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Chapter 20 – Resonance Introductory Circuit Analysis Robert L. Boylestad

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20.1 – Introduction Knowing the fundamentals of a resonant (or tuned) circuit is very important to the operation of a wide variety of electrical and electronic systems in use today. The resonant circuit is a combination of R, L and C elements having a frequency response characteristic.

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Introduction The response is a maximum for the frequency f r. When the response is at or near the maximum, the circuit is said to be in a state of resonance. There are two types of resonant circuits – series and parallel. The frequency to the far left or right or resonance have very low voltage or current levels and, for all practical purposes, have little effect on the systems response.

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20.2 – Series Resonant Circuit A resonant circuit (series or parallel) must have an inductive and capacitive element. A resistive element will always be present due to the internal resistance of the source (R S ), the internal. resistance of the inductor (R I ), and any added resistance to control the shape of the response curve (R design ).

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Series Resonant Circuit Total impedance of the network at any frequency is Z T = R + j X L j X C = R + (X L – X C ) Resonance will occur when X L = X C Removing the reactive component from the total impedance equation, total impedance is simply Z T S = R The resonant frequency can be determined in terms of the inductance and capacitance by

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Series Resonant Circuit The input voltage and current are in phase at resonance. The current is the same through the capacitor and inductor, the voltage across each is equal in magnitude but 180° out of phase at resonance. And, since X L = X C, the magnitude of V L equals V C at resonance:

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Series Resonant Circuit The average power to the resistor at resonance is equal to I 2 R, and the reactive power to the capacitor and inductor are I 2 X C and I 2 X L, respectively. The power triangle at resonance shows that the total apparent power is equal to the average power dissipated by the resistor since Q L = Q C. The power factor of the circuit at resonance is:

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Series Resonant Circuit Even thought the total reactive power at any instant is equal to zero, energy is still being absorbed and released by the inductor and capacitor at resonance.

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20.3 – The Quality Factor (Q) The quality factor (Q) of a series resonant circuit is defined as the ratio of the reactive power of either the inductor or the capacitor to the average power of the resistor at resonance. The quality factor is also an indication of how much energy is placed in storage compared to that dissipated. The lower the level of dissipation for the same reactive power, the larger the Q S factor and the more concentrated and intense the region of resonance.

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The Quality Factor (Q) Substituting for an inductive reactance at resonance gives us If the resistance R is just the resistance of the coil (R l ), we can speak of the Q of the coil, where

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The Quality Factor (Q) The Q of a coil is usually provided by the manufacturer. As the frequency increases the effective resistance of the coil also increases, due primarily to the skin effect phenomena, causing the Q 1 to decrease. The capacitive effect between windings will increase, further reducing the Q 1 of the coil. Q 1 must be specified for a particular frequency or frequency range. Q S in terms of circuit parameters: For series circuits used in communication systems Q S is usually greater than 1.

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20.4 – Z T Versus Frequency From previous discussions we found that total impedance of a series R-L-C circuit at any frequency is determined by Z T = R + j X L j X C or Z T = R + j(X L – X C ) The total-impedance-versus-frequency curve for the series resonant circuit can be found by applying the impedance-versus- frequency curve for each element of the equation:

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Z T Versus Frequency The phase angle associated with the total impedance: For a series resonant circuit

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20.5 – Selectivity If we now plot the magnitude of the current I=E/Z T versus frequency for a fixed applied voltage E, we obtain a curve that is actually the inverse of the impedance-versus- frequency curve. Z T is not absolutely symmetrical about the resonant frequency, and the curve of I vs f is also non-symmetrical about resonance.

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Selectivity The frequencies corresponding to of the maximum current are called the band frequencies, cutoff frequencies, or half-power frequencies (ƒ 1, ƒ 2 ). Half-power frequencies are those frequencies at which the power delivered is one-half that delivered at resonant frequency. The range of frequencies between the two are referred to as bandwidth (abbreviated BW) of the resonant circuit. Since the resonant circuit is adjusted to select a band of frequencies it is called a selectivity curve.

