omega}=0.633\\text{ A}\\\\[\/latex] at 10.0 kHz<\/p>\r\n\r\n<h4><strong>Discussion for (a)<\/strong><\/h4>\r\nThe current at 60.0 Hz is the same (to three digits) as found for the capacitor alone in Example 2 from <a href=\".\/chapter\/23-11-reactance-inductive-and-capacitive\/\" target=\"_blank\">Reactance, Inductive, and Capacitive<\/a>. The capacitor dominates at low frequency. The current at 10.0 kHz is only slightly different from that found for the inductor alone in Example 1 from <a href=\".\/chapter\/23-11-reactance-inductive-and-capacitive\/\" target=\"_blank\">Reactance, Inductive, and Capacitive<\/a>. The inductor dominates at high frequency.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div title=\"Resonance in RLC Series AC Circuits\">\r\n<div>\r\n<div>\r\n<div>\r\n<h2>Resonance in <em>RLC<\/em> Series AC Circuits<\/h2>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nHow does an <em>RLC<\/em> circuit behave as a function of the frequency of the driving voltage source? Combining Ohm\u2019s law, <em>I<\/em><sub>rms\u00a0<\/sub>=\u00a0<em>V<\/em><sub>rms<\/sub>\/<em>Z<\/em>, and the expression for impedance <em>Z<\/em> from [latex]Z=\\sqrt{{R}^{2}+\\left({{X}_{L}}-{{X}_{C}}\\right)^{2}}\\\\[\/latex]\u00a0gives\r\n<p style=\"text-align: center;\">[latex]{I}_{\\text{rms}}=\\frac{{V}_{\\text{rms}}}{\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}}\\\\[\/latex]<\/p>\r\nThe reactances vary with frequency, with <em>X<\/em><sub><em>L<\/em><\/sub> large at high frequencies and <em>X<\/em><sub><em>C<\/em><\/sub> large at low frequencies, as we have seen in three previous examples. At some intermediate frequency <em>f<\/em><sub>0<\/sub>, the reactances will be equal and cancel, giving <em><em>Z\u00a0<\/em>=\u00a0<em>R<\/em><\/em> \u2014 this is a minimum value for impedance, and a maximum value for <em>I<\/em><sub>rms<\/sub> results. We can get an expression for <em>f<\/em><sub>0<\/sub> by taking\r\n<div style=\"text-align: center;\" title=\"Equation 23.87.\"><em>X<\/em><sub><em>L\u00a0<\/em><\/sub>=\u00a0<em>X<\/em><sub><em>C<\/em><\/sub>.<\/div>\r\nSubstituting the definitions of <em>X<\/em><sub><em>L<\/em><\/sub> and <em>X<\/em><sub><em>C<\/em><\/sub>,\r\n<p style=\"text-align: center;\">[latex]2\\pi f_{0}L=\\frac{1}{2\\pi f_{0}C}\\\\[\/latex].<\/p>\r\nSolving this expression for <em>f<\/em><sub>0<\/sub> yields\r\n<p style=\"text-align: center;\">[latex]{f}_{0}=\\frac{1}{2\\pi \\sqrt{LC}}\\\\[\/latex],<\/p>\r\nwhere <em>f<\/em><sub>0<\/sub> is the <em> resonant frequency<\/em> of an <em>RLC<\/em> series circuit. This is also the <em>natural frequency<\/em> at which the circuit would oscillate if not driven by the voltage source. At <em>f<\/em><sub>0<\/sub>, the effects of the inductor and capacitor cancel, so that <em><em>Z\u00a0<\/em>=\u00a0<em>R<\/em><\/em>, and <em>I<\/em><sub>rms<\/sub> is a maximum.\r\n\r\nResonance in AC circuits is analogous to mechanical resonance, where resonance is defined to be a forced oscillation\u2014in this case, forced by the voltage source\u2014at the natural frequency of the system. The receiver in a radio is an <em>RLC<\/em> circuit that oscillates best at its <em>f<\/em><sub>0<\/sub>. A variable capacitor is often used to adjust <em>f<\/em><sub>0<\/sub> to receive a desired frequency and to reject others. Figure 3\u00a0is a graph of current as a function of frequency, illustrating a resonant peak in <em>I<\/em><sub>rms<\/sub> at <em>f<\/em><sub>0<\/sub>. The two curves are for two different circuits, which differ only in the amount of resistance in them. The peak is lower and broader for the higher-resistance circuit. Thus the higher-resistance circuit does not resonate as strongly and would not be as selective in a radio receiver, for example.\r\n<div title=\"Figure 23.51.\">\r\n<div>\r\n<div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"225\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20111012\/Figure_24_12_03a.jpg\" alt=\"The figure describes a graph of current I versus frequency f. Current I r m s is plotted along Y axis and frequency f is plotted along X axis. Two curves are shown. The upper curve is for small resistance and lower curve is for large resistance. Both the curves have a smooth rise and a fall. The peaks are marked for frequency f zero. The curve for smaller resistance has a higher value of peak than the curve for large resistance.\" width=\"225\" height=\"285\" \/> Figure 3. A graph of current versus frequency for two RLC series circuits differing only in the amount of resistance. Both have a resonance at <em>f<\/em><sub>0<\/sub>, but that for the higher resistance is lower and broader. The driving AC voltage source has a fixed amplitude <em>V<\/em><sub>0<\/sub>.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div title=\"Example 23.13. Calculating Resonant Frequency and Current\">\r\n<div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2. Calculating Resonant Frequency and Current<\/h3>\r\n<div>\r\n\r\nFor the same <em>RLC<\/em> series circuit having a 40.0 \u03a9 resistor, a 3.00 mH inductor, and a 5.00 \u03bcF capacitor: (a) Find the resonant frequency. (b) Calculate <em>I<\/em><sub>rms<\/sub> at resonance if <em>V<\/em><sub>rms<\/sub> is 120 V.\r\n\r\n<strong>Strategy<\/strong>\r\n\r\nThe resonant frequency is found by using the expression in [latex]{f}_{0}=\\frac{1}{2\\pi\\sqrt{LC}}\\\\[\/latex]. The current at that frequency is the same as if the resistor alone were in the circuit.\r\n\r\n<strong>Solution for (a)<\/strong>\r\n\r\nEntering the given values for <em>L<\/em> and <em>C<\/em> into the expression given for <em>f<\/em><sub>0<\/sub> in [latex]{f}_{0}=\\frac{1}{2\\pi\\sqrt{LC}}\\\\[\/latex]\u00a0yields\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}{f}_{0}&amp;=&amp;\\frac{1}{2\\pi\\sqrt{LC}}\\\\ &amp; =&amp; \\frac{1}{2\\pi \\sqrt{\\left(3.00\\times 10^{-3}\\text{ H}\\right)\\left(5.00\\times 10^{-6}\\text{ F}\\right)}}=1.30\\text{ kHz}\\end{array}\\\\[\/latex]<\/p>\r\n\r\n<h4><strong>Discussion for (a)<\/strong><\/h4>\r\nWe see that the resonant frequency is between 60.0 Hz and 10.0 kHz, the two frequencies chosen in earlier examples. This was to be expected, since the capacitor dominated at the low frequency and the inductor dominated at the high frequency. Their effects are the same at this intermediate frequency.\r\n<h4><strong>Solution for (b)<\/strong><\/h4>\r\nThe current is given by Ohm\u2019s law. At resonance, the two reactances are equal and cancel, so that the impedance equals the resistance alone. Thus,\r\n<p style=\"text-align: center;\">[latex]{I}_{\\text{rms}}=\\frac{{V}_{\\text{rms}}}{Z}=\\frac{120\\text{ V}}{40.0\\text{ }\\Omega}=3.00\\text{ A}\\\\[\/latex].<\/p>\r\n\r\n<h4><strong>Discussion for (b)<\/strong><\/h4>\r\nAt resonance, the current is greater than at the higher and lower frequencies considered for the same circuit in the preceding example.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n<h2>Power in <em>RLC<\/em> Series AC Circuits<\/h2>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div title=\"Power in RLC Series AC Circuits\">\r\n\r\nIf current varies with frequency in an <em>RLC<\/em> circuit, then the power delivered to it also varies with frequency. But the average power is not simply current times voltage, as it is in purely resistive circuits. As was seen in Figure 2, voltage and current are out of phase in an <em>RLC<\/em> circuit. There is a <em> phase angle<\/em><em>\u03d5<\/em> between the source voltage <em>V<\/em> and the current <em>I<\/em>, which can be found from\r\n<p style=\"text-align: center;\">[latex]\\cos\\varphi =\\frac{R}{Z}\\\\[\/latex]<\/p>\r\nFor example, at the resonant frequency or in a purely resistive circuit <em><em>Z\u00a0<\/em>=\u00a0<em>R<\/em><\/em>, so that [latex]\\text{cos}\\varphi =1\\\\[\/latex]. This implies that <em>\u03d5\u00a0<\/em>= 0\u00ba and that voltage and current are in phase, as expected for resistors. At other frequencies, average power is less than at resonance. This is both because voltage and current are out of phase and because <em>I<\/em><sub>rms<\/sub> is lower. The fact that source voltage and current are out of phase affects the power delivered to the circuit. It can be shown that the <em><em>average power<\/em><\/em> is\r\n<p style=\"text-align: center;\">[latex]{P}_{\\text{ave}}={I}_{\\text{rms}}{V}_{\\text{rms}}\\cos\\varphi\\\\[\/latex],<\/p>\r\nThus\u00a0<span id=\"MathJax-Span-114477\" class=\"mtext\">cos\u00a0<\/span><span id=\"MathJax-Span-114478\" class=\"mspace\"><\/span><em><span id=\"MathJax-Span-114479\" class=\"mi\">\u03d5<\/span><\/em>\u00a0is called the <em> power factor<\/em>, which can range from 0 to 1. Power factors near 1 are desirable when designing an efficient motor, for example. At the resonant frequency,\u00a0<span id=\"MathJax-Span-114477\" class=\"mtext\">cos\u00a0<\/span><span id=\"MathJax-Span-114478\" class=\"mspace\"><\/span><em><span id=\"MathJax-Span-114479\" class=\"mi\">\u03d5 = <\/span><\/em><span id=\"MathJax-Span-114479\" class=\"mi\">1<\/span>.\r\n<div title=\"Example 23.14. Calculating the Power Factor and Power\">\r\n<div class=\"textbox examples\">\r\n<h3>Example 3. Calculating the Power Factor and Power<\/h3>\r\n<div>\r\n\r\nFor the same <em>RLC<\/em> series circuit having a 40.0 \u03a9 resistor, a 3.00 mH inductor, a 5.00 \u03bcF capacitor, and a voltage source with a <em>V<\/em><sub>rms<\/sub> of 120 V: (a) Calculate the power factor and phase angle for\u00a0<span id=\"MathJax-Span-114519\" class=\"mrow\"><span id=\"MathJax-Span-114520\" class=\"mi\"><em>f<\/em>\u00a0<\/span><span id=\"MathJax-Span-114521\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-114522\" class=\"mtext\">60<\/span><\/span><span id=\"MathJax-Span-114523\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-114524\" class=\"mn\">0<\/span><span id=\"MathJax-Span-114525\" class=\"mi\"><\/span><span id=\"MathJax-Span-114526\" class=\"mtext\">Hz<\/span>. (b) What is the average power at 50.0 Hz? (c) Find the average power at the circuit\u2019s resonant frequency.\r\n<h4><strong>Strategy and Solution for (a)<\/strong><\/h4>\r\nThe power factor at 60.0 Hz is found from\r\n<p style=\"text-align: center;\">[latex]\\cos\\varphi =\\frac{R}{Z}\\\\[\/latex].<\/p>\r\nWe know <em>Z<\/em> = 531 \u03a9 from <em>Example 1:\u00a0Calculating Impedance and Current<\/em>, so that\r\n<p style=\"text-align: center;\">[latex]\\cos\\varphi =\\frac{40.0\\text{ }\\Omega}{531\\text{ }\\Omega}=0.0753 \\text{ at }60.0\\text{ Hz}\\\\[\/latex].<\/p>\r\nThis small value indicates the voltage and current are significantly out of phase. In fact, the phase angle is\r\n<p style=\"text-align: center;\">[latex]\\varphi ={\\cos}^{-1} 0.0753=\\text{85.7\u00ba}\\text{ at }60.0\\text{ Hz}\\\\[\/latex].<\/p>\r\n\r\n<h4><strong>Discussion for (a)<\/strong><\/h4>\r\nThe phase angle is close to 90\u00ba, consistent with the fact that the capacitor dominates the circuit at this low frequency (a pure <em>RC<\/em> circuit has its voltage and current 90\u00ba out of phase).\r\n<h4><strong>Strategy and Solution for (b)<\/strong><\/h4>\r\nThe average power at 60.0 Hz is\r\n<p style=\"text-align: center;\"><span id=\"MathJax-Span-114620\" class=\"mrow\"><span id=\"MathJax-Span-114621\" class=\"semantics\"><span id=\"MathJax-Span-114622\" class=\"mrow\"><span id=\"MathJax-Span-114623\" class=\"mrow\"><span id=\"MathJax-Span-114624\" class=\"mrow\"><span id=\"MathJax-Span-114625\" class=\"mrow\"><span id=\"MathJax-Span-114626\" class=\"mrow\"><span id=\"MathJax-Span-114627\" class=\"msub\"><em><span id=\"MathJax-Span-114628\" class=\"mi\">P<\/span><\/em><sub><span id=\"MathJax-Span-114629\" class=\"mrow\"><span id=\"MathJax-Span-114630\" class=\"mtext\">ave\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-114631\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-114632\" class=\"msub\"><em><span id=\"MathJax-Span-114633\" class=\"mi\">I<\/span><\/em><sub><span id=\"MathJax-Span-114634\" class=\"mrow\"><span id=\"MathJax-Span-114635\" class=\"mtext\">rms<\/span><\/span><\/sub><\/span><\/span><span id=\"MathJax-Span-114636\" class=\"msub\"><em><span id=\"MathJax-Span-114637\" class=\"mi\">V<\/span><\/em><sub><span id=\"MathJax-Span-114638\" class=\"mrow\"><span id=\"MathJax-Span-114639\" class=\"mtext\">rms\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-114640\" class=\"mtext\">cos\u00a0<\/span><span id=\"MathJax-Span-114641\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114642\" class=\"mi\"><em>\u03d5<\/em>.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\r\n<em>I<\/em><sub>rms<\/sub> was found to be 0.226 A in <em>Example 1:\u00a0Calculating Impedance and Current<\/em>. Entering the known values gives\r\n<p style=\"text-align: center;\"><span id=\"MathJax-Span-114656\" class=\"mrow\"><span id=\"MathJax-Span-114657\" class=\"semantics\"><span id=\"MathJax-Span-114658\" class=\"mrow\"><span id=\"MathJax-Span-114659\" class=\"mrow\"><span id=\"MathJax-Span-114660\" class=\"mrow\"><span id=\"MathJax-Span-114661\" class=\"mrow\"><span id=\"MathJax-Span-114662\" class=\"mrow\"><span id=\"MathJax-Span-114663\" class=\"msub\"><em><span id=\"MathJax-Span-114664\" class=\"mi\">P<\/span><\/em><sub><span id=\"MathJax-Span-114665\" class=\"mrow\"><span id=\"MathJax-Span-114666\" class=\"mtext\">ave\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-114667\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-114668\" class=\"mo\">(<\/span><\/span><span id=\"MathJax-Span-114669\" class=\"mn\">0<\/span><span id=\"MathJax-Span-114670\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-114671\" class=\"mtext\">226\u00a0<\/span><span id=\"MathJax-Span-114672\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114673\" class=\"mtext\">A<\/span><span id=\"MathJax-Span-114674\" class=\"mo\">)<\/span><span id=\"MathJax-Span-114675\" class=\"mo\">(<\/span><span id=\"MathJax-Span-114676\" class=\"mtext\">120\u00a0<\/span><span id=\"MathJax-Span-114677\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114678\" class=\"mtext\">V<\/span><span id=\"MathJax-Span-114679\" class=\"mo\">)<\/span><span id=\"MathJax-Span-114680\" class=\"mo\">(<\/span><span id=\"MathJax-Span-114681\" class=\"mn\">0<\/span><span id=\"MathJax-Span-114682\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-114683\" class=\"mtext\">0753<\/span><span id=\"MathJax-Span-114684\" class=\"mrow\"><span id=\"MathJax-Span-114685\" class=\"mo\">)\u00a0<\/span><span id=\"MathJax-Span-114686\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-114687\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-114688\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-114689\" class=\"mtext\">04\u00a0<\/span><span id=\"MathJax-Span-114690\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114691\" class=\"mtext\">W at 60.0 Hz.