Parallel Resonance

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Parallel Resonance:

Parallel Resonance ET 242 Circuit Analysis II E lectrical and T elecommunication Engineering Technology Professor Jang

Acknowledgement:

Acknowledgement I want to express my gratitude to Prentice Hall giving me the permission to use instructor’s material for developing this module. I would like to thank the Department of Electrical and Telecommunications Engineering Technology of NYCCT for giving me support to commence and complete this module. I hope this module is helpful to enhance our students’ academic performance.

OUTLINES:

OUTLINES Introduction to Parallel Resonance Selectivity Curve Parallel Resonance Circuit Unity Power Factor ( f p ) Effect of Q L ≥ 10 Examples ET 242 Circuit Analysis II – Parallel Resonance Boylestad 2 Key Words : Resonance, Unity Power Factor, Selective Curve, Quality Factor

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Parallel Resonance Circuit - Introduction The basic format of the series resonant circuit is a series R-L-C combination in series with an applied voltage source. The parallel resonant circuit has the basic configuration in Fig. 20.21, a parallel R-L-C combination in parallel with an applied current source. ET 242 Circuit Analysis II – Parallel Resonance Boylestad 3 If the practical equivalent in Fig. 20.22 had the format in Fig. 20.21, the analysis would be as direct and lucid as that experience for series resonance. However, in the practical world, the internal resistance of the coil must be placed in series with the inductor, as shown in Fig.20.22. Figure 20.22 Practical parallel L-C network. Figure 20.21 Ideal parallel resonant network.

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ET162 Circuit Analysis – Parallel Resonance Boylestad 4 The first effort is to find a parallel network equivalent for the series R-L branch in Fig.20.22 using the technique in earlier section. That is Figure 20.23 Equivalent parallel network for a series R-L combination. Parallel Resonant Circuit – Unity Power Factor, f p Figure 20.25 Substituting R = R s //R p for the network in Fig. 20.24.

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ET 242 Circuit Analysis II – Parallel Resonance Boylestad 5 Where f p is the resonant frequency of a parallel resonant circuit (for F p = 1 ) and f s is the resonant frequency as determined by X L = X C for series resonance. Note that unlike a series resonant circuit, the resonant frequency f p is a function of resistance (in this case R l ). Parallel Resonant Circuit – Maximum Impedance, f m At f = f p the input impedance of a parallel resonant circuit will be near its maximum value but not quite its maximum value due to the frequency dependence of R p . The frequency at which impedance occurs is defined by fm and is slightly more than f p , as demonstrated in Fig. 20.26. Figure 20.26 Z T versus frequency for the parallel resonant circuit.

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Parallel Resonant Circuit – Selectivity Curve ET 242 Circuit Analysis II – Parallel Resonance Boylestad 6 The frequency f m is determined by differentiating the general equation for Z T with respect to frequency and then determining the frequency at which the resulting equation is equal to zero. The resulting equation, however, is the following: Note the similarities with Eq. (20.31). Since square root factor of Eq. (20.32) is always more than the similar factor of Eq. (20.31), fm is always closer to fs and more than f p . In general, f s > f m > f p Once fm is determined, the network in Fig. 20.25 can be used to determine the magnitude and phase angle of the total impedance at the resonance condition simply by substituting f = f m and performing the required calculations. That is Z T m = R // X L p // X C f =f m Since the current I of the current source is constant for any value of Z T or frequency, the voltage across the parallel circuit will have the same shape as the total impedance Z T , as shown in Fig. 20.27. For parallel circuit, the resonance curve of interest in V C derives from electronic considerations that often place the capacitor at the input to another stage of a network. Figure 20.27 Defining the shape of the V p (f) curve.

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ET 242 Circuit Analysis II – Parallel Resonance Boylestad 7 Since the voltage across parallel elements is the same, V C = V p = IZ T The resonant value of V C is therefore determined by the value of Z T m and magnitude of the current source I. The quality factor of the parallel resonant circuit continues to be determined as following; For the ideal current source (Rs = ∞ Ω ) or when R s is sufficiently large compared to R p , we can make the following approximation: In general, the bandwidth is still related to the resonant frequency and the quality factor by The cutoff frequencies f 1 and f 2 can be determined using the equivalent network and the quality factor by

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The effect of R l , L , and C on the shape of the parallel resonance curve, as shown in Fig. 20.28 for the input impedance, is quite similar to their effect on the series resonance curve. Whether or not R l is zero, the parallel resonant circuit frequently appears in a network schematic as shown in Fig. 20.28. At resonance, an increase in R l or decrease in the ratio L/R results in a decrease in the resonant impedance, with a corresponding increase in the current. Figure 20.28 Effect of R 1 , L, and, C on the parallel resonance curve. ET 242 Circuit Analysis II – Parallel Resonance Boylestad 8 Parallel Resonant Circuit – Effect of Q L ≥ 10 The analysis of parallel resonant circuits is significantly more complex than encountered for series circuits. However, this is not the case since, for the majority of parallel resonant circuits, the quality factor of the coil Q l is sufficiently large to permit a number of approximations that simplify the required analysis. Effect of Q L ≥ 10 – Inductive Resistance, X L p

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ET 242 Circuit Analysis II – Parallel Resonance Boylestad 9 Effect of Q L ≥ 10 – Resonant Frequency, f p (Unity Power Factor) Effect of Q L ≥ 10 – Resonant Frequency, f m (Maximum V C ) R p

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ET 242 Circuit Analysis II – Parallel Resonance Boylestad 10 Figure 20.31 Establishing the relationship between I C and I L and current I T . Z T p Q p BW I L and I C A portion of Fig. 20.30 is reproduced in Fig. 20.31, with I T defined as shown

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ET 242 Circuit Analysis II – Power for AC Circuits Boylestad 11 Ex. 20-6 Given the parallel network in Fig. 20.32 composed of “ideal” elements: a. Determine the resonant frequency f p . b. Find the total impedance at resonance c. Calculate the quality factor, bandwidth, and cutoff frequencies f 1 and f 2 of the system. d. Find the voltage V C at resonance. e. Determine the currents I L and I C at resonance. Figure 20.32 Example 20.6.

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ET 242 Circuit Analysis II – Parallel Resonance Boylestad 12 Ex. 20-7 For the parallel resonant circuit in Fig. 20.33 with R s = ∞ Ω : a. Determine f p , f m , and f p , and compare their levels. b. Calculate the maximum impedance and the magnitude of the voltage V C at f m . c. Determine the quality factor Q p . d. Calculate the bandwidth. e. Compare the above results with those obtained using the equations associated with Q l ≥ 10. Figure 20.33 Example 20.7.

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ET 242 Circuit Analysis II – Parallel Resonance Boylestad 13 Ex. 20-8 For the network in Fig. 20.34 with f p provided: a. Determine Q l . b. Determine R p . c. Calculate Z T p . d. Find C at resonance. e. Find Q p . f. Calculate the BW and cutoff frequencies. Figure 20.34 Example 20.8.

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Figure 20.35 Example 20.9. ET 242 Circuit Analysis II – Series Resonance Boylestad 14 Ex. 20-10 Repeat Example 20.9, but ignore the effects of R s , and compare results.

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