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Selectivity The shape of the curve depends on each element of the series R-L-C circuit. If resistance is made smaller with a fixed inductance and capacitance, the bandwidth decreases and the selectivity increases. If the ratio L/C increases with fixed resistance, the bandwidth again decreases with an increase in selectivity.

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Selectivity In terms of Q S, if R is larger for the same X L, then Q S is less, as determined by the equation Q S = S L/R. A small Q S, therefore, is associated with a resonant curve having a large bandwidth and a small selectivity, while a large Q S indicates the opposite. For circuits where Q S 10, a widely accepted approximation is that the resonant frequency bisects the bandwidth and that the resonant curve is symmetrical about the resonant frequency.

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20.6 – V R, V L, and V C Plotting the magnitude (effective value) of the voltages V R, V L, and V C and the current I versus frequency for the series resonant circuit on the same set of axes The V R curve has the same shape as the I curve and the peak value equal to the magnitude of the input voltage E.

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V R, V L, and V C The V C curve builds up slowly at first from a value equal to the input voltage since the reactance of the capacitor is infinite (open circuit) at zero frequency and the reactance of the inductor is zero (short circuit) at this frequency. As the Frequency increases, 1/ C of the equation becomes smaller, but I increases at a rate faster than that at which 1/ C drops.

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V R, V L, and V C The curve for V L increases steadily from zero to the resonant frequency since both quantities L and I of the equation V L = IX L =(I )( L) increases over this frequency range. At resonance, I has reached its maximum value, but L is still rising. V L will reach its maximum value after resonance.

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V R, V L, and V C Review V C and V L are at their maximum values at or near resonance (depending on Q S ). At very low frequencies, V C is very close to the source voltage and V L is very close to zero volts, whereas at very high frequencies, V L approaches the source voltage and V C approaches zero volts. Both V R and I peak at the resonance frequency and have the same shape.

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20.8 – Parallel Resonance The parallel resonant circuit has the basic configuration at right. For the parallel resonant circuit impedance is relatively high at resonance, but in the practical world the internal resistance of the coil ( R l ) must be placed in series with the inductor as shown.

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Parallel Resonance Utility Power Factor, f P For unity power factor, reactive components must equal zero. 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 is a function of resistance.

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Parallel Resonance Maximum Impedance The frequency at which maximum impedance will occur is defined by f m and is slightly more than f p

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20.9 – Selectivity Curve for Parallel Resonant Circuits The parallel resonance circuit exhibits maximum impedance at resonance (f m ) unlike the series resonant circuit, which experiences minimum resistance levels at resonance. For parallel circuits, the resonance curve of interest is that of the V C across the capacitor. Since the voltage across parallel elements is the same:

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Selectivity Curve for Parallel Resonant Circuits The quality factor of the parallel resonant circuit continues to be determined by the ratio of the reactive power to the real power: The bandwidth is still related to the resonant frequency and the quality factor: The cutoff frequencies of f 1 and f 2 can be determined using the equivalent network and the utility power condition for resonance.

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Selectivity Curve for Parallel Resonant Circuits At low frequencies the capacitive reactance is high, and the inductive reactance is low. Hence, the total impedance at low frequencies will be inductive. At high frequencies, the network is capacitive. At resonance ( f p ), the network appears resistive. The phase plot is the inverse of that appearing for the series resonant circuit.

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20.10 – Effect of Q L 10 For the majority of parallel resonant circuits the quality factor of the coil Q I is sufficiently large to permit a number of approximations that simplify the analysis For Q L 10 Inductive Reactance, X L p Resonant Frequency, f P (Utility Power Factor)

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Effect of Q L 10 –Resonant Frequency, f m (Maximum V C ) –Parallel resistance R P –Total impedance Z T p

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Effect of Q L 10 –Quality Q P –Bandwidth BW –Inductor current I L and capacitor current I C

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Summary Table For the future, the analysis of a parallel resonant network might proceed as follows: Determine f s to obtain some idea of the resonant frequency. Calculate an approximate Q I using the f s from above, and compare it to the condition Q I 10. If the condition is satisfied, the approximate approach should be the chosen path unless a high degree of accuracy is required. If Q I is less than 10, the approximate approach can be applied, but it must be understood that the smaller the level of Q l, the less accurate the solution.

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THE END

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