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\r\n\r\n<h4><strong>Strategy and Solution for (c)<\/strong><\/h4>\r\nAt the resonant frequency, we know\u00a0<span id=\"MathJax-Span-114477\" class=\"mtext\">cos\u00a0<\/span><span id=\"MathJax-Span-114478\" class=\"mspace\"><\/span><em><span id=\"MathJax-Span-114479\" class=\"mi\">\u03d5 = <\/span><\/em><span id=\"MathJax-Span-114479\" class=\"mi\">1<\/span>, and <em>I<\/em><sub>rms<\/sub> was found to be 6.00 A in <em>Example 3: Calculating Resonant Frequency and Current<\/em>. Thus,\u00a0<span id=\"MathJax-Element-7251-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-114718\" class=\"math\"><span id=\"MathJax-Span-114719\" class=\"mrow\"><span id=\"MathJax-Span-114720\" class=\"semantics\"><span id=\"MathJax-Span-114721\" class=\"mrow\"><span id=\"MathJax-Span-114722\" class=\"mrow\"><span id=\"MathJax-Span-114723\" class=\"mrow\"><span id=\"MathJax-Span-114724\" class=\"mrow\"><span id=\"MathJax-Span-114725\" class=\"mrow\"><span id=\"MathJax-Span-114726\" class=\"msub\"><em><span id=\"MathJax-Span-114727\" class=\"mi\">P<\/span><\/em><sub><span id=\"MathJax-Span-114728\" class=\"mrow\"><span id=\"MathJax-Span-114729\" class=\"mtext\">ave\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-114730\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-114731\" class=\"mo\">(<\/span><\/span><span id=\"MathJax-Span-114732\" class=\"mn\">3<\/span><span id=\"MathJax-Span-114733\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-114734\" class=\"mtext\">00\u00a0<\/span><span id=\"MathJax-Span-114735\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114736\" class=\"mtext\">A<\/span><span id=\"MathJax-Span-114737\" class=\"mo\">)<\/span><span id=\"MathJax-Span-114738\" class=\"mo\">(<\/span><span id=\"MathJax-Span-114739\" class=\"mtext\">120\u00a0<\/span><span id=\"MathJax-Span-114740\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114741\" class=\"mtext\">V<\/span><span id=\"MathJax-Span-114742\" class=\"mo\">)<\/span><span id=\"MathJax-Span-114743\" class=\"mo\">(<\/span><span id=\"MathJax-Span-114744\" class=\"mn\">1<\/span><span id=\"MathJax-Span-114745\" class=\"mrow\"><span id=\"MathJax-Span-114746\" class=\"mo\">)\u00a0<\/span><span id=\"MathJax-Span-114747\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-114748\" class=\"mtext\">360\u00a0<\/span><\/span><span id=\"MathJax-Span-114749\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114750\" class=\"mtext\">W<\/span><\/span><\/span><span id=\"MathJax-Span-114751\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> at resonance (1.30 kHz)\r\n<h4><strong>Discussion<\/strong><\/h4>\r\nBoth the current and the power factor are greater at resonance, producing significantly greater power than at higher and lower frequencies.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nPower delivered to an <em>RLC<\/em> series AC circuit is dissipated by the resistance alone. The inductor and capacitor have energy input and output but do not dissipate it out of the circuit. Rather they transfer energy back and forth to one another, with the resistor dissipating exactly what the voltage source puts into the circuit. This assumes no significant electromagnetic radiation from the inductor and capacitor, such as radio waves. Such radiation can happen and may even be desired, as we will see in the next chapter on electromagnetic radiation, but it can also be suppressed as is the case in this chapter. The circuit is analogous to the wheel of a car driven over a corrugated road as shown in Figure 4. The regularly spaced bumps in the road are analogous to the voltage source, driving the wheel up and down. The shock absorber is analogous to the resistance damping and limiting the amplitude of the oscillation. Energy within the system goes back and forth between kinetic (analogous to maximum current, and energy stored in an inductor) and potential energy stored in the car spring (analogous to no current, and energy stored in the electric field of a capacitor). The amplitude of the wheels\u2019 motion is a maximum if the bumps in the road are hit at the resonant frequency.\r\n<div title=\"Figure 23.52.\">\r\n<div>\r\n<div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"225\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20111023\/Figure_24_12_04a.jpg\" alt=\"The figure describes the path of motion of a wheel of a car. The front wheel of a car is shown. A shock absorber attached to the wheel is also shown. The path of motion is shown as vertically up and down.\" width=\"225\" height=\"389\" \/> Figure 4. The forced but damped motion of the wheel on the car spring is analogous to an<em> RLC<\/em> series AC circuit. The shock absorber damps the motion and dissipates energy, analogous to the resistance in an <em>RLC<\/em> circuit. The mass and spring determine the resonant frequency.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nA pure <em>LC<\/em> circuit with negligible resistance oscillates at <em>f<\/em><sub>0<\/sub>, the same resonant frequency as an <em>RLC<\/em> circuit. It can serve as a frequency standard or clock circuit\u2014for example, in a digital wristwatch. With a very small resistance, only a very small energy input is necessary to maintain the oscillations. The circuit is analogous to a car with no shock absorbers. Once it starts oscillating, it continues at its natural frequency for some time. Figure 5\u00a0shows the analogy between an <em>LC<\/em> circuit and a mass on a spring.\r\n<div title=\"Figure 23.53.\">\r\n<div>\r\n<div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"350\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20111024\/Figure_24_12_05a.jpg\" alt=\"The figure describes four stages of an L C oscillation circuit compared to a mass oscillating on a spring. Part a of the figure shows a mass attached to a horizontal spring. The spring is attached to a fixed support on the left. The mass is at rest as shown by velocity v equals zero. The energy of the spring is shown as potential energy. This is compared with a circuit containing a capacitor C and inductor L connected together. The energy is shown as stored in the electric field E of the capacitor between the plates. One plate is shown to have a negative polarity and other plate is shown to have a positive polarity. Part b of the figure shows a mass attached to a horizontal spring which is attached to a fixed support on the left. The mass is shown to move horizontal toward the fixed support with velocity v. The energy here is stored as the kinetic energy of the spring. This is compared with a circuit containing a capacitor C and inductor L connected together. A current is shown in the circuit and energy is stored as magnetic field B in the inductor. Part c of the figure shows a mass attached to a horizontal spring which is attached to a fixed support on the left. The spring is shown as not stretched and the energy is shown as potential energy of the spring. The mass is show to have displaced toward left. This is compared with a circuit containing a capacitor C and inductor L connected together. The energy is shown as stored in the electric field E of the capacitor between the plates. One plate is shown to have a negative polarity and other plate is shown to have a positive polarity. But the polarities are reverse of the first case in part a. Part d of the figure shows a mass attached to a horizontal spring which is attached to a fixed support on the left. The mass is shown to move toward right with velocity v. the energy of the spring is kinetic energy. This is compared with a circuit containing a capacitor C and inductor L connected together. A current is shown in the circuit opposite to that in part b and energy is stored as magnetic field B in the inductor.\" width=\"350\" height=\"1089\" \/> Figure 5. An LC circuit is analogous to a mass oscillating on a spring with no friction and no driving force. Energy moves back and forth between the inductor and capacitor, just as it moves from kinetic to potential in the mass-spring system.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox\">\r\n<div>\r\n<h2><strong>PhET Explorations: Circuit Construction Kit (AC+DC), Virtual Lab<\/strong><\/h2>\r\n<div>Build circuits with capacitors, inductors, resistors and AC or DC voltage sources, and inspect them using lab instruments such as voltmeters and ammeters.<\/div>\r\n<\/div>\r\n<div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"300\"]<a href=\"http:\/\/phet.colorado.edu\/sims\/circuit-construction-kit\/circuit-construction-kit-ac_en.jnlp\"><img src=\"http:\/\/phet.colorado.edu\/sims\/circuit-construction-kit\/circuit-construction-kit-ac-600.png\" alt=\"Circuit Construction Kit (AC+DC)\" width=\"300\" height=\"197\" \/><\/a> Click to download the simulation. Run using Java.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>Section Summary<\/h2>\r\n<ul>\r\n\t<li>The AC analogy to resistance is impedance <em>Z<\/em>, the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm\u2019s law:\r\n<div style=\"text-align: center;\">[latex]{I}_{0}=\\frac{{V}_{0}}{Z}\\text{ or }{I}_{\\text{rms}}=\\frac{{V}_{\\text{rms}}}{Z}\\\\[\/latex],<\/div>\r\nwhere <em>I<\/em><sub>o<\/sub>\u00a0is the peak current and <em>V<\/em><sub>o<\/sub>\u00a0is the peak source voltage.<\/li>\r\n\t<li>Impedance has units of ohms and is given by [latex]Z=\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\[\/latex].<\/li>\r\n\t<li>The resonant frequency\u00a0<span id=\"MathJax-Element-7896-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123708\" class=\"math\"><span id=\"MathJax-Span-123709\" class=\"mrow\"><span id=\"MathJax-Span-123710\" class=\"semantics\"><span id=\"MathJax-Span-123711\" class=\"mrow\"><span id=\"MathJax-Span-123712\" class=\"mrow\"><span id=\"MathJax-Span-123713\" class=\"mrow\"><span id=\"MathJax-Span-123714\" class=\"msub\"><em><span id=\"MathJax-Span-123715\" class=\"mi\">f<\/span><\/em><sub><span id=\"MathJax-Span-123716\" class=\"mrow\"><span id=\"MathJax-Span-123717\" class=\"mn\">0<\/span><\/span><\/sub><\/span><\/span><span id=\"MathJax-Span-123718\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, at which <span id=\"MathJax-Element-7897-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123719\" class=\"math\"><span id=\"MathJax-Span-123720\" class=\"mrow\"><span id=\"MathJax-Span-123721\" class=\"semantics\"><span id=\"MathJax-Span-123722\" class=\"mrow\"><span id=\"MathJax-Span-123723\" class=\"mrow\"><span id=\"MathJax-Span-123724\" class=\"mrow\"><span id=\"MathJax-Span-123725\" class=\"mrow\"><em><span id=\"MathJax-Span-123726\" class=\"msub\"><span id=\"MathJax-Span-123727\" class=\"mi\">X<\/span><span id=\"MathJax-Span-123728\" class=\"mrow\"><span id=\"MathJax-Span-123729\" class=\"mi\"><sub>L<\/sub>\u00a0<\/span><\/span><\/span><\/em><span id=\"MathJax-Span-123730\" class=\"mo\">=\u00a0<\/span><em><span id=\"MathJax-Span-123731\" class=\"msub\"><span id=\"MathJax-Span-123732\" class=\"mi\">X<\/span><sub><span id=\"MathJax-Span-123733\" class=\"mrow\"><span id=\"MathJax-Span-123734\" class=\"mi\">C<\/span><\/span><\/sub><\/span><\/em><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, is\r\n<div style=\"text-align: center;\">[latex]{f}_{0}=\\frac{1}{2\\pi \\sqrt{LC}}\\\\[\/latex]<\/div><\/li>\r\n\t<li>In an AC circuit, there is a phase angle\u00a0<em data-effect=\"italics\"><span id=\"MathJax-Element-7899-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123762\" class=\"math\"><span id=\"MathJax-Span-123763\" class=\"mrow\"><span id=\"MathJax-Span-123764\" class=\"semantics\"><span id=\"MathJax-Span-123765\" class=\"mrow\"><span id=\"MathJax-Span-123766\" class=\"mrow\"><span id=\"MathJax-Span-123767\" class=\"mrow\"><span id=\"MathJax-Span-123768\" class=\"mi\">\u03d5<\/span><\/span><span id=\"MathJax-Span-123769\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/em> between source voltage <span id=\"MathJax-Element-7900-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123770\" class=\"math\"><span id=\"MathJax-Span-123771\" class=\"mrow\"><span id=\"MathJax-Span-123772\" class=\"semantics\"><span id=\"MathJax-Span-123773\" class=\"mrow\"><span id=\"MathJax-Span-123774\" class=\"mrow\"><em><span id=\"MathJax-Span-123775\" class=\"mrow\"><span id=\"MathJax-Span-123776\" class=\"mi\">V<\/span><\/span><\/em><span id=\"MathJax-Span-123777\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and the current <span id=\"MathJax-Element-7901-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123778\" class=\"math\"><span id=\"MathJax-Span-123779\" class=\"mrow\"><span id=\"MathJax-Span-123780\" class=\"semantics\"><span id=\"MathJax-Span-123781\" class=\"mrow\"><span id=\"MathJax-Span-123782\" class=\"mrow\"><em><span id=\"MathJax-Span-123783\" class=\"mrow\"><span id=\"MathJax-Span-123784\" class=\"mi\">I<\/span><\/span><\/em><\/span><\/span><\/span><\/span><\/span><\/span>, which can be found from\r\n<div style=\"text-align: center;\">[latex]\\text{cos}\\varphi =\\frac{R}{Z}\\\\[\/latex],<\/div><\/li>\r\n\t<li><span id=\"MathJax-Span-123812\" class=\"mi\"><em>\u03d5<\/em>\u00a0<\/span><span id=\"MathJax-Span-123813\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-123814\" class=\"mn\">0\u00ba<\/span> for a purely resistive circuit or an <em>RLC<\/em> circuit at resonance.<\/li>\r\n\t<li>The average power delivered to an <em>RLC<\/em> circuit is affected by the phase angle and is given by\r\n<div style=\"text-align: center;\">[latex]{P}_{\\text{ave}}={I}_{\\text{rms}}{V}_{\\text{rms}}\\cos\\varphi\\\\[\/latex],<\/div>\r\n<span id=\"MathJax-Element-7905-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123845\" class=\"math\"><span id=\"MathJax-Span-123846\" class=\"mrow\"><span id=\"MathJax-Span-123847\" class=\"semantics\"><span id=\"MathJax-Span-123848\" class=\"mrow\"><span id=\"MathJax-Span-123849\" class=\"mrow\"><span id=\"MathJax-Span-123850\" class=\"mrow\"><span id=\"MathJax-Span-123851\" class=\"mrow\"><span id=\"MathJax-Span-123852\" class=\"mtext\">cos\u00a0<\/span><span id=\"MathJax-Span-123853\" class=\"mspace\"><\/span><em><span id=\"MathJax-Span-123854\" class=\"mi\">\u03d5<\/span><\/em><\/span><\/span><span id=\"MathJax-Span-123855\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is called the power factor, which ranges from 0 to 1.<\/li>\r\n<\/ul>\r\n<section>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Conceptual Questions<\/h3>\r\n<div>\r\n<div>\r\n\r\n1. Does the resonant frequency of an AC circuit depend on the peak voltage of the AC source? Explain why or why not.\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n\r\n2. Suppose you have a motor with a power factor significantly less than 1. Explain why it would be better to improve the power factor as a method of improving the motor\u2019s output, rather than to increase the voltage input.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section>\r\n<div class=\"textbox exercises\">\r\n<h3>Problems &amp; Exercises<\/h3>\r\n<div>\r\n<div>\r\n\r\n1. An <em>RL<\/em> circuit consists of a 40.0 \u03a9 resistor and a 3.00 mH inductor. (a) Find its impedance <em>Z<\/em>\u00a0at 60.0 Hz and 10.0 kHz. (b) Compare these values of\u00a0<em>Z<\/em> with those found in <em>Example 1:\u00a0Calculating Impedance and Current<\/em>\u00a0in which there was also a capacitor.\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n\r\n2. An <em>RC<\/em> circuit consists of a\u00a0<span id=\"MathJax-Element-7913-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123891\" class=\"math\"><span id=\"MathJax-Span-123892\" class=\"mrow\"><span id=\"MathJax-Span-123893\" class=\"semantics\"><span id=\"MathJax-Span-123894\" class=\"mrow\"><span id=\"MathJax-Span-123895\" class=\"mi\">40.0 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> resistor and a <span id=\"MathJax-Element-7914-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123896\" class=\"math\"><span id=\"MathJax-Span-123897\" class=\"mrow\"><span id=\"MathJax-Span-123898\" class=\"semantics\"><span id=\"MathJax-Span-123899\" class=\"mrow\"><span id=\"MathJax-Span-123900\" class=\"mtext\">5.00 \u03bcF<\/span><\/span><\/span><\/span><\/span><\/span> capacitor. (a) Find its impedance at 60.0 Hz and 10.0 kHz. (b) Compare these values of <em>Z<\/em>\u00a0with those found in\u00a0<em>Example 1:\u00a0Calculating Impedance and Current<\/em>, in which there was also an inductor.