Resonance Topics Covered in Chapter : The Resonance Effect 25-2: Series Resonance 25-3: Parallel Resonance 25-4: Resonant Frequency: Chapter 25 © 2007 The McGraw-Hill Companies, Inc. All rights reserved.

Topics Covered in Chapter 25  25-5: Q Magnification Factor of Resonant Circuit  25-6: Bandwidth of Resonant Circuit  25-7: Tuning  25-8: Mistuning  25-9: Analysis of Parallel Resonant Circuits  25-10: Damping of Parallel Resonant Circuits  25-11: Choosing L and C for a Resonant Circuit McGraw-Hill© 2007 The McGraw-Hill Companies, Inc. All rights reserved.

25-1: The Resonance Effect  Inductive reactance increases as the frequency is increased, but capacitive reactance decreases with higher frequencies.  Because of these opposite characteristics, for any LC combination, there must be a frequency at which the X L equals the X C ; one increases while the other decreases.  This case of equal and opposite reactances is called resonance, and the ac circuit is then a resonant circuit.  The frequency at which X L = X C is the resonant frequency.

25-1: The Resonance Effect Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 25-1:  The most common application of resonance in rf circuits is called tuning.  In Fig. 25-1, the LC circuit is resonant at 1000 kHz.  The result is maximum output at 1000 kHz, compared with lower or higher frequencies.

25-2: Series Resonance  At the resonant frequency, the inductive reactance and capacitive reactance are equal.  In a series ac circuit, inductive reactance leads by 90°, compared with the zero reference angle of the resistance, and capacitive reactance lags by 90°.  X L and X C are 180° out of phase.  The opposite reactances cancel each other completely when they are equal.

25-2: Series Resonance Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Series Resonant Circuit L C where: f r = resonant frequency in Hz L = inductance in henrys C = capacitance in farads

25-2: Series Resonance Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig. 25-2:  Fig (b) shows X L and X C equal, resulting in a net reactance of zero ohms.  The only opposition to current is the coil resistance r s, which limits how low the series resistance in the circuit can be.

25-2: Series Resonance Resonant Rise in V L and V C 5 A R = 4  X C = 31  X L = 31  20 V 5 kHz I r = 20/4 = 5 A V L = I × X L = 155 V V C = I × X C = 155 V Note: The reactive voltages are phasor opposites and they cancel (V X L +V X C = 0).

25-2: Series Resonance Resonant Rise in V L and V C 4  0.25  F 4 mH Q = 32 5 A 20 V 5 kHz V L = I × X L = 640 V V C = I × X C = 640 V 32 × 20 V = 640 V V L = I × X L = 155 V V C = I × X C = 155 V 7.8 × 20 V = 155 V R = 4  L 20 V 5 kHz 5 A Q =  F 1 mH QV S = V X

25-2: Series Resonance Frequency Response 20 V f 4 Ω 1 μF 1 mH Frequency in kHz Current in A = 1 2 π 1× 10 −3 × 1× 10 −6 = 5.03 kHz

25-3: Parallel Resonance  When L and C are in parallel and X L equals X C, the reactive branch currents are equal and opposite at resonance.  Then they cancel each other to produce minimum current in the main line.  Since the line current is minimum, the impedance is maximum.

25-3: Parallel Resonance where: f r = resonant frequency in Hz L = inductance in henrys C = capacitance in farads L C Parallel Resonant Circuit LC f r π 2 1 = [Ideal; no resistance] Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

25-3: Parallel Resonance  Fig. 25-6

25-3: Parallel Resonance 20 V R = 1 k  C = 1  F L = 1 mH Frequency Response Frequency in kHz I T in A InductiveCapacitive Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

25-4: Resonant Frequency  The formula for the resonant frequency is derived from X L = X C.  For any series or parallel LC circuit, the f r equal to is the resonant frequency that makes the inductive and capacitive reactances equal.

25-5: Q Magnification Factor of Resonant Circuit  The quality, or figure of merit, of the resonant circuit, in sharpness of resonance, is indicated by the factor Q.  The higher the ratio of the reactance at resonance to the series resistance, the higher the Q and the sharper the resonance effect.  The Q of the resonant circuit can be considered a magnification factor that determines how much the voltage across L or C is increased by the resonant rise of current in a series circuit.

25-5: Q Magnification Factor of Resonant Circuit Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Q is often established by coil resistance == rSrS XLXL Q = 20 V 5.03 kHz C = 1  F L = 1 mH r S = 1 

25-5: Q Magnification Factor of Resonant Circuit 4  20 V 1  F 1 mH 4  20 V 0.25  F 4 mH Frequency in kHz Current in A Half-power point Q = 7.8Q = 32 Increasing the L/C Ratio Raises the Q Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

25-6: Bandwidth of Resonant Circuit  When we say that an LC circuit is resonant at one frequency, this is true for the maximum resonance effect.  Other frequencies close to f r also are effective.  The width of the resonant band of frequencies centered around f r is called the bandwidth of the tuned circuit.

25-6: Bandwidth of Resonant Circuit Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Fig :

25-7: Tuning Fig  Tuning means obtaining resonance at different frequencies by varying either L or C.  As illustrated in Fig , the variable capacitance C can be adjusted to tune the series LC circuit to resonance at any one of five different frequencies.

25-7: Tuning Fig  Fig illustrates a typical application of resonant circuits in tuning a receiver to the carrier frequency of a desired radio station.  The tuning is done by the air capacitor C, which can be varied from 360 pF to 40 pF.

25-8: Mistuning  When the frequency of the input voltage and the resonant frequency of a series LC circuit are not the same, the mistuned circuit has very little output compared with the Q rise in voltage at resonance.  Similarly, when a parallel circuit is mistuned, it does not have a high value of impedance  The net reactance off-resonance makes the LC circuit either inductive or capacitive.

25-9: Analysis of Parallel Resonant Circuits Fig  Parallel resonance is more complex than series resonance because the reactive branch currents are not exactly equal when X L equals X C.  The coil has its series resistance r s in the X L branch, whereas the capacitor has only X C in its branch.  For high-Q circuits, we consider r s negligible.

25-9: Analysis of Parallel Resonant Circuits  In low-Q circuits, the inductive branch must be analyzed as a complex impedance with X L and r s in series.  This impedance is in parallel with X C, as shown in Fig  The total impedance Z EQ can then be calculated by using complex numbers. Fig

25-10: Damping of Parallel Resonant Circuits Fig  In Fig (a), the shunt R P across L and C is a damping resistance because it lowers the Q of the tuned circuit.  The R P may represent the resistance of the external source driving the parallel resonant circuit, or R p can be an actual resistor.  Using the parallel R P to reduce Q is better than increasing r s.

25-11: Choosing L and C for a Resonant Circuit  A known value for either L or C is needed to calculate the other.  In some cases, particularly at very high frequencies, C must be the minimum possible value.  At medium frequencies, we can choose L for the general case when an X L of 1000 Ω is desirable and can be obtained.  For resonance at 159 kHz with a 1-mH L, the required C is μF.  This value of C can be calculated for an X C of 1000 Ω, equal to X L at the f r of 159 kHz.