1	Chapter 21 Resonance
2	Series Resonance Simple series resonant circuit Has an ac source, an inductor, a capacitor, and possibly a resistor Z_T = R + jX_L – jX_C = R + j(X_L – X_C) Resonance occurs when X_L = X_C At resonance, Z_T= R
3	Series Resonance Response curves for a series resonant circuit
4	Series Resonance
5	Series Resonance Since X_L = wL = 2pfL and X_C = 1/wC = 1/2pfC for resonance set X_L= X_C Solve for the series resonant frequency f_s
6	Series Resonance At resonance Impedance of a series resonant circuit is small and the current is large I = E/Z_T = E/R
7	Series Resonance At resonance V_R = IR V_L = IX_L V_C = IX_C
8	Series Resonance At resonance, average power is P = I²R Reactive powers dissipated by inductor and capacitor are I²X Reactive powers are equal and opposite at resonance
9	The Quality Factor,Q Q = reactive power/average power Q may be expressed in terms of inductor or capacitor For an inductor, Q_coil= X_L/R_coil
10	The Quality Factor,Q Q is often greater than 1 Voltages across inductors and capacitors can be larger than source voltage
11	The Quality Factor,Q This is true even though the sum of the two voltages algebraically is zero
12	Impedance of a Series Resonant Circuit Impedance of a series resonant circuit varies with frequency
13	Bandwidth Bandwidth of a circuit Difference between frequencies at which circuit delivers half of the maximum power Frequencies, f₁ and f₂ Half-power frequencies or the cutoff frequencies
14	Bandwidth A circuit with a narrow bandwidth High selectivity If the bandwidth is wide Low selectivity
15	Bandwidth Cutoff frequencies Found by evaluating frequencies at which the power dissipated by the circuit is half of the maximum power
16	Bandwidth
17	Bandwidth From BW = f₂- f₁ BW = R/L When expression is multiplied by w on top and bottom BW = w_s/Q (rad/sec) or BW = f_s/Q (Hz)
18	Series-to-Parallel Conversion For analysis of parallel resonant circuits Necessary to convert a series inductor and its resistance to a parallel equivalent circuit
19	Series-to-Parallel Conversion If Q of a circuit is greater than or equal to 10 Approximations may be made Resistance of parallel network is approximately Q² larger than resistance of series network R_P » Q²R_S X_LP » X_LS
20	Parallel Resonance Parallel resonant circuit Has X_C and equivalents of inductive reactance and its series resistor, X_LP and R_S At resonance X_C = X_LP
21	Parallel Resonance Two reactances cancel each other at resonance Cause an open circuit for that portion Z_T = R_P at resonance
22	Parallel Resonance Response curves for a parallel resonant circuit
23	Parallel Resonance From X_C = X_LP Resonant frequency is found to be
24	Parallel Resonance If (L/C) >> R Term under the radical is approximately equal to 1 If (L/C) ³ 100R Resonant frequency becomes
25	Parallel Resonance Because reactances cancel Voltage is V = IR Impedance is maximum at resonance Q = R/X_C If resistance of coil is the only resistance present Circuit Q will be that of the inductor
26	Parallel Resonance Circuit currents are
27	Parallel Resonance Magnitudes of currents through the inductor and capacitor May be much larger than the current source
28	Bandwidth Cutoff frequencies are
29	Bandwidth BW = w₂ - w₁= 1/RC If Q ³ 10 Selectivity curve becomes symmetrical around w_P
30	Bandwidth Equation of bandwidth becomes Same for both series and parallel circuits