\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n\r\n3. An <em>LC<\/em> circuit consists of a\u00a0<span id=\"MathJax-Element-7916-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123906\" class=\"math\"><span id=\"MathJax-Span-123907\" class=\"mrow\"><span id=\"MathJax-Span-123908\" class=\"semantics\"><span id=\"MathJax-Span-123909\" class=\"mrow\"><span id=\"MathJax-Span-123910\" class=\"mrow\"><span id=\"MathJax-Span-123911\" class=\"mrow\"><span id=\"MathJax-Span-123912\" class=\"mrow\"><span id=\"MathJax-Span-123913\" class=\"mn\">3<\/span><span id=\"MathJax-Span-123914\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-123915\" class=\"mtext\">00<\/span><span id=\"MathJax-Span-123916\" class=\"mspace\"><\/span><span id=\"MathJax-Span-123917\" class=\"mtext\">mH<\/span><\/span><\/span><span id=\"MathJax-Span-123918\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> inductor and a <span id=\"MathJax-Element-7917-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123919\" class=\"math\"><span id=\"MathJax-Span-123920\" class=\"mrow\"><span id=\"MathJax-Span-123921\" class=\"semantics\"><span id=\"MathJax-Span-123922\" class=\"mrow\"><span id=\"MathJax-Span-123923\" class=\"mrow\"><span id=\"MathJax-Span-123924\" class=\"mrow\"><span id=\"MathJax-Span-123925\" class=\"mrow\"><span id=\"MathJax-Span-123926\" class=\"mn\">5<\/span><span id=\"MathJax-Span-123927\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-123928\" class=\"mtext\">00\u00a0<\/span><span id=\"MathJax-Span-123929\" class=\"mspace\"><\/span><span id=\"MathJax-Span-123930\" class=\"mi\">\u03bcF<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> capacitor. (a) Find its impedance at 60.0 Hz and 10.0 kHz. (b) Compare these values of\u00a0<em>Z<\/em> with those found in\u00a0<em>Example 1:\u00a0Calculating Impedance and Current<\/em>\u00a0in which there was also a resistor.\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n\r\n4. What is the resonant frequency of a 0.500 mH inductor connected to a 40.0 \u03bcF capacitor?\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n\r\n5. To receive AM radio, you want an <em>RLC<\/em> circuit that can be made to resonate at any frequency between 500 and 1650 kHz. This is accomplished with a fixed\u00a0<span id=\"MathJax-Element-7922-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123955\" class=\"math\"><span id=\"MathJax-Span-123956\" class=\"mrow\"><span id=\"MathJax-Span-123957\" class=\"semantics\"><span id=\"MathJax-Span-123958\" class=\"mrow\"><span id=\"MathJax-Span-123959\" class=\"mtext\">1.00 \u03bcH<\/span><\/span><\/span><\/span><\/span><\/span> inductor connected to a variable capacitor. What range of capacitance is needed?\r\n\r\n<\/div>\r\n<div>\r\n\r\n6. Suppose you have a supply of inductors ranging from 1.00 nH to 10.0 H, and capacitors ranging from 1.00 pF to 0.100 F. What is the range of resonant frequencies that can be achieved from combinations of a single inductor and a single capacitor?\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n\r\n7. What capacitance do you need to produce a resonant frequency of 1.00 GHz, when using an 8.00 nH inductor?\r\n\r\n<\/div>\r\n<div>\r\n\r\n8. What inductance do you need to produce a resonant frequency of 60.0 Hz, when using a 2.00 \u03bcF capacitor?\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n\r\n9. The lowest frequency in the FM radio band is 88.0 MHz. (a) What inductance is needed to produce this resonant frequency if it is connected to a 2.50 pF capacitor? (b) The capacitor is variable, to allow the resonant frequency to be adjusted to as high as 108 MHz. What must the capacitance be at this frequency?\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n\r\n10. An <em>RLC<\/em> series circuit has a\u00a0<span id=\"MathJax-Element-7925-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123970\" class=\"math\"><span id=\"MathJax-Span-123971\" class=\"mrow\"><span id=\"MathJax-Span-123972\" class=\"semantics\"><span id=\"MathJax-Span-123973\" class=\"mrow\"><span id=\"MathJax-Span-123974\" class=\"mi\">2.50 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> resistor, a <span id=\"MathJax-Element-7926-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123975\" class=\"math\"><span id=\"MathJax-Span-123976\" class=\"mrow\"><span id=\"MathJax-Span-123977\" class=\"semantics\"><span id=\"MathJax-Span-123978\" class=\"mrow\"><span id=\"MathJax-Span-123979\" class=\"mi\">100 \u03bcH<\/span><\/span><\/span><\/span><\/span><\/span> inductor, and an <span id=\"MathJax-Element-7927-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123980\" class=\"math\"><span id=\"MathJax-Span-123981\" class=\"mrow\"><span id=\"MathJax-Span-123982\" class=\"semantics\"><span id=\"MathJax-Span-123983\" class=\"mrow\"><span id=\"MathJax-Span-123984\" class=\"mi\">80.0 \u03bcF\u00a0<\/span><\/span><\/span><\/span><\/span><\/span>capacitor.(a) Find the circuit\u2019s impedance at 120 Hz. (b) Find the circuit\u2019s impedance at 5.00 kHz. (c) If the voltage source has <span id=\"MathJax-Element-7928-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123985\" class=\"math\"><span id=\"MathJax-Span-123986\" class=\"mrow\"><span id=\"MathJax-Span-123987\" class=\"semantics\"><span id=\"MathJax-Span-123988\" class=\"mrow\"><span id=\"MathJax-Span-123989\" class=\"mrow\"><span id=\"MathJax-Span-123990\" class=\"mrow\"><span id=\"MathJax-Span-123991\" class=\"mrow\"><span id=\"MathJax-Span-123992\" class=\"mrow\"><span id=\"MathJax-Span-123993\" class=\"msub\"><em><span id=\"MathJax-Span-123994\" class=\"mi\">V<\/span><\/em><sub><span id=\"MathJax-Span-123995\" class=\"mrow\"><span id=\"MathJax-Span-123996\" class=\"mtext\">rms\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-123997\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-123998\" class=\"mn\">5<\/span><\/span><span id=\"MathJax-Span-123999\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-124000\" class=\"mtext\">60\u00a0<\/span><span id=\"MathJax-Span-124001\" class=\"mspace\"><\/span><span id=\"MathJax-Span-124002\" class=\"mtext\">V<\/span><\/span><\/span><span id=\"MathJax-Span-124003\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, what is <span id=\"MathJax-Element-7929-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124004\" class=\"math\"><span id=\"MathJax-Span-124005\" class=\"mrow\"><span id=\"MathJax-Span-124006\" class=\"semantics\"><span id=\"MathJax-Span-124007\" class=\"mrow\"><span id=\"MathJax-Span-124008\" class=\"mrow\"><span id=\"MathJax-Span-124009\" class=\"mrow\"><span id=\"MathJax-Span-124010\" class=\"msub\"><em><span id=\"MathJax-Span-124011\" class=\"mi\">I<\/span><\/em><sub><span id=\"MathJax-Span-124012\" class=\"mrow\"><span id=\"MathJax-Span-124013\" class=\"mtext\">rms<\/span><\/span><\/sub><\/span><\/span><span id=\"MathJax-Span-124014\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> at each frequency? (d) What is the resonant frequency of the circuit? (e) What is\u00a0<em><span id=\"MathJax-Span-124011\" class=\"mi\">I<\/span><\/em><sub><span id=\"MathJax-Span-124012\" class=\"mrow\"><span id=\"MathJax-Span-124013\" class=\"mtext\">rms<\/span><\/span><\/sub>\u00a0at resonance?\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n\r\n11. An <em>RLC<\/em> series circuit has a\u00a0<span id=\"MathJax-Element-7931-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124026\" class=\"math\"><span id=\"MathJax-Span-124027\" class=\"mrow\"><span id=\"MathJax-Span-124028\" class=\"semantics\"><span id=\"MathJax-Span-124029\" class=\"mrow\"><span id=\"MathJax-Span-124030\" class=\"mi\">1.00 k\u03a9<\/span><\/span><\/span><\/span><\/span><\/span> resistor, a <span id=\"MathJax-Element-7932-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124031\" class=\"math\"><span id=\"MathJax-Span-124032\" class=\"mrow\"><span id=\"MathJax-Span-124033\" class=\"semantics\"><span id=\"MathJax-Span-124034\" class=\"mrow\"><span id=\"MathJax-Span-124035\" class=\"mi\">150 \u03bcH<\/span><\/span><\/span><\/span><\/span><\/span> inductor, and a 25.0 nF capacitor. (a) Find the circuit\u2019s impedance at 500 Hz. (b) Find the circuit\u2019s impedance at 7.50 kHz. (c) If the voltage source has <span id=\"MathJax-Element-7933-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124036\" class=\"math\"><span id=\"MathJax-Span-124037\" class=\"mrow\"><span id=\"MathJax-Span-124038\" class=\"semantics\"><span id=\"MathJax-Span-124039\" class=\"mrow\"><span id=\"MathJax-Span-124040\" class=\"mrow\"><span id=\"MathJax-Span-124041\" class=\"mrow\"><span id=\"MathJax-Span-124042\" class=\"mrow\"><span id=\"MathJax-Span-124043\" class=\"mrow\"><span id=\"MathJax-Span-124044\" class=\"msub\"><em><span id=\"MathJax-Span-124045\" class=\"mi\">V<\/span><\/em><sub><span id=\"MathJax-Span-124046\" class=\"mrow\"><span id=\"MathJax-Span-124047\" class=\"mtext\">rms\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-124048\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-124049\" class=\"mtext\">408\u00a0<\/span><\/span><span id=\"MathJax-Span-124050\" class=\"mspace\"><\/span><span id=\"MathJax-Span-124051\" class=\"mtext\">V<\/span><\/span><\/span><span id=\"MathJax-Span-124052\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, what is <span id=\"MathJax-Element-7934-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124053\" class=\"math\"><span id=\"MathJax-Span-124054\" class=\"mrow\"><span id=\"MathJax-Span-124055\" class=\"semantics\"><span id=\"MathJax-Span-124056\" class=\"mrow\"><span id=\"MathJax-Span-124057\" class=\"mrow\"><span id=\"MathJax-Span-124058\" class=\"mrow\"><span id=\"MathJax-Span-124059\" class=\"msub\"><em><span id=\"MathJax-Span-124060\" class=\"mi\">I<\/span><\/em><sub><span id=\"MathJax-Span-124061\" class=\"mrow\"><span id=\"MathJax-Span-124062\" class=\"mtext\">rms<\/span><\/span><\/sub><\/span><\/span><span id=\"MathJax-Span-124063\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> at each frequency? (d) What is the resonant frequency of the circuit? (e) What is\u00a0<em><span id=\"MathJax-Span-124060\" class=\"mi\">I<\/span><\/em><sub><span id=\"MathJax-Span-124061\" class=\"mrow\"><span id=\"MathJax-Span-124062\" class=\"mtext\">rms<\/span><\/span><\/sub>\u00a0at resonance?\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n\r\n12. An <em>RLC<\/em> series circuit has a\u00a0<span id=\"MathJax-Element-7938-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124085\" class=\"math\"><span id=\"MathJax-Span-124086\" class=\"mrow\"><span id=\"MathJax-Span-124087\" class=\"semantics\"><span id=\"MathJax-Span-124088\" class=\"mrow\"><span id=\"MathJax-Span-124089\" class=\"mi\">2.50 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> resistor, a <span id=\"MathJax-Element-7939-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124090\" class=\"math\"><span id=\"MathJax-Span-124091\" class=\"mrow\"><span id=\"MathJax-Span-124092\" class=\"semantics\"><span id=\"MathJax-Span-124093\" class=\"mrow\"><span id=\"MathJax-Span-124094\" class=\"mi\">100 \u03bcH<\/span><\/span><\/span><\/span><\/span><\/span> inductor, and an <span id=\"MathJax-Element-7940-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124095\" class=\"math\"><span id=\"MathJax-Span-124096\" class=\"mrow\"><span id=\"MathJax-Span-124097\" class=\"semantics\"><span id=\"MathJax-Span-124098\" class=\"mrow\"><span id=\"MathJax-Span-124099\" class=\"mi\">80.0 \u03bcF\u00a0<\/span><\/span><\/span><\/span><\/span><\/span>capacitor. (a) Find the power factor at <span id=\"MathJax-Element-7941-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124100\" class=\"math\"><span id=\"MathJax-Span-124101\" class=\"mrow\"><span id=\"MathJax-Span-124102\" class=\"semantics\"><span id=\"MathJax-Span-124103\" class=\"mrow\"><span id=\"MathJax-Span-124104\" class=\"mrow\"><span id=\"MathJax-Span-124105\" class=\"mi\"><em>f<\/em>\u00a0<\/span><span id=\"MathJax-Span-124106\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-124107\" class=\"mi\">120 Hz<\/span><\/span><\/span><\/span><\/span><\/span><\/span>. (b) What is the phase angle at 120 Hz? (c) What is the average power at 120 Hz? (d) Find the average power at the circuit\u2019s resonant frequency.\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n\r\n13. An <em>RLC<\/em> series circuit has a\u00a0<span id=\"MathJax-Element-7942-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124108\" class=\"math\"><span id=\"MathJax-Span-124109\" class=\"mrow\"><span id=\"MathJax-Span-124110\" class=\"semantics\"><span id=\"MathJax-Span-124111\" class=\"mrow\"><span id=\"MathJax-Span-124112\" class=\"mi\">1.00 k\u03a9<\/span><\/span><\/span><\/span><\/span><\/span> resistor, a <span id=\"MathJax-Element-7943-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124113\" class=\"math\"><span id=\"MathJax-Span-124114\" class=\"mrow\"><span id=\"MathJax-Span-124115\" class=\"semantics\"><span id=\"MathJax-Span-124116\" class=\"mrow\"><span id=\"MathJax-Span-124117\" class=\"mi\">150 \u03bcH<\/span><\/span><\/span><\/span><\/span><\/span> inductor, and a 25.0 nF capacitor. (a) Find the power factor at <span id=\"MathJax-Element-7944-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124118\" class=\"math\"><span id=\"MathJax-Span-124119\" class=\"mrow\"><span id=\"MathJax-Span-124120\" class=\"semantics\"><span id=\"MathJax-Span-124121\" class=\"mrow\"><span id=\"MathJax-Span-124122\" class=\"mrow\"><em><span id=\"MathJax-Span-124123\" class=\"mi\">f\u00a0<\/span><\/em><span id=\"MathJax-Span-124124\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-124125\" class=\"mi\">7.50 Hz<\/span><\/span><\/span><\/span><\/span><\/span><\/span>. (b) What is the phase angle at this frequency? (c) What is the average power at this frequency? (d) Find the average power at the circuit\u2019s resonant frequency.\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n\r\n14. An <em>RLC<\/em> series circuit has a\u00a0<span id=\"MathJax-Element-7946-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124131\" class=\"math\"><span id=\"MathJax-Span-124132\" class=\"mrow\"><span id=\"MathJax-Span-124133\" class=\"semantics\"><span id=\"MathJax-Span-124134\" class=\"mrow\"><span id=\"MathJax-Span-124135\" class=\"mi\">200 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> resistor and a 25.0 mH inductor. At 8000 Hz, the phase angle is <span id=\"MathJax-Element-7947-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124136\" class=\"math\"><span id=\"MathJax-Span-124137\" class=\"mrow\"><span id=\"MathJax-Span-124138\" class=\"semantics\"><span id=\"MathJax-Span-124139\" class=\"mrow\"><span id=\"MathJax-Span-124140\" class=\"mi\">45.0\u00ba<\/span><\/span><\/span><\/span><\/span><\/span>. (a) What is the impedance? (b) Find the circuit\u2019s capacitance. (c) If\u00a0<span id=\"MathJax-Element-7948-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124141\" class=\"math\"><span id=\"MathJax-Span-124142\" class=\"mrow\"><span id=\"MathJax-Span-124143\" class=\"semantics\"><span id=\"MathJax-Span-124144\" class=\"mrow\"><span id=\"MathJax-Span-124145\" class=\"mrow\"><span id=\"MathJax-Span-124146\" class=\"mrow\"><span id=\"MathJax-Span-124147\" class=\"mrow\"><span id=\"MathJax-Span-124148\" class=\"mrow\"><span id=\"MathJax-Span-124149\" class=\"msub\"><em><span id=\"MathJax-Span-124150\" class=\"mi\">V<\/span><\/em><sub><span id=\"MathJax-Span-124151\" class=\"mrow\"><span id=\"MathJax-Span-124152\" class=\"mtext\">rms\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-124153\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-124154\" class=\"mtext\">408\u00a0<\/span><\/span><span id=\"MathJax-Span-124155\" class=\"mspace\"><\/span><span id=\"MathJax-Span-124156\" class=\"mtext\">V<\/span><\/span><\/span><span id=\"MathJax-Span-124157\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is applied, what is the average power supplied?\r\n\r\n<\/div>\r\n<\/div>\r\n<div>\r\n<div>\r\n\r\n15. Referring to\u00a0Example 3. Calculating the Power Factor and Power, find the average power at 10.0 kHz.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div>\r\n<h2>Glossary<\/h2>\r\n<dl><dt>impedance:<\/dt><dd>the AC analogue to resistance in a DC circuit; it is the combined effect of resistance, inductive reactance, and capacitive reactance in the form [latex]Z=\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\[\/latex]<\/dd><\/dl><dl><dt>resonant frequency:<\/dt><dd>the frequency at which the impedance in a circuit is at a minimum, and also the frequency at which the circuit would oscillate if not driven by a voltage source; calculated by [latex]{f}_{0}=\\frac{1}{2\\pi \\sqrt{\\text{LC}}}\\\\[\/latex]<\/dd><\/dl><dl><dt>phase angle:<\/dt><dd>denoted by\u00a0<em><span id=\"MathJax-Span-124232\" class=\"mi\">\u03d5<\/span><\/em>, the amount by which the voltage and current are out of phase with each other in a circuit<\/dd><\/dl><dl><dt>power factor:<\/dt><dd>the amount by which the power delivered in the circuit is less than the theoretical maximum of the circuit due to voltage and current being out of phase; calculated by\u00a0<span id=\"MathJax-Span-124224\" class=\"mrow\"><span id=\"MathJax-Span-124225\" class=\"semantics\"><span id=\"MathJax-Span-124226\" class=\"mrow\"><span id=\"MathJax-Span-124227\" class=\"mrow\"><span id=\"MathJax-Span-124228\" class=\"mrow\"><span id=\"MathJax-Span-124229\" class=\"mrow\"><span id=\"MathJax-Span-124230\" class=\"mtext\">cos\u00a0<\/span><span id=\"MathJax-Span-124231\" class=\"mspace\"><\/span><em><span id=\"MathJax-Span-124232\" class=\"mi\">\u03d5<\/span><\/em><\/span><\/span><span id=\"MathJax-Span-124233\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/dd><\/dl>\r\n<div class=\"textbox exercises\">\r\n<h3>Selected Solutions to Problems &amp; Exercises<\/h3>\r\n1. \u00a0(a) <span id=\"MathJax-Element-7909-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123871\" class=\"math\"><span id=\"MathJax-Span-123872\" class=\"mrow\"><span id=\"MathJax-Span-123873\" class=\"semantics\"><span id=\"MathJax-Span-123874\" class=\"mrow\"><span id=\"MathJax-Span-123875\" class=\"mi\">40.02 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> at 60.0 Hz, <span id=\"MathJax-Element-7910-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123876\" class=\"math\"><span id=\"MathJax-Span-123877\" class=\"mrow\"><span id=\"MathJax-Span-123878\" class=\"semantics\"><span id=\"MathJax-Span-123879\" class=\"mrow\"><span id=\"MathJax-Span-123880\" class=\"mi\">193 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> at 10.0 kHz\u00a0(b) At 60 Hz, with a capacitor, <span id=\"MathJax-Element-7911-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123881\" class=\"math\"><span id=\"MathJax-Span-123882\" class=\"mrow\"><span id=\"MathJax-Span-123883\" class=\"semantics\"><span id=\"MathJax-Span-123884\" class=\"mrow\"><span id=\"MathJax-Span-123885\" class=\"mi\">Z = 531 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span>, over 13 times as high as without the capacitor. The capacitor makes a large difference at low frequencies. At 10 kHz, with a capacitor <span id=\"MathJax-Element-7912-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123886\" class=\"math\"><span id=\"MathJax-Span-123887\" class=\"mrow\"><span id=\"MathJax-Span-123888\" class=\"semantics\"><span id=\"MathJax-Span-123889\" class=\"mrow\"><span id=\"MathJax-Span-123890\" class=\"mi\">Z = 190 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span>, about the same as without the capacitor. The capacitor has a smaller effect at high frequencies.\r\n\r\n3.\u00a0(a) <span id=\"MathJax-Element-7919-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123940\" class=\"math\"><span id=\"MathJax-Span-123941\" class=\"mrow\"><span id=\"MathJax-Span-123942\" class=\"semantics\"><span id=\"MathJax-Span-123943\" class=\"mrow\"><span id=\"MathJax-Span-123944\" class=\"mi\">529 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> at 60.0 Hz, <span id=\"MathJax-Element-7920-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123945\" class=\"math\"><span id=\"MathJax-Span-123946\" class=\"mrow\"><span id=\"MathJax-Span-123947\" class=\"semantics\"><span id=\"MathJax-Span-123948\" class=\"mrow\"><span id=\"MathJax-Span-123949\" class=\"mi\">185 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> at 10.0 kHz\u00a0(b) These values are close to those obtained in\u00a0<em>Example 1:\u00a0Calculating Impedance and Current<\/em>\u00a0because at low frequency the capacitor dominates and at high frequency the inductor dominates. So in both cases the resistor makes little contribution to the total impedance.\r\n\r\n5.\u00a09.30 nF to 101 nF\r\n\r\n7. 3.17 pF\r\n\r\n9. (a) <span id=\"MathJax-Element-7924-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123965\" class=\"math\"><span id=\"MathJax-Span-123966\" class=\"mrow\"><span id=\"MathJax-Span-123967\" class=\"semantics\"><span id=\"MathJax-Span-123968\" class=\"mrow\"><span id=\"MathJax-Span-123969\" class=\"mi\">1.31 \u03bcH\u00a0<\/span><\/span><\/span><\/span><\/span><\/span>(b) 1.66 pF\r\n\r\n11.\u00a0(a) <span id=\"MathJax-Element-7936-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124075\" class=\"math\"><span id=\"MathJax-Span-124076\" class=\"mrow\"><span id=\"MathJax-Span-124077\" class=\"semantics\"><span id=\"MathJax-Span-124078\" class=\"mrow\"><span id=\"MathJax-Span-124079\" class=\"mi\">12.8 k\u03a9\u00a0<\/span><\/span><\/span><\/span><\/span><\/span>(b) <span id=\"MathJax-Element-7937-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124080\" class=\"math\"><span id=\"MathJax-Span-124081\" class=\"mrow\"><span id=\"MathJax-Span-124082\" class=\"semantics\"><span id=\"MathJax-Span-124083\" class=\"mrow\"><span id=\"MathJax-Span-124084\" class=\"mi\">1.31 k\u03a9\u00a0<\/span><\/span><\/span><\/span><\/span><\/span>(c) 31.9 mA at 500 Hz, 312 mA at 7.50 kHz\u00a0(d) 82.2 kHz\u00a0(e) 0.408 A\r\n\r\n13.\u00a0(a) 0.159\u00a0(b) <span id=\"MathJax-Element-7945-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124126\" class=\"math\"><span id=\"MathJax-Span-124127\" class=\"mrow\"><span id=\"MathJax-Span-124128\" class=\"semantics\"><span id=\"MathJax-Span-124129\" class=\"mrow\"><span id=\"MathJax-Span-124130\" class=\"mi\">80.9\u00ba\u00a0<\/span><\/span><\/span><\/span><\/span><\/span>(c) 26.4 W\u00a0(d) 166 W\r\n\r\n15.\u00a016.0 W\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>","rendered":"<div>\n<div>\n<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<div>\n<div>\n<ul>\n<li>Calculate the impedance, phase angle, resonant frequency, power, power factor, voltage, and\/or current in a RLC series circuit.<\/li>\n<li>Draw the circuit diagram for an RLC series circuit.<\/li>\n<li>Explain the significance of the resonant frequency.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div title=\"Impedance\">\n<div>\n<div>\n<div>\n<h2>Impedance<\/h2>\n<\/div>\n<\/div>\n<\/div>\n<p>When alone in an AC circuit, inductors, capacitors, and resistors all impede current. How do they behave when all three occur together? Interestingly, their individual resistances in ohms do not simply add. Because inductors and capacitors behave in opposite ways, they partially to totally cancel each other\u2019s effect. Figure 1\u00a0shows an <em>RLC <\/em>series circuit with an AC voltage source, the behavior of which is the subject of this section. The crux of the analysis of an <em>RLC<\/em> circuit is the frequency dependence of <em>X<\/em><sub><em>L<\/em><\/sub> and <em>X<\/em><sub><em>C<\/em><\/sub>, and the effect they have on the phase of voltage versus current (established in the preceding section). These give rise to the frequency dependence of the circuit, with important \u201cresonance\u201d features that are the basis of many applications, such as radio tuners.<\/p>\n<div title=\"Figure 23.49.\">\n<div>\n<div>\n<div style=\"width: 285px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20110957\/Figure_24_12_01a.jpg\" alt=\"The figure describes an R LC series circuit. It shows a resistor R connected in series with an inductor L, connected to a capacitor C in series to an A C source V. The voltage of the A C source is given by V equals V zero sine two pi f t. The voltage across R is V R, across L is V L and across C is V C.\" width=\"275\" height=\"417\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. An RLC series circuit with an AC voltage source.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>The combined effect of resistance <em>R<\/em>, inductive reactance <em>X<\/em><sub><em>L<\/em><\/sub>, and capacitive reactance <em>X<\/em><sub><em>C<\/em><\/sub> is defined to be <em> impedance<\/em>, an AC analogue to resistance in a DC circuit. Current, voltage, and impedance in an <em>RLC<\/em> circuit are related by an AC version of Ohm\u2019s law:<\/p>\n<p style=\"text-align: center;\">[latex]{I}_{0}=\\frac{{V}_{0}}{Z}\\text{ or }{I}_{\\text{rms}}=\\frac{{V}_{\\text{rms}}}{Z}\\\\[\/latex].<\/p>\n<p>Here <em>I<\/em><sub>0<\/sub> is the peak current, <em>V<\/em><sub>0<\/sub> the peak source voltage, and <em>Z<\/em> is the impedance of the circuit. The units of impedance are ohms, and its effect on the circuit is as you might expect: the greater the impedance, the smaller the current. To get an expression for <em>Z<\/em> in terms of <em>R<\/em> , <em>X<\/em><sub><em>L<\/em><\/sub>, and <em>X<\/em><sub><em>C<\/em><\/sub>, we will now examine how the voltages across the various components are related to the source voltage. Those voltages are labeled <em>V<\/em><sub><em>R<\/em><\/sub>, <em>V<\/em><sub><em>L<\/em><\/sub>, and <em>V<\/em><sub><em>C<\/em><\/sub> in Figure 1. Conservation of charge requires current to be the same in each part of the circuit at all times, so that we can say the currents in <em>R<\/em>, <em>L<\/em>, and <em>C<\/em> are equal and in phase. But we know from the preceding section that the voltage across the inductor <em>V<\/em><sub><em>L<\/em><\/sub> leads the current by one-fourth of a cycle, the voltage across the capacitor <em>V<\/em><sub><em>C<\/em><\/sub> follows the current by one-fourth of a cycle, and the voltage across the resistor <em>V<\/em><sub><em>R<\/em><\/sub> is exactly in phase with the current. Figure 2\u00a0shows these relationships in one graph, as well as showing the total voltage around the circuit <em>V\u00a0<\/em>=\u00a0<em>V<\/em><sub><em>R\u00a0<\/em><\/sub>+\u00a0<em>V<\/em><sub><em>L\u00a0<\/em><\/sub>+\u00a0<em>V<\/em><sub><em>C<\/em><\/sub>, where all four voltages are the instantaneous values. According to Kirchhoff\u2019s loop rule, the total voltage around the circuit <em>V<\/em> is also the voltage of the source. You can see from Figure 2\u00a0that while <em>V<\/em><sub><em>R<\/em><\/sub> is in phase with the current, <em>V<\/em><sub><em>L<\/em><\/sub> leads by 90\u00ba, and <em>V<\/em><sub><em>C<\/em><\/sub> follows by 90\u00ba. Thus <em>V<\/em><sub><em>L<\/em><\/sub> and <em>V<\/em><sub><em>C<\/em><\/sub> are 180\u00ba out of phase (crest to trough) and tend to cancel, although not completely unless they have the same magnitude. Since the peak voltages are not aligned (not in phase), the peak voltage <em>V<\/em><sub>0<\/sub> of the source does <em>not<\/em> equal the sum of the peak voltages across <em>R<\/em>, <em>L<\/em>, and <em>C<\/em>. The actual relationship is<\/p>\n<p style=\"text-align: center;\">[latex]{V}_{0}=\\sqrt{{{{V}_{0R}}^{2}+\\left({V}_{0L}-{V}_{0C}\\right)^{2}}}\\\\[\/latex],<\/p>\n<p>where <em>V<\/em><sub>0<em>R<\/em><\/sub>, <em>V<\/em><sub>0<em>L<\/em><\/sub>, and <em>V<\/em><sub>0<em>C<\/em><\/sub> are the peak voltages across <em>R<\/em>, <em>L<\/em>, and <em>C<\/em>, respectively. Now, using Ohm\u2019s law and definitions from <a title=\"23.11. Reactance, Inductive and Capacitive\" href=\".\/chapter\/23-11-reactance-inductive-and-capacitive\/\" target=\"_blank\">Reactance, Inductive and Capacitive<\/a>, we substitute <em>V<\/em><sub>0\u00a0<\/sub>=\u00a0<em>I<\/em><sub>0<\/sub><em>Z<\/em> into the above, as well as <em>V<\/em><sub>0<em>R\u00a0<\/em><\/sub>=\u00a0<em>I<\/em><sub>0<\/sub><em>R<\/em>, <em>V<\/em><sub>0<em>L\u00a0<\/em><\/sub>=\u00a0<em>I<\/em><sub>0<\/sub><em>X<\/em><sub><em>L<\/em><\/sub>, and <em>V<\/em><sub>0<em>C\u00a0<\/em><\/sub>=\u00a0<em>I<\/em><sub>0<\/sub><em>X<\/em><sub><em>C<\/em><\/sub>, yielding<\/p>\n<p style=\"text-align: center;\">[latex]{I}_{0}Z=\\sqrt{{{I}_{0}}^{2}{R}^{2}+\\left({I}_{0}{X}_{L}-{I}_{0}{X}_{C}\\right)^{2}}={I}_{0}\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\[\/latex]<\/p>\n<p><em>I<\/em><sub>0<\/sub> cancels to yield an expression for <em>Z<\/em>:<\/p>\n<p style=\"text-align: center;\">[latex]Z=\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\[\/latex],<\/p>\n<p>which is the impedance of an <em>RLC<\/em> series AC circuit. For circuits without a resistor, take <em>R<\/em> = 0; for those without an inductor, take <em>X<\/em><sub><em>L\u00a0<\/em><\/sub>= 0; and for those without a capacitor, take <em>X<\/em><sub><em>C\u00a0<\/em><\/sub>= 0.<\/p>\n<div title=\"Figure 23.50.\">\n<div>\n<div>\n<div style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20111001\/Figure_24_12_02a.jpg\" alt=\"The figure shows graphs showing the relationships of the voltages in an RLC circuit to the current. It has five graphs on the left and two graphs on the right. The first graph on the right is for current I versus time t. Current is plotted along Y axis and time is along X axis. The curve is a smooth progressive sine wave. The second graph is on the right is for voltage V R versus time t. Voltage V R is plotted along Y axis and time is along X axis. The curve is a smooth progressive sine wave. The third graph is on the right is for voltage V L versus time t. Voltage V L is plotted along Y axis and time is along X axis. The curve is a smooth progressive cosine wave. The fourth graph is on the right is for voltage V C versus time t. Voltage V C is plotted along Y axis and time t is along X axis. The curve is a smooth progressive cosine wave starting from negative Y axis. The fifth graph shows the voltage V verses time t for the R L C circuit. Voltage V is plotted along Y axis and time t is along X axis. The curve is a smooth progressive sine wave starting from a point near to origin on negative X axis. The first and the fifth graphs are again shown on the right and their amplitudes and phases compared. The current graph is shown to have a lesser amplitude.\" width=\"300\" height=\"476\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. This graph shows the relationships of the voltages in an <em>RLC<\/em> circuit to the current. The voltages across the circuit elements add to equal the voltage of the source, which is seen to be out of phase with the current.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div title=\"Example 23.12. Calculating Impedance and Current\">\n<div class=\"textbox examples\">\n<h3>Example 1. Calculating Impedance and Current<\/h3>\n<div>\n<p>An <em>RLC <\/em>series circuit has a 40.0 \u03a9 resistor, a 3.00 mH inductor, and a 5.00 \u03bcF capacitor. (a) Find the circuit\u2019s impedance at 60.0 Hz and 10.0 kHz, noting that these frequencies and the values for <em>L<\/em> and <em>C<\/em> are the same as in Example 1 and Example 2 from <a href=\".\/chapter\/23-11-reactance-inductive-and-capacitive\/\" target=\"_blank\">Reactance, Inductive, and Capacitive<\/a>.\u00a0(b) If the voltage source has <em>V<\/em><sub>rms\u00a0<\/sub>= 120 V, what is <em>I<\/em><sub>rms<\/sub> at each frequency?<\/p>\n<h4><strong>Strategy<\/strong><\/h4>\n<p>For each frequency, we use [latex]Z=\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\[\/latex]\u00a0to find the impedance and then Ohm\u2019s law to find current. We can take advantage of the results of the previous two examples rather than calculate the reactances again.<\/p>\n<h4><strong>Solution for (a)<\/strong><\/h4>\n<p>At 60.0 Hz, the values of the reactances were found in Example 1 from <a href=\".\/chapter\/23-11-reactance-inductive-and-capacitive\/\" target=\"_blank\">Reactance, Inductive, and Capacitive<\/a>\u00a0to be\u00a0<span id=\"MathJax-Element-7174-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-113330\" class=\"math\"><span id=\"MathJax-Span-113331\" class=\"mrow\"><span id=\"MathJax-Span-113332\" class=\"semantics\"><span id=\"MathJax-Span-113333\" class=\"mrow\"><span id=\"MathJax-Span-113334\" class=\"mrow\"><span id=\"MathJax-Span-113335\" class=\"mrow\"><span id=\"MathJax-Span-113336\" class=\"mrow\"><span id=\"MathJax-Span-113337\" class=\"mrow\"><em><span id=\"MathJax-Span-113338\" class=\"msub\"><span id=\"MathJax-Span-113339\" class=\"mi\">X<\/span><sub><span id=\"MathJax-Span-113340\" class=\"mrow\"><span id=\"MathJax-Span-113341\" class=\"mi\">L\u00a0<\/span><\/span><\/sub><\/span><\/em><span id=\"MathJax-Span-113342\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-113343\" class=\"mn\">1<\/span><\/span><span id=\"MathJax-Span-113344\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-113345\" class=\"mtext\">13\u00a0<\/span><span id=\"MathJax-Span-113346\" class=\"mspace\"><\/span><span id=\"MathJax-Span-113347\" class=\"mo\">\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u00a0and in Example 2 from <a href=\".\/chapter\/23-11-reactance-inductive-and-capacitive\/\" target=\"_blank\">Reactance, Inductive, and Capacitive<\/a>\u00a0to be\u00a0<span id=\"MathJax-Element-7175-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-113349\" class=\"math\"><span id=\"MathJax-Span-113350\" class=\"mrow\"><span id=\"MathJax-Span-113351\" class=\"semantics\"><span id=\"MathJax-Span-113352\" class=\"mrow\"><span id=\"MathJax-Span-113353\" class=\"mrow\"><span id=\"MathJax-Span-113354\" class=\"mrow\"><span id=\"MathJax-Span-113355\" class=\"mrow\"><span id=\"MathJax-Span-113356\" class=\"mrow\"><em><span id=\"MathJax-Span-113357\" class=\"msub\"><span id=\"MathJax-Span-113358\" class=\"mi\">X<\/span><span id=\"MathJax-Span-113359\" class=\"mrow\"><span id=\"MathJax-Span-113360\" class=\"mi\"><sub>C<\/sub>\u00a0<\/span><\/span><\/span><\/em><span id=\"MathJax-Span-113361\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-113362\" class=\"mtext\">531\u00a0<\/span><\/span><span id=\"MathJax-Span-113363\" class=\"mspace\"><\/span><span id=\"MathJax-Span-113364\" class=\"mo\">\u03a9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>. Entering these and the given 40.0 \u03a9 for resistance into [latex]Z=\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\[\/latex]\u00a0yields<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}Z & =& \\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\ & =& \\sqrt{\\left(40.0\\text{ }\\Omega\\right)^{2}+\\left(1.13\\text{ }\\Omega -531\\text{ }\\Omega\\right)^{2}}\\\\ & =& 531\\text{ }\\Omega\\text{ at }60.0\\text{ Hz}\\end{array}\\\\[\/latex]<\/p>\n<p>Similarly, at 10.0 kHz, <em>X<sub>L\u00a0<\/sub><\/em>= 188 \u03a9\u00a0and\u00a0<em>X<sub>C\u00a0<\/sub><\/em>= 3.18 \u03a9, so that<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}Z & =& \\sqrt{\\left(40.0\\text{ }\\Omega\\right)^{2}+\\left(188\\text{ }\\Omega -3.18\\text{ }\\Omega\\right)^{2}}\\\\ & =& 190\\text{ }\\Omega\\text{ at }10.0\\text{ kHz}\\end{array}\\\\[\/latex]<\/p>\n<h4><strong>Discussion for (a)<\/strong><\/h4>\n<p>In both cases, the result is nearly the same as the largest value, and the impedance is definitely not the sum of the individual values. It is clear that <em>X<\/em><sub><em>L<\/em><\/sub> dominates at high frequency and <em>X<\/em><sub><em>C<\/em><\/sub> dominates at low frequency.<\/p>\n<h4><strong>Solution for (b)<\/strong><\/h4>\n<p>The current <em>I<\/em><sub>rms<\/sub> can be found using the AC version of Ohm\u2019s law in the equation <em>I<\/em><sub>rms<\/sub>=<em>V<\/em><sub>rms<\/sub>\/<em>Z<\/em>:<\/p>\n<p style=\"text-align: center;\">[latex]{I}_{\\text{rms}}=\\frac{{V}_{\\text{rms}}}{Z}=\\frac{120\\text{V}}{531\\text{ }\\Omega}=0.226\\text{ A}\\\\[\/latex] at 60.0 Hz<\/p>\n<p>Finally, at 10.0 kHz, we find<\/p>\n<p style=\"text-align: center;\">[latex]{I}_{\\text{rms}}=\\frac{{V}_{\\text{rms}}}{Z}=\\frac{120\\text{ V}}{190\\text{ }\\Omega}=0.633\\text{ A}\\\\[\/latex] at 10.0 kHz<\/p>\n<h4><strong>Discussion for (a)<\/strong><\/h4>\n<p>The current at 60.0 Hz is the same (to three digits) as found for the capacitor alone in Example 2 from <a href=\".\/chapter\/23-11-reactance-inductive-and-capacitive\/\" target=\"_blank\">Reactance, Inductive, and Capacitive<\/a>. The capacitor dominates at low frequency. The current at 10.0 kHz is only slightly different from that found for the inductor alone in Example 1 from <a href=\".\/chapter\/23-11-reactance-inductive-and-capacitive\/\" target=\"_blank\">Reactance, Inductive, and Capacitive<\/a>. The inductor dominates at high frequency.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div title=\"Resonance in RLC Series AC Circuits\">\n<div>\n<div>\n<div>\n<h2>Resonance in <em>RLC<\/em> Series AC Circuits<\/h2>\n<\/div>\n<\/div>\n<\/div>\n<p>How does an <em>RLC<\/em> circuit behave as a function of the frequency of the driving voltage source? Combining Ohm\u2019s law, <em>I<\/em><sub>rms\u00a0<\/sub>=\u00a0<em>V<\/em><sub>rms<\/sub>\/<em>Z<\/em>, and the expression for impedance <em>Z<\/em> from [latex]Z=\\sqrt{{R}^{2}+\\left({{X}_{L}}-{{X}_{C}}\\right)^{2}}\\\\[\/latex]\u00a0gives<\/p>\n<p style=\"text-align: center;\">[latex]{I}_{\\text{rms}}=\\frac{{V}_{\\text{rms}}}{\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}}\\\\[\/latex]<\/p>\n<p>The reactances vary with frequency, with <em>X<\/em><sub><em>L<\/em><\/sub> large at high frequencies and <em>X<\/em><sub><em>C<\/em><\/sub> large at low frequencies, as we have seen in three previous examples. At some intermediate frequency <em>f<\/em><sub>0<\/sub>, the reactances will be equal and cancel, giving <em><em>Z\u00a0<\/em>=\u00a0<em>R<\/em><\/em> \u2014 this is a minimum value for impedance, and a maximum value for <em>I<\/em><sub>rms<\/sub> results. We can get an expression for <em>f<\/em><sub>0<\/sub> by taking<\/p>\n<div style=\"text-align: center;\" title=\"Equation 23.87.\"><em>X<\/em><sub><em>L\u00a0<\/em><\/sub>=\u00a0<em>X<\/em><sub><em>C<\/em><\/sub>.<\/div>\n<p>Substituting the definitions of <em>X<\/em><sub><em>L<\/em><\/sub> and <em>X<\/em><sub><em>C<\/em><\/sub>,<\/p>\n<p style=\"text-align: center;\">[latex]2\\pi f_{0}L=\\frac{1}{2\\pi f_{0}C}\\\\[\/latex].<\/p>\n<p>Solving this expression for <em>f<\/em><sub>0<\/sub> yields<\/p>\n<p style=\"text-align: center;\">[latex]{f}_{0}=\\frac{1}{2\\pi \\sqrt{LC}}\\\\[\/latex],<\/p>\n<p>where <em>f<\/em><sub>0<\/sub> is the <em> resonant frequency<\/em> of an <em>RLC<\/em> series circuit. This is also the <em>natural frequency<\/em> at which the circuit would oscillate if not driven by the voltage source. At <em>f<\/em><sub>0<\/sub>, the effects of the inductor and capacitor cancel, so that <em><em>Z\u00a0<\/em>=\u00a0<em>R<\/em><\/em>, and <em>I<\/em><sub>rms<\/sub> is a maximum.<\/p>\n<p>Resonance in AC circuits is analogous to mechanical resonance, where resonance is defined to be a forced oscillation\u2014in this case, forced by the voltage source\u2014at the natural frequency of the system. The receiver in a radio is an <em>RLC<\/em> circuit that oscillates best at its <em>f<\/em><sub>0<\/sub>. A variable capacitor is often used to adjust <em>f<\/em><sub>0<\/sub> to receive a desired frequency and to reject others. Figure 3\u00a0is a graph of current as a function of frequency, illustrating a resonant peak in <em>I<\/em><sub>rms<\/sub> at <em>f<\/em><sub>0<\/sub>. The two curves are for two different circuits, which differ only in the amount of resistance in them. The peak is lower and broader for the higher-resistance circuit. Thus the higher-resistance circuit does not resonate as strongly and would not be as selective in a radio receiver, for example.<\/p>\n<div title=\"Figure 23.51.\">\n<div>\n<div>\n<div style=\"width: 235px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20111012\/Figure_24_12_03a.jpg\" alt=\"The figure describes a graph of current I versus frequency f. Current I r m s is plotted along Y axis and frequency f is plotted along X axis. Two curves are shown. The upper curve is for small resistance and lower curve is for large resistance. Both the curves have a smooth rise and a fall. The peaks are marked for frequency f zero. The curve for smaller resistance has a higher value of peak than the curve for large resistance.\" width=\"225\" height=\"285\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. A graph of current versus frequency for two RLC series circuits differing only in the amount of resistance. Both have a resonance at <em>f<\/em><sub>0<\/sub>, but that for the higher resistance is lower and broader. The driving AC voltage source has a fixed amplitude <em>V<\/em><sub>0<\/sub>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div title=\"Example 23.13. Calculating Resonant Frequency and Current\">\n<div>\n<div class=\"textbox examples\">\n<h3>Example 2. Calculating Resonant Frequency and Current<\/h3>\n<div>\n<p>For the same <em>RLC<\/em> series circuit having a 40.0 \u03a9 resistor, a 3.00 mH inductor, and a 5.00 \u03bcF capacitor: (a) Find the resonant frequency. (b) Calculate <em>I<\/em><sub>rms<\/sub> at resonance if <em>V<\/em><sub>rms<\/sub> is 120 V.<\/p>\n<p><strong>Strategy<\/strong><\/p>\n<p>The resonant frequency is found by using the expression in [latex]{f}_{0}=\\frac{1}{2\\pi\\sqrt{LC}}\\\\[\/latex]. The current at that frequency is the same as if the resistor alone were in the circuit.<\/p>\n<p><strong>Solution for (a)<\/strong><\/p>\n<p>Entering the given values for <em>L<\/em> and <em>C<\/em> into the expression given for <em>f<\/em><sub>0<\/sub> in [latex]{f}_{0}=\\frac{1}{2\\pi\\sqrt{LC}}\\\\[\/latex]\u00a0yields<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll}{f}_{0}&=&\\frac{1}{2\\pi\\sqrt{LC}}\\\\ & =& \\frac{1}{2\\pi \\sqrt{\\left(3.00\\times 10^{-3}\\text{ H}\\right)\\left(5.00\\times 10^{-6}\\text{ F}\\right)}}=1.30\\text{ kHz}\\end{array}\\\\[\/latex]<\/p>\n<h4><strong>Discussion for (a)<\/strong><\/h4>\n<p>We see that the resonant frequency is between 60.0 Hz and 10.0 kHz, the two frequencies chosen in earlier examples. This was to be expected, since the capacitor dominated at the low frequency and the inductor dominated at the high frequency. Their effects are the same at this intermediate frequency.<\/p>\n<h4><strong>Solution for (b)<\/strong><\/h4>\n<p>The current is given by Ohm\u2019s law. At resonance, the two reactances are equal and cancel, so that the impedance equals the resistance alone. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]{I}_{\\text{rms}}=\\frac{{V}_{\\text{rms}}}{Z}=\\frac{120\\text{ V}}{40.0\\text{ }\\Omega}=3.00\\text{ A}\\\\[\/latex].<\/p>\n<h4><strong>Discussion for (b)<\/strong><\/h4>\n<p>At resonance, the current is greater than at the higher and lower frequencies considered for the same circuit in the preceding example.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div>\n<h2>Power in <em>RLC<\/em> Series AC Circuits<\/h2>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div title=\"Power in RLC Series AC Circuits\">\n<p>If current varies with frequency in an <em>RLC<\/em> circuit, then the power delivered to it also varies with frequency. But the average power is not simply current times voltage, as it is in purely resistive circuits. As was seen in Figure 2, voltage and current are out of phase in an <em>RLC<\/em> circuit. There is a <em> phase angle<\/em><em>\u03d5<\/em> between the source voltage <em>V<\/em> and the current <em>I<\/em>, which can be found from<\/p>\n<p style=\"text-align: center;\">[latex]\\cos\\varphi =\\frac{R}{Z}\\\\[\/latex]<\/p>\n<p>For example, at the resonant frequency or in a purely resistive circuit <em><em>Z\u00a0<\/em>=\u00a0<em>R<\/em><\/em>, so that [latex]\\text{cos}\\varphi =1\\\\[\/latex]. This implies that <em>\u03d5\u00a0<\/em>= 0\u00ba and that voltage and current are in phase, as expected for resistors. At other frequencies, average power is less than at resonance. This is both because voltage and current are out of phase and because <em>I<\/em><sub>rms<\/sub> is lower. The fact that source voltage and current are out of phase affects the power delivered to the circuit. It can be shown that the <em><em>average power<\/em><\/em> is<\/p>\n<p style=\"text-align: center;\">[latex]{P}_{\\text{ave}}={I}_{\\text{rms}}{V}_{\\text{rms}}\\cos\\varphi\\\\[\/latex],<\/p>\n<p>Thus\u00a0<span id=\"MathJax-Span-114477\" class=\"mtext\">cos\u00a0<\/span><span id=\"MathJax-Span-114478\" class=\"mspace\"><\/span><em><span id=\"MathJax-Span-114479\" class=\"mi\">\u03d5<\/span><\/em>\u00a0is called the <em> power factor<\/em>, which can range from 0 to 1. Power factors near 1 are desirable when designing an efficient motor, for example. At the resonant frequency,\u00a0<span id=\"MathJax-Span-114477\" class=\"mtext\">cos\u00a0<\/span><span id=\"MathJax-Span-114478\" class=\"mspace\"><\/span><em><span id=\"MathJax-Span-114479\" class=\"mi\">\u03d5 = <\/span><\/em><span id=\"MathJax-Span-114479\" class=\"mi\">1<\/span>.<\/p>\n<div title=\"Example 23.14. Calculating the Power Factor and Power\">\n<div class=\"textbox examples\">\n<h3>Example 3. Calculating the Power Factor and Power<\/h3>\n<div>\n<p>For the same <em>RLC<\/em> series circuit having a 40.0 \u03a9 resistor, a 3.00 mH inductor, a 5.00 \u03bcF capacitor, and a voltage source with a <em>V<\/em><sub>rms<\/sub> of 120 V: (a) Calculate the power factor and phase angle for\u00a0<span id=\"MathJax-Span-114519\" class=\"mrow\"><span id=\"MathJax-Span-114520\" class=\"mi\"><em>f<\/em>\u00a0<\/span><span id=\"MathJax-Span-114521\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-114522\" class=\"mtext\">60<\/span><\/span><span id=\"MathJax-Span-114523\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-114524\" class=\"mn\">0<\/span><span id=\"MathJax-Span-114525\" class=\"mi\"><\/span><span id=\"MathJax-Span-114526\" class=\"mtext\">Hz<\/span>. (b) What is the average power at 50.0 Hz? (c) Find the average power at the circuit\u2019s resonant frequency.<\/p>\n<h4><strong>Strategy and Solution for (a)<\/strong><\/h4>\n<p>The power factor at 60.0 Hz is found from<\/p>\n<p style=\"text-align: center;\">[latex]\\cos\\varphi =\\frac{R}{Z}\\\\[\/latex].<\/p>\n<p>We know <em>Z<\/em> = 531 \u03a9 from <em>Example 1:\u00a0Calculating Impedance and Current<\/em>, so that<\/p>\n<p style=\"text-align: center;\">[latex]\\cos\\varphi =\\frac{40.0\\text{ }\\Omega}{531\\text{ }\\Omega}=0.0753 \\text{ at }60.0\\text{ Hz}\\\\[\/latex].<\/p>\n<p>This small value indicates the voltage and current are significantly out of phase. In fact, the phase angle is<\/p>\n<p style=\"text-align: center;\">[latex]\\varphi ={\\cos}^{-1} 0.0753=\\text{85.7\u00ba}\\text{ at }60.0\\text{ Hz}\\\\[\/latex].<\/p>\n<h4><strong>Discussion for (a)<\/strong><\/h4>\n<p>The phase angle is close to 90\u00ba, consistent with the fact that the capacitor dominates the circuit at this low frequency (a pure <em>RC<\/em> circuit has its voltage and current 90\u00ba out of phase).<\/p>\n<h4><strong>Strategy and Solution for (b)<\/strong><\/h4>\n<p>The average power at 60.0 Hz is<\/p>\n<p style=\"text-align: center;\"><span id=\"MathJax-Span-114620\" class=\"mrow\"><span id=\"MathJax-Span-114621\" class=\"semantics\"><span id=\"MathJax-Span-114622\" class=\"mrow\"><span id=\"MathJax-Span-114623\" class=\"mrow\"><span id=\"MathJax-Span-114624\" class=\"mrow\"><span id=\"MathJax-Span-114625\" class=\"mrow\"><span id=\"MathJax-Span-114626\" class=\"mrow\"><span id=\"MathJax-Span-114627\" class=\"msub\"><em><span id=\"MathJax-Span-114628\" class=\"mi\">P<\/span><\/em><sub><span id=\"MathJax-Span-114629\" class=\"mrow\"><span id=\"MathJax-Span-114630\" class=\"mtext\">ave\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-114631\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-114632\" class=\"msub\"><em><span id=\"MathJax-Span-114633\" class=\"mi\">I<\/span><\/em><sub><span id=\"MathJax-Span-114634\" class=\"mrow\"><span id=\"MathJax-Span-114635\" class=\"mtext\">rms<\/span><\/span><\/sub><\/span><\/span><span id=\"MathJax-Span-114636\" class=\"msub\"><em><span id=\"MathJax-Span-114637\" class=\"mi\">V<\/span><\/em><sub><span id=\"MathJax-Span-114638\" class=\"mrow\"><span id=\"MathJax-Span-114639\" class=\"mtext\">rms\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-114640\" class=\"mtext\">cos\u00a0<\/span><span id=\"MathJax-Span-114641\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114642\" class=\"mi\"><em>\u03d5<\/em>.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><em>I<\/em><sub>rms<\/sub> was found to be 0.226 A in <em>Example 1:\u00a0Calculating Impedance and Current<\/em>. Entering the known values gives<\/p>\n<p style=\"text-align: center;\"><span id=\"MathJax-Span-114656\" class=\"mrow\"><span id=\"MathJax-Span-114657\" class=\"semantics\"><span id=\"MathJax-Span-114658\" class=\"mrow\"><span id=\"MathJax-Span-114659\" class=\"mrow\"><span id=\"MathJax-Span-114660\" class=\"mrow\"><span id=\"MathJax-Span-114661\" class=\"mrow\"><span id=\"MathJax-Span-114662\" class=\"mrow\"><span id=\"MathJax-Span-114663\" class=\"msub\"><em><span id=\"MathJax-Span-114664\" class=\"mi\">P<\/span><\/em><sub><span id=\"MathJax-Span-114665\" class=\"mrow\"><span id=\"MathJax-Span-114666\" class=\"mtext\">ave\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-114667\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-114668\" class=\"mo\">(<\/span><\/span><span id=\"MathJax-Span-114669\" class=\"mn\">0<\/span><span id=\"MathJax-Span-114670\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-114671\" class=\"mtext\">226\u00a0<\/span><span id=\"MathJax-Span-114672\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114673\" class=\"mtext\">A<\/span><span id=\"MathJax-Span-114674\" class=\"mo\">)<\/span><span id=\"MathJax-Span-114675\" class=\"mo\">(<\/span><span id=\"MathJax-Span-114676\" class=\"mtext\">120\u00a0<\/span><span id=\"MathJax-Span-114677\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114678\" class=\"mtext\">V<\/span><span id=\"MathJax-Span-114679\" class=\"mo\">)<\/span><span id=\"MathJax-Span-114680\" class=\"mo\">(<\/span><span id=\"MathJax-Span-114681\" class=\"mn\">0<\/span><span id=\"MathJax-Span-114682\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-114683\" class=\"mtext\">0753<\/span><span id=\"MathJax-Span-114684\" class=\"mrow\"><span id=\"MathJax-Span-114685\" class=\"mo\">)\u00a0<\/span><span id=\"MathJax-Span-114686\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-114687\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-114688\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-114689\" class=\"mtext\">04\u00a0<\/span><span id=\"MathJax-Span-114690\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114691\" class=\"mtext\">W at 60.0 Hz.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<h4><strong>Strategy and Solution for (c)<\/strong><\/h4>\n<p>At the resonant frequency, we know\u00a0<span id=\"MathJax-Span-114477\" class=\"mtext\">cos\u00a0<\/span><span id=\"MathJax-Span-114478\" class=\"mspace\"><\/span><em><span id=\"MathJax-Span-114479\" class=\"mi\">\u03d5 = <\/span><\/em><span id=\"MathJax-Span-114479\" class=\"mi\">1<\/span>, and <em>I<\/em><sub>rms<\/sub> was found to be 6.00 A in <em>Example 3: Calculating Resonant Frequency and Current<\/em>. Thus,\u00a0<span id=\"MathJax-Element-7251-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-114718\" class=\"math\"><span id=\"MathJax-Span-114719\" class=\"mrow\"><span id=\"MathJax-Span-114720\" class=\"semantics\"><span id=\"MathJax-Span-114721\" class=\"mrow\"><span id=\"MathJax-Span-114722\" class=\"mrow\"><span id=\"MathJax-Span-114723\" class=\"mrow\"><span id=\"MathJax-Span-114724\" class=\"mrow\"><span id=\"MathJax-Span-114725\" class=\"mrow\"><span id=\"MathJax-Span-114726\" class=\"msub\"><em><span id=\"MathJax-Span-114727\" class=\"mi\">P<\/span><\/em><sub><span id=\"MathJax-Span-114728\" class=\"mrow\"><span id=\"MathJax-Span-114729\" class=\"mtext\">ave\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-114730\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-114731\" class=\"mo\">(<\/span><\/span><span id=\"MathJax-Span-114732\" class=\"mn\">3<\/span><span id=\"MathJax-Span-114733\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-114734\" class=\"mtext\">00\u00a0<\/span><span id=\"MathJax-Span-114735\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114736\" class=\"mtext\">A<\/span><span id=\"MathJax-Span-114737\" class=\"mo\">)<\/span><span id=\"MathJax-Span-114738\" class=\"mo\">(<\/span><span id=\"MathJax-Span-114739\" class=\"mtext\">120\u00a0<\/span><span id=\"MathJax-Span-114740\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114741\" class=\"mtext\">V<\/span><span id=\"MathJax-Span-114742\" class=\"mo\">)<\/span><span id=\"MathJax-Span-114743\" class=\"mo\">(<\/span><span id=\"MathJax-Span-114744\" class=\"mn\">1<\/span><span id=\"MathJax-Span-114745\" class=\"mrow\"><span id=\"MathJax-Span-114746\" class=\"mo\">)\u00a0<\/span><span id=\"MathJax-Span-114747\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-114748\" class=\"mtext\">360\u00a0<\/span><\/span><span id=\"MathJax-Span-114749\" class=\"mspace\"><\/span><span id=\"MathJax-Span-114750\" class=\"mtext\">W<\/span><\/span><\/span><span id=\"MathJax-Span-114751\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> at resonance (1.30 kHz)<\/p>\n<h4><strong>Discussion<\/strong><\/h4>\n<p>Both the current and the power factor are greater at resonance, producing significantly greater power than at higher and lower frequencies.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Power delivered to an <em>RLC<\/em> series AC circuit is dissipated by the resistance alone. The inductor and capacitor have energy input and output but do not dissipate it out of the circuit. Rather they transfer energy back and forth to one another, with the resistor dissipating exactly what the voltage source puts into the circuit. This assumes no significant electromagnetic radiation from the inductor and capacitor, such as radio waves. Such radiation can happen and may even be desired, as we will see in the next chapter on electromagnetic radiation, but it can also be suppressed as is the case in this chapter. The circuit is analogous to the wheel of a car driven over a corrugated road as shown in Figure 4. The regularly spaced bumps in the road are analogous to the voltage source, driving the wheel up and down. The shock absorber is analogous to the resistance damping and limiting the amplitude of the oscillation. Energy within the system goes back and forth between kinetic (analogous to maximum current, and energy stored in an inductor) and potential energy stored in the car spring (analogous to no current, and energy stored in the electric field of a capacitor). The amplitude of the wheels\u2019 motion is a maximum if the bumps in the road are hit at the resonant frequency.<\/p>\n<div title=\"Figure 23.52.\">\n<div>\n<div>\n<div style=\"width: 235px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20111023\/Figure_24_12_04a.jpg\" alt=\"The figure describes the path of motion of a wheel of a car. The front wheel of a car is shown. A shock absorber attached to the wheel is also shown. The path of motion is shown as vertically up and down.\" width=\"225\" height=\"389\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. The forced but damped motion of the wheel on the car spring is analogous to an<em> RLC<\/em> series AC circuit. The shock absorber damps the motion and dissipates energy, analogous to the resistance in an <em>RLC<\/em> circuit. The mass and spring determine the resonant frequency.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>A pure <em>LC<\/em> circuit with negligible resistance oscillates at <em>f<\/em><sub>0<\/sub>, the same resonant frequency as an <em>RLC<\/em> circuit. It can serve as a frequency standard or clock circuit\u2014for example, in a digital wristwatch. With a very small resistance, only a very small energy input is necessary to maintain the oscillations. The circuit is analogous to a car with no shock absorbers. Once it starts oscillating, it continues at its natural frequency for some time. Figure 5\u00a0shows the analogy between an <em>LC<\/em> circuit and a mass on a spring.<\/p>\n<div title=\"Figure 23.53.\">\n<div>\n<div>\n<div style=\"width: 360px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/222\/2014\/12\/20111024\/Figure_24_12_05a.jpg\" alt=\"The figure describes four stages of an L C oscillation circuit compared to a mass oscillating on a spring. Part a of the figure shows a mass attached to a horizontal spring. The spring is attached to a fixed support on the left. The mass is at rest as shown by velocity v equals zero. The energy of the spring is shown as potential energy. This is compared with a circuit containing a capacitor C and inductor L connected together. The energy is shown as stored in the electric field E of the capacitor between the plates. One plate is shown to have a negative polarity and other plate is shown to have a positive polarity. Part b of the figure shows a mass attached to a horizontal spring which is attached to a fixed support on the left. The mass is shown to move horizontal toward the fixed support with velocity v. The energy here is stored as the kinetic energy of the spring. This is compared with a circuit containing a capacitor C and inductor L connected together. A current is shown in the circuit and energy is stored as magnetic field B in the inductor. Part c of the figure shows a mass attached to a horizontal spring which is attached to a fixed support on the left. The spring is shown as not stretched and the energy is shown as potential energy of the spring. The mass is show to have displaced toward left. This is compared with a circuit containing a capacitor C and inductor L connected together. The energy is shown as stored in the electric field E of the capacitor between the plates. One plate is shown to have a negative polarity and other plate is shown to have a positive polarity. But the polarities are reverse of the first case in part a. Part d of the figure shows a mass attached to a horizontal spring which is attached to a fixed support on the left. The mass is shown to move toward right with velocity v. the energy of the spring is kinetic energy. This is compared with a circuit containing a capacitor C and inductor L connected together. A current is shown in the circuit opposite to that in part b and energy is stored as magnetic field B in the inductor.\" width=\"350\" height=\"1089\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5. An LC circuit is analogous to a mass oscillating on a spring with no friction and no driving force. Energy moves back and forth between the inductor and capacitor, just as it moves from kinetic to potential in the mass-spring system.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div>\n<div class=\"textbox\">\n<div>\n<h2><strong>PhET Explorations: Circuit Construction Kit (AC+DC), Virtual Lab<\/strong><\/h2>\n<div>Build circuits with capacitors, inductors, resistors and AC or DC voltage sources, and inspect them using lab instruments such as voltmeters and ammeters.<\/div>\n<\/div>\n<div>\n<div style=\"width: 310px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/phet.colorado.edu\/sims\/circuit-construction-kit\/circuit-construction-kit-ac_en.jnlp\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/phet.colorado.edu\/sims\/circuit-construction-kit\/circuit-construction-kit-ac-600.png\" alt=\"Circuit Construction Kit (AC+DC)\" width=\"300\" height=\"197\" \/><\/a><\/p>\n<p class=\"wp-caption-text\">Click to download the simulation. Run using Java.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Section Summary<\/h2>\n<ul>\n<li>The AC analogy to resistance is impedance <em>Z<\/em>, the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm\u2019s law:\n<div style=\"text-align: center;\">[latex]{I}_{0}=\\frac{{V}_{0}}{Z}\\text{ or }{I}_{\\text{rms}}=\\frac{{V}_{\\text{rms}}}{Z}\\\\[\/latex],<\/div>\n<p>where <em>I<\/em><sub>o<\/sub>\u00a0is the peak current and <em>V<\/em><sub>o<\/sub>\u00a0is the peak source voltage.<\/li>\n<li>Impedance has units of ohms and is given by [latex]Z=\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\[\/latex].<\/li>\n<li>The resonant frequency\u00a0<span id=\"MathJax-Element-7896-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123708\" class=\"math\"><span id=\"MathJax-Span-123709\" class=\"mrow\"><span id=\"MathJax-Span-123710\" class=\"semantics\"><span id=\"MathJax-Span-123711\" class=\"mrow\"><span id=\"MathJax-Span-123712\" class=\"mrow\"><span id=\"MathJax-Span-123713\" class=\"mrow\"><span id=\"MathJax-Span-123714\" class=\"msub\"><em><span id=\"MathJax-Span-123715\" class=\"mi\">f<\/span><\/em><sub><span id=\"MathJax-Span-123716\" class=\"mrow\"><span id=\"MathJax-Span-123717\" class=\"mn\">0<\/span><\/span><\/sub><\/span><\/span><span id=\"MathJax-Span-123718\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, at which <span id=\"MathJax-Element-7897-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123719\" class=\"math\"><span id=\"MathJax-Span-123720\" class=\"mrow\"><span id=\"MathJax-Span-123721\" class=\"semantics\"><span id=\"MathJax-Span-123722\" class=\"mrow\"><span id=\"MathJax-Span-123723\" class=\"mrow\"><span id=\"MathJax-Span-123724\" class=\"mrow\"><span id=\"MathJax-Span-123725\" class=\"mrow\"><em><span id=\"MathJax-Span-123726\" class=\"msub\"><span id=\"MathJax-Span-123727\" class=\"mi\">X<\/span><span id=\"MathJax-Span-123728\" class=\"mrow\"><span id=\"MathJax-Span-123729\" class=\"mi\"><sub>L<\/sub>\u00a0<\/span><\/span><\/span><\/em><span id=\"MathJax-Span-123730\" class=\"mo\">=\u00a0<\/span><em><span id=\"MathJax-Span-123731\" class=\"msub\"><span id=\"MathJax-Span-123732\" class=\"mi\">X<\/span><sub><span id=\"MathJax-Span-123733\" class=\"mrow\"><span id=\"MathJax-Span-123734\" class=\"mi\">C<\/span><\/span><\/sub><\/span><\/em><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, is\n<div style=\"text-align: center;\">[latex]{f}_{0}=\\frac{1}{2\\pi \\sqrt{LC}}\\\\[\/latex]<\/div>\n<\/li>\n<li>In an AC circuit, there is a phase angle\u00a0<em data-effect=\"italics\"><span id=\"MathJax-Element-7899-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123762\" class=\"math\"><span id=\"MathJax-Span-123763\" class=\"mrow\"><span id=\"MathJax-Span-123764\" class=\"semantics\"><span id=\"MathJax-Span-123765\" class=\"mrow\"><span id=\"MathJax-Span-123766\" class=\"mrow\"><span id=\"MathJax-Span-123767\" class=\"mrow\"><span id=\"MathJax-Span-123768\" class=\"mi\">\u03d5<\/span><\/span><span id=\"MathJax-Span-123769\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/em> between source voltage <span id=\"MathJax-Element-7900-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123770\" class=\"math\"><span id=\"MathJax-Span-123771\" class=\"mrow\"><span id=\"MathJax-Span-123772\" class=\"semantics\"><span id=\"MathJax-Span-123773\" class=\"mrow\"><span id=\"MathJax-Span-123774\" class=\"mrow\"><em><span id=\"MathJax-Span-123775\" class=\"mrow\"><span id=\"MathJax-Span-123776\" class=\"mi\">V<\/span><\/span><\/em><span id=\"MathJax-Span-123777\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> and the current <span id=\"MathJax-Element-7901-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123778\" class=\"math\"><span id=\"MathJax-Span-123779\" class=\"mrow\"><span id=\"MathJax-Span-123780\" class=\"semantics\"><span id=\"MathJax-Span-123781\" class=\"mrow\"><span id=\"MathJax-Span-123782\" class=\"mrow\"><em><span id=\"MathJax-Span-123783\" class=\"mrow\"><span id=\"MathJax-Span-123784\" class=\"mi\">I<\/span><\/span><\/em><\/span><\/span><\/span><\/span><\/span><\/span>, which can be found from\n<div style=\"text-align: center;\">[latex]\\text{cos}\\varphi =\\frac{R}{Z}\\\\[\/latex],<\/div>\n<\/li>\n<li><span id=\"MathJax-Span-123812\" class=\"mi\"><em>\u03d5<\/em>\u00a0<\/span><span id=\"MathJax-Span-123813\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-123814\" class=\"mn\">0\u00ba<\/span> for a purely resistive circuit or an <em>RLC<\/em> circuit at resonance.<\/li>\n<li>The average power delivered to an <em>RLC<\/em> circuit is affected by the phase angle and is given by\n<div style=\"text-align: center;\">[latex]{P}_{\\text{ave}}={I}_{\\text{rms}}{V}_{\\text{rms}}\\cos\\varphi\\\\[\/latex],<\/div>\n<p><span id=\"MathJax-Element-7905-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123845\" class=\"math\"><span id=\"MathJax-Span-123846\" class=\"mrow\"><span id=\"MathJax-Span-123847\" class=\"semantics\"><span id=\"MathJax-Span-123848\" class=\"mrow\"><span id=\"MathJax-Span-123849\" class=\"mrow\"><span id=\"MathJax-Span-123850\" class=\"mrow\"><span id=\"MathJax-Span-123851\" class=\"mrow\"><span id=\"MathJax-Span-123852\" class=\"mtext\">cos\u00a0<\/span><span id=\"MathJax-Span-123853\" class=\"mspace\"><\/span><em><span id=\"MathJax-Span-123854\" class=\"mi\">\u03d5<\/span><\/em><\/span><\/span><span id=\"MathJax-Span-123855\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is called the power factor, which ranges from 0 to 1.<\/li>\n<\/ul>\n<section>\n<div class=\"textbox key-takeaways\">\n<h3>Conceptual Questions<\/h3>\n<div>\n<div>\n<p>1. Does the resonant frequency of an AC circuit depend on the peak voltage of the AC source? Explain why or why not.<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>2. Suppose you have a motor with a power factor significantly less than 1. Explain why it would be better to improve the power factor as a method of improving the motor\u2019s output, rather than to increase the voltage input.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section>\n<div class=\"textbox exercises\">\n<h3>Problems &amp; Exercises<\/h3>\n<div>\n<div>\n<p>1. An <em>RL<\/em> circuit consists of a 40.0 \u03a9 resistor and a 3.00 mH inductor. (a) Find its impedance <em>Z<\/em>\u00a0at 60.0 Hz and 10.0 kHz. (b) Compare these values of\u00a0<em>Z<\/em> with those found in <em>Example 1:\u00a0Calculating Impedance and Current<\/em>\u00a0in which there was also a capacitor.<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>2. An <em>RC<\/em> circuit consists of a\u00a0<span id=\"MathJax-Element-7913-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123891\" class=\"math\"><span id=\"MathJax-Span-123892\" class=\"mrow\"><span id=\"MathJax-Span-123893\" class=\"semantics\"><span id=\"MathJax-Span-123894\" class=\"mrow\"><span id=\"MathJax-Span-123895\" class=\"mi\">40.0 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> resistor and a <span id=\"MathJax-Element-7914-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123896\" class=\"math\"><span id=\"MathJax-Span-123897\" class=\"mrow\"><span id=\"MathJax-Span-123898\" class=\"semantics\"><span id=\"MathJax-Span-123899\" class=\"mrow\"><span id=\"MathJax-Span-123900\" class=\"mtext\">5.00 \u03bcF<\/span><\/span><\/span><\/span><\/span><\/span> capacitor. (a) Find its impedance at 60.0 Hz and 10.0 kHz. (b) Compare these values of <em>Z<\/em>\u00a0with those found in\u00a0<em>Example 1:\u00a0Calculating Impedance and Current<\/em>, in which there was also an inductor.<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>3. An <em>LC<\/em> circuit consists of a\u00a0<span id=\"MathJax-Element-7916-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123906\" class=\"math\"><span id=\"MathJax-Span-123907\" class=\"mrow\"><span id=\"MathJax-Span-123908\" class=\"semantics\"><span id=\"MathJax-Span-123909\" class=\"mrow\"><span id=\"MathJax-Span-123910\" class=\"mrow\"><span id=\"MathJax-Span-123911\" class=\"mrow\"><span id=\"MathJax-Span-123912\" class=\"mrow\"><span id=\"MathJax-Span-123913\" class=\"mn\">3<\/span><span id=\"MathJax-Span-123914\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-123915\" class=\"mtext\">00<\/span><span id=\"MathJax-Span-123916\" class=\"mspace\"><\/span><span id=\"MathJax-Span-123917\" class=\"mtext\">mH<\/span><\/span><\/span><span id=\"MathJax-Span-123918\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> inductor and a <span id=\"MathJax-Element-7917-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123919\" class=\"math\"><span id=\"MathJax-Span-123920\" class=\"mrow\"><span id=\"MathJax-Span-123921\" class=\"semantics\"><span id=\"MathJax-Span-123922\" class=\"mrow\"><span id=\"MathJax-Span-123923\" class=\"mrow\"><span id=\"MathJax-Span-123924\" class=\"mrow\"><span id=\"MathJax-Span-123925\" class=\"mrow\"><span id=\"MathJax-Span-123926\" class=\"mn\">5<\/span><span id=\"MathJax-Span-123927\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-123928\" class=\"mtext\">00\u00a0<\/span><span id=\"MathJax-Span-123929\" class=\"mspace\"><\/span><span id=\"MathJax-Span-123930\" class=\"mi\">\u03bcF<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> capacitor. (a) Find its impedance at 60.0 Hz and 10.0 kHz. (b) Compare these values of\u00a0<em>Z<\/em> with those found in\u00a0<em>Example 1:\u00a0Calculating Impedance and Current<\/em>\u00a0in which there was also a resistor.<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>4. What is the resonant frequency of a 0.500 mH inductor connected to a 40.0 \u03bcF capacitor?<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>5. To receive AM radio, you want an <em>RLC<\/em> circuit that can be made to resonate at any frequency between 500 and 1650 kHz. This is accomplished with a fixed\u00a0<span id=\"MathJax-Element-7922-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123955\" class=\"math\"><span id=\"MathJax-Span-123956\" class=\"mrow\"><span id=\"MathJax-Span-123957\" class=\"semantics\"><span id=\"MathJax-Span-123958\" class=\"mrow\"><span id=\"MathJax-Span-123959\" class=\"mtext\">1.00 \u03bcH<\/span><\/span><\/span><\/span><\/span><\/span> inductor connected to a variable capacitor. What range of capacitance is needed?<\/p>\n<\/div>\n<div>\n<p>6. Suppose you have a supply of inductors ranging from 1.00 nH to 10.0 H, and capacitors ranging from 1.00 pF to 0.100 F. What is the range of resonant frequencies that can be achieved from combinations of a single inductor and a single capacitor?<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>7. What capacitance do you need to produce a resonant frequency of 1.00 GHz, when using an 8.00 nH inductor?<\/p>\n<\/div>\n<div>\n<p>8. What inductance do you need to produce a resonant frequency of 60.0 Hz, when using a 2.00 \u03bcF capacitor?<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>9. The lowest frequency in the FM radio band is 88.0 MHz. (a) What inductance is needed to produce this resonant frequency if it is connected to a 2.50 pF capacitor? (b) The capacitor is variable, to allow the resonant frequency to be adjusted to as high as 108 MHz. What must the capacitance be at this frequency?<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>10. An <em>RLC<\/em> series circuit has a\u00a0<span id=\"MathJax-Element-7925-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123970\" class=\"math\"><span id=\"MathJax-Span-123971\" class=\"mrow\"><span id=\"MathJax-Span-123972\" class=\"semantics\"><span id=\"MathJax-Span-123973\" class=\"mrow\"><span id=\"MathJax-Span-123974\" class=\"mi\">2.50 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> resistor, a <span id=\"MathJax-Element-7926-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123975\" class=\"math\"><span id=\"MathJax-Span-123976\" class=\"mrow\"><span id=\"MathJax-Span-123977\" class=\"semantics\"><span id=\"MathJax-Span-123978\" class=\"mrow\"><span id=\"MathJax-Span-123979\" class=\"mi\">100 \u03bcH<\/span><\/span><\/span><\/span><\/span><\/span> inductor, and an <span id=\"MathJax-Element-7927-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123980\" class=\"math\"><span id=\"MathJax-Span-123981\" class=\"mrow\"><span id=\"MathJax-Span-123982\" class=\"semantics\"><span id=\"MathJax-Span-123983\" class=\"mrow\"><span id=\"MathJax-Span-123984\" class=\"mi\">80.0 \u03bcF\u00a0<\/span><\/span><\/span><\/span><\/span><\/span>capacitor.(a) Find the circuit\u2019s impedance at 120 Hz. (b) Find the circuit\u2019s impedance at 5.00 kHz. (c) If the voltage source has <span id=\"MathJax-Element-7928-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123985\" class=\"math\"><span id=\"MathJax-Span-123986\" class=\"mrow\"><span id=\"MathJax-Span-123987\" class=\"semantics\"><span id=\"MathJax-Span-123988\" class=\"mrow\"><span id=\"MathJax-Span-123989\" class=\"mrow\"><span id=\"MathJax-Span-123990\" class=\"mrow\"><span id=\"MathJax-Span-123991\" class=\"mrow\"><span id=\"MathJax-Span-123992\" class=\"mrow\"><span id=\"MathJax-Span-123993\" class=\"msub\"><em><span id=\"MathJax-Span-123994\" class=\"mi\">V<\/span><\/em><sub><span id=\"MathJax-Span-123995\" class=\"mrow\"><span id=\"MathJax-Span-123996\" class=\"mtext\">rms\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-123997\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-123998\" class=\"mn\">5<\/span><\/span><span id=\"MathJax-Span-123999\" class=\"mtext\">.<\/span><span id=\"MathJax-Span-124000\" class=\"mtext\">60\u00a0<\/span><span id=\"MathJax-Span-124001\" class=\"mspace\"><\/span><span id=\"MathJax-Span-124002\" class=\"mtext\">V<\/span><\/span><\/span><span id=\"MathJax-Span-124003\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, what is <span id=\"MathJax-Element-7929-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124004\" class=\"math\"><span id=\"MathJax-Span-124005\" class=\"mrow\"><span id=\"MathJax-Span-124006\" class=\"semantics\"><span id=\"MathJax-Span-124007\" class=\"mrow\"><span id=\"MathJax-Span-124008\" class=\"mrow\"><span id=\"MathJax-Span-124009\" class=\"mrow\"><span id=\"MathJax-Span-124010\" class=\"msub\"><em><span id=\"MathJax-Span-124011\" class=\"mi\">I<\/span><\/em><sub><span id=\"MathJax-Span-124012\" class=\"mrow\"><span id=\"MathJax-Span-124013\" class=\"mtext\">rms<\/span><\/span><\/sub><\/span><\/span><span id=\"MathJax-Span-124014\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> at each frequency? (d) What is the resonant frequency of the circuit? (e) What is\u00a0<em><span id=\"MathJax-Span-124011\" class=\"mi\">I<\/span><\/em><sub><span id=\"MathJax-Span-124012\" class=\"mrow\"><span id=\"MathJax-Span-124013\" class=\"mtext\">rms<\/span><\/span><\/sub>\u00a0at resonance?<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>11. An <em>RLC<\/em> series circuit has a\u00a0<span id=\"MathJax-Element-7931-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124026\" class=\"math\"><span id=\"MathJax-Span-124027\" class=\"mrow\"><span id=\"MathJax-Span-124028\" class=\"semantics\"><span id=\"MathJax-Span-124029\" class=\"mrow\"><span id=\"MathJax-Span-124030\" class=\"mi\">1.00 k\u03a9<\/span><\/span><\/span><\/span><\/span><\/span> resistor, a <span id=\"MathJax-Element-7932-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124031\" class=\"math\"><span id=\"MathJax-Span-124032\" class=\"mrow\"><span id=\"MathJax-Span-124033\" class=\"semantics\"><span id=\"MathJax-Span-124034\" class=\"mrow\"><span id=\"MathJax-Span-124035\" class=\"mi\">150 \u03bcH<\/span><\/span><\/span><\/span><\/span><\/span> inductor, and a 25.0 nF capacitor. (a) Find the circuit\u2019s impedance at 500 Hz. (b) Find the circuit\u2019s impedance at 7.50 kHz. (c) If the voltage source has <span id=\"MathJax-Element-7933-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124036\" class=\"math\"><span id=\"MathJax-Span-124037\" class=\"mrow\"><span id=\"MathJax-Span-124038\" class=\"semantics\"><span id=\"MathJax-Span-124039\" class=\"mrow\"><span id=\"MathJax-Span-124040\" class=\"mrow\"><span id=\"MathJax-Span-124041\" class=\"mrow\"><span id=\"MathJax-Span-124042\" class=\"mrow\"><span id=\"MathJax-Span-124043\" class=\"mrow\"><span id=\"MathJax-Span-124044\" class=\"msub\"><em><span id=\"MathJax-Span-124045\" class=\"mi\">V<\/span><\/em><sub><span id=\"MathJax-Span-124046\" class=\"mrow\"><span id=\"MathJax-Span-124047\" class=\"mtext\">rms\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-124048\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-124049\" class=\"mtext\">408\u00a0<\/span><\/span><span id=\"MathJax-Span-124050\" class=\"mspace\"><\/span><span id=\"MathJax-Span-124051\" class=\"mtext\">V<\/span><\/span><\/span><span id=\"MathJax-Span-124052\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, what is <span id=\"MathJax-Element-7934-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124053\" class=\"math\"><span id=\"MathJax-Span-124054\" class=\"mrow\"><span id=\"MathJax-Span-124055\" class=\"semantics\"><span id=\"MathJax-Span-124056\" class=\"mrow\"><span id=\"MathJax-Span-124057\" class=\"mrow\"><span id=\"MathJax-Span-124058\" class=\"mrow\"><span id=\"MathJax-Span-124059\" class=\"msub\"><em><span id=\"MathJax-Span-124060\" class=\"mi\">I<\/span><\/em><sub><span id=\"MathJax-Span-124061\" class=\"mrow\"><span id=\"MathJax-Span-124062\" class=\"mtext\">rms<\/span><\/span><\/sub><\/span><\/span><span id=\"MathJax-Span-124063\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> at each frequency? (d) What is the resonant frequency of the circuit? (e) What is\u00a0<em><span id=\"MathJax-Span-124060\" class=\"mi\">I<\/span><\/em><sub><span id=\"MathJax-Span-124061\" class=\"mrow\"><span id=\"MathJax-Span-124062\" class=\"mtext\">rms<\/span><\/span><\/sub>\u00a0at resonance?<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>12. An <em>RLC<\/em> series circuit has a\u00a0<span id=\"MathJax-Element-7938-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124085\" class=\"math\"><span id=\"MathJax-Span-124086\" class=\"mrow\"><span id=\"MathJax-Span-124087\" class=\"semantics\"><span id=\"MathJax-Span-124088\" class=\"mrow\"><span id=\"MathJax-Span-124089\" class=\"mi\">2.50 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> resistor, a <span id=\"MathJax-Element-7939-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124090\" class=\"math\"><span id=\"MathJax-Span-124091\" class=\"mrow\"><span id=\"MathJax-Span-124092\" class=\"semantics\"><span id=\"MathJax-Span-124093\" class=\"mrow\"><span id=\"MathJax-Span-124094\" class=\"mi\">100 \u03bcH<\/span><\/span><\/span><\/span><\/span><\/span> inductor, and an <span id=\"MathJax-Element-7940-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124095\" class=\"math\"><span id=\"MathJax-Span-124096\" class=\"mrow\"><span id=\"MathJax-Span-124097\" class=\"semantics\"><span id=\"MathJax-Span-124098\" class=\"mrow\"><span id=\"MathJax-Span-124099\" class=\"mi\">80.0 \u03bcF\u00a0<\/span><\/span><\/span><\/span><\/span><\/span>capacitor. (a) Find the power factor at <span id=\"MathJax-Element-7941-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124100\" class=\"math\"><span id=\"MathJax-Span-124101\" class=\"mrow\"><span id=\"MathJax-Span-124102\" class=\"semantics\"><span id=\"MathJax-Span-124103\" class=\"mrow\"><span id=\"MathJax-Span-124104\" class=\"mrow\"><span id=\"MathJax-Span-124105\" class=\"mi\"><em>f<\/em>\u00a0<\/span><span id=\"MathJax-Span-124106\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-124107\" class=\"mi\">120 Hz<\/span><\/span><\/span><\/span><\/span><\/span><\/span>. (b) What is the phase angle at 120 Hz? (c) What is the average power at 120 Hz? (d) Find the average power at the circuit\u2019s resonant frequency.<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>13. An <em>RLC<\/em> series circuit has a\u00a0<span id=\"MathJax-Element-7942-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124108\" class=\"math\"><span id=\"MathJax-Span-124109\" class=\"mrow\"><span id=\"MathJax-Span-124110\" class=\"semantics\"><span id=\"MathJax-Span-124111\" class=\"mrow\"><span id=\"MathJax-Span-124112\" class=\"mi\">1.00 k\u03a9<\/span><\/span><\/span><\/span><\/span><\/span> resistor, a <span id=\"MathJax-Element-7943-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124113\" class=\"math\"><span id=\"MathJax-Span-124114\" class=\"mrow\"><span id=\"MathJax-Span-124115\" class=\"semantics\"><span id=\"MathJax-Span-124116\" class=\"mrow\"><span id=\"MathJax-Span-124117\" class=\"mi\">150 \u03bcH<\/span><\/span><\/span><\/span><\/span><\/span> inductor, and a 25.0 nF capacitor. (a) Find the power factor at <span id=\"MathJax-Element-7944-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124118\" class=\"math\"><span id=\"MathJax-Span-124119\" class=\"mrow\"><span id=\"MathJax-Span-124120\" class=\"semantics\"><span id=\"MathJax-Span-124121\" class=\"mrow\"><span id=\"MathJax-Span-124122\" class=\"mrow\"><em><span id=\"MathJax-Span-124123\" class=\"mi\">f\u00a0<\/span><\/em><span id=\"MathJax-Span-124124\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-124125\" class=\"mi\">7.50 Hz<\/span><\/span><\/span><\/span><\/span><\/span><\/span>. (b) What is the phase angle at this frequency? (c) What is the average power at this frequency? (d) Find the average power at the circuit\u2019s resonant frequency.<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>14. An <em>RLC<\/em> series circuit has a\u00a0<span id=\"MathJax-Element-7946-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124131\" class=\"math\"><span id=\"MathJax-Span-124132\" class=\"mrow\"><span id=\"MathJax-Span-124133\" class=\"semantics\"><span id=\"MathJax-Span-124134\" class=\"mrow\"><span id=\"MathJax-Span-124135\" class=\"mi\">200 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> resistor and a 25.0 mH inductor. At 8000 Hz, the phase angle is <span id=\"MathJax-Element-7947-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124136\" class=\"math\"><span id=\"MathJax-Span-124137\" class=\"mrow\"><span id=\"MathJax-Span-124138\" class=\"semantics\"><span id=\"MathJax-Span-124139\" class=\"mrow\"><span id=\"MathJax-Span-124140\" class=\"mi\">45.0\u00ba<\/span><\/span><\/span><\/span><\/span><\/span>. (a) What is the impedance? (b) Find the circuit\u2019s capacitance. (c) If\u00a0<span id=\"MathJax-Element-7948-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124141\" class=\"math\"><span id=\"MathJax-Span-124142\" class=\"mrow\"><span id=\"MathJax-Span-124143\" class=\"semantics\"><span id=\"MathJax-Span-124144\" class=\"mrow\"><span id=\"MathJax-Span-124145\" class=\"mrow\"><span id=\"MathJax-Span-124146\" class=\"mrow\"><span id=\"MathJax-Span-124147\" class=\"mrow\"><span id=\"MathJax-Span-124148\" class=\"mrow\"><span id=\"MathJax-Span-124149\" class=\"msub\"><em><span id=\"MathJax-Span-124150\" class=\"mi\">V<\/span><\/em><sub><span id=\"MathJax-Span-124151\" class=\"mrow\"><span id=\"MathJax-Span-124152\" class=\"mtext\">rms\u00a0<\/span><\/span><\/sub><\/span><span id=\"MathJax-Span-124153\" class=\"mo\">=\u00a0<\/span><span id=\"MathJax-Span-124154\" class=\"mtext\">408\u00a0<\/span><\/span><span id=\"MathJax-Span-124155\" class=\"mspace\"><\/span><span id=\"MathJax-Span-124156\" class=\"mtext\">V<\/span><\/span><\/span><span id=\"MathJax-Span-124157\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span> is applied, what is the average power supplied?<\/p>\n<\/div>\n<\/div>\n<div>\n<div>\n<p>15. Referring to\u00a0Example 3. Calculating the Power Factor and Power, find the average power at 10.0 kHz.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<h2>Glossary<\/h2>\n<dl>\n<dt>impedance:<\/dt>\n<dd>the AC analogue to resistance in a DC circuit; it is the combined effect of resistance, inductive reactance, and capacitive reactance in the form [latex]Z=\\sqrt{{R}^{2}+\\left({X}_{L}-{X}_{C}\\right)^{2}}\\\\[\/latex]<\/dd>\n<\/dl>\n<dl>\n<dt>resonant frequency:<\/dt>\n<dd>the frequency at which the impedance in a circuit is at a minimum, and also the frequency at which the circuit would oscillate if not driven by a voltage source; calculated by [latex]{f}_{0}=\\frac{1}{2\\pi \\sqrt{\\text{LC}}}\\\\[\/latex]<\/dd>\n<\/dl>\n<dl>\n<dt>phase angle:<\/dt>\n<dd>denoted by\u00a0<em><span id=\"MathJax-Span-124232\" class=\"mi\">\u03d5<\/span><\/em>, the amount by which the voltage and current are out of phase with each other in a circuit<\/dd>\n<\/dl>\n<dl>\n<dt>power factor:<\/dt>\n<dd>the amount by which the power delivered in the circuit is less than the theoretical maximum of the circuit due to voltage and current being out of phase; calculated by\u00a0<span id=\"MathJax-Span-124224\" class=\"mrow\"><span id=\"MathJax-Span-124225\" class=\"semantics\"><span id=\"MathJax-Span-124226\" class=\"mrow\"><span id=\"MathJax-Span-124227\" class=\"mrow\"><span id=\"MathJax-Span-124228\" class=\"mrow\"><span id=\"MathJax-Span-124229\" class=\"mrow\"><span id=\"MathJax-Span-124230\" class=\"mtext\">cos\u00a0<\/span><span id=\"MathJax-Span-124231\" class=\"mspace\"><\/span><em><span id=\"MathJax-Span-124232\" class=\"mi\">\u03d5<\/span><\/em><\/span><\/span><span id=\"MathJax-Span-124233\" class=\"mrow\"><\/span><\/span><\/span><\/span><\/span><\/dd>\n<\/dl>\n<div class=\"textbox exercises\">\n<h3>Selected Solutions to Problems &amp; Exercises<\/h3>\n<p>1. \u00a0(a) <span id=\"MathJax-Element-7909-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123871\" class=\"math\"><span id=\"MathJax-Span-123872\" class=\"mrow\"><span id=\"MathJax-Span-123873\" class=\"semantics\"><span id=\"MathJax-Span-123874\" class=\"mrow\"><span id=\"MathJax-Span-123875\" class=\"mi\">40.02 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> at 60.0 Hz, <span id=\"MathJax-Element-7910-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123876\" class=\"math\"><span id=\"MathJax-Span-123877\" class=\"mrow\"><span id=\"MathJax-Span-123878\" class=\"semantics\"><span id=\"MathJax-Span-123879\" class=\"mrow\"><span id=\"MathJax-Span-123880\" class=\"mi\">193 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> at 10.0 kHz\u00a0(b) At 60 Hz, with a capacitor, <span id=\"MathJax-Element-7911-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123881\" class=\"math\"><span id=\"MathJax-Span-123882\" class=\"mrow\"><span id=\"MathJax-Span-123883\" class=\"semantics\"><span id=\"MathJax-Span-123884\" class=\"mrow\"><span id=\"MathJax-Span-123885\" class=\"mi\">Z = 531 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span>, over 13 times as high as without the capacitor. The capacitor makes a large difference at low frequencies. At 10 kHz, with a capacitor <span id=\"MathJax-Element-7912-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123886\" class=\"math\"><span id=\"MathJax-Span-123887\" class=\"mrow\"><span id=\"MathJax-Span-123888\" class=\"semantics\"><span id=\"MathJax-Span-123889\" class=\"mrow\"><span id=\"MathJax-Span-123890\" class=\"mi\">Z = 190 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span>, about the same as without the capacitor. The capacitor has a smaller effect at high frequencies.<\/p>\n<p>3.\u00a0(a) <span id=\"MathJax-Element-7919-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123940\" class=\"math\"><span id=\"MathJax-Span-123941\" class=\"mrow\"><span id=\"MathJax-Span-123942\" class=\"semantics\"><span id=\"MathJax-Span-123943\" class=\"mrow\"><span id=\"MathJax-Span-123944\" class=\"mi\">529 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> at 60.0 Hz, <span id=\"MathJax-Element-7920-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123945\" class=\"math\"><span id=\"MathJax-Span-123946\" class=\"mrow\"><span id=\"MathJax-Span-123947\" class=\"semantics\"><span id=\"MathJax-Span-123948\" class=\"mrow\"><span id=\"MathJax-Span-123949\" class=\"mi\">185 \u03a9<\/span><\/span><\/span><\/span><\/span><\/span> at 10.0 kHz\u00a0(b) These values are close to those obtained in\u00a0<em>Example 1:\u00a0Calculating Impedance and Current<\/em>\u00a0because at low frequency the capacitor dominates and at high frequency the inductor dominates. So in both cases the resistor makes little contribution to the total impedance.<\/p>\n<p>5.\u00a09.30 nF to 101 nF<\/p>\n<p>7. 3.17 pF<\/p>\n<p>9. (a) <span id=\"MathJax-Element-7924-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-123965\" class=\"math\"><span id=\"MathJax-Span-123966\" class=\"mrow\"><span id=\"MathJax-Span-123967\" class=\"semantics\"><span id=\"MathJax-Span-123968\" class=\"mrow\"><span id=\"MathJax-Span-123969\" class=\"mi\">1.31 \u03bcH\u00a0<\/span><\/span><\/span><\/span><\/span><\/span>(b) 1.66 pF<\/p>\n<p>11.\u00a0(a) <span id=\"MathJax-Element-7936-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124075\" class=\"math\"><span id=\"MathJax-Span-124076\" class=\"mrow\"><span id=\"MathJax-Span-124077\" class=\"semantics\"><span id=\"MathJax-Span-124078\" class=\"mrow\"><span id=\"MathJax-Span-124079\" class=\"mi\">12.8 k\u03a9\u00a0<\/span><\/span><\/span><\/span><\/span><\/span>(b) <span id=\"MathJax-Element-7937-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124080\" class=\"math\"><span id=\"MathJax-Span-124081\" class=\"mrow\"><span id=\"MathJax-Span-124082\" class=\"semantics\"><span id=\"MathJax-Span-124083\" class=\"mrow\"><span id=\"MathJax-Span-124084\" class=\"mi\">1.31 k\u03a9\u00a0<\/span><\/span><\/span><\/span><\/span><\/span>(c) 31.9 mA at 500 Hz, 312 mA at 7.50 kHz\u00a0(d) 82.2 kHz\u00a0(e) 0.408 A<\/p>\n<p>13.\u00a0(a) 0.159\u00a0(b) <span id=\"MathJax-Element-7945-Frame\" class=\"MathJax\"><span id=\"MathJax-Span-124126\" class=\"math\"><span id=\"MathJax-Span-124127\" class=\"mrow\"><span id=\"MathJax-Span-124128\" class=\"semantics\"><span id=\"MathJax-Span-124129\" class=\"mrow\"><span id=\"MathJax-Span-124130\" class=\"mi\">80.9\u00ba\u00a0<\/span><\/span><\/span><\/span><\/span><\/span>(c) 26.4 W\u00a0(d) 166 W<\/p>\n<p>15.\u00a016.0 W<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-5160\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Physics. <strong>Authored by<\/strong>: OpenStax College. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/031da8d3-b525-429c-80cf-6c8ed997733a\/College_Physics\">http:\/\/cnx.org\/contents\/031da8d3-b525-429c-80cf-6c8ed997733a\/College_Physics<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Located at License<\/li><li>PhET Interactive Simulations . <strong>Provided by<\/strong>: University of Colorado Boulder . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/phet.colorado.edu\">http:\/\/phet.colorado.edu<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":1,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Physics\",\"author\":\"OpenStax College\",\"organization\":\"\",\"url\":\"http:\/\/cnx.org\/contents\/031da8d3-b525-429c-80cf-6c8ed997733a\/College_Physics\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Located at License\"},{\"type\":\"cc\",\"description\":\"PhET Interactive Simulations \",\"author\":\"\",\"organization\":\"University of Colorado Boulder \",\"url\":\"http:\/\/phet.colorado.edu\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-5160","chapter","type-chapter","status-publish","hentry"],"part":7663,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters\/5160","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":24,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters\/5160\/revisions"}],"predecessor-version":[{"id":12098,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters\/5160\/revisions\/12098"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/parts\/7663"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapters\/5160\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/media?parent=5160"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/pressbooks\/v2\/chapter-type?post=5160"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/contributor?post=5160"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-physics\/wp-json\/wp\/v2\/license?post=5160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}