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Chapter 1 Resonance 1.0.1 Introduction: Most of the transmission lines, electrical circuits and communication networks are made up of network elements like resistor R,inductor, L and Capacitor C. The impedance of the inductor and capacitor depends on the frequency of the applied sinusoidal voltage to the network. As we vary the frequency of the supply the network impedance is purely resistive in which the impedance of the inductor is equal to the impedance of the capacitor. The phenomenon in which the the complex circuit behaves like a pure resistive is called resonance. The frequency at which resonance takes place is called the frequency of resonance ωr (radians/sec) or fr . (Hz) Resonance may occur by varying frequency of the applied voltage to the complex network or by varying inductance, L or Capacitance, C, by keeping the frequency constant. Under resonance the following conditions occur in the circuit. • The impedance of the inductance is equal to the impedance Capacitance. • The phase of the current in the circuit is in phase with the applied voltage. • Maximum current will flow in the circuit. • The voltage across the capacitor or inductor is I × XC or I × XL where I is the current at resonance and XC or XL is the impedance of the circuit. • The total power is dissipated in the resistor and the absorbed average power is maximum. ———————————————————————————————————————- 1.1 Series Resonance At resonance the imaginary part is zero Consider a series circuit consists of resistor, inductor and capacitor as shown in Figure 1.1. XL − XC 1 ωr L − ωr C L C = 0 = 0 ωr L = + vi (t ) - i(t ) + vo (t ) R - ωr2 = ωr = Figure 1.1: Series resonance circuit 1 ωr C 1 LC 1 √ radians/sec LC The impedance of the circuit is fr = Z = R + j(XL − XC ) 1 1 √ Hz 2π LC

1.1. SERIES RESONANCE Chapter 1. Resonance The plot of the frequency response of series circuit is as shown in Figure 1.2. At resonant frequency ωr the current is maximum. I ω1 = Im Im 2 1 R , c=− L LC q s 4 R 2 ± + R 2 R 1 L LC =− ± + 2 2L 2L LC a = 1, b = −R L Frequency is always positive ω1 = − ω ω1 ωr ω2 R + 2L s Figure 1.2: Frequency response of series circuit Z 2 + 1 LC In terms of frequency f1 f1 = 1 R − + 2π 2L s R 2L 2 + 1 LC XL Impedance Z R 2L At frequency ω2 the circuit impedance XL > XC R 0 XL − XC 1 ω2 L − − ω2 C ω22 LC − 1 ω2 C 2 ω2 LC − Rω2 C − 1 R 1 ω22 − ω2 − L LC ω ωr -X C Figure 1.3: Frequency response of impedance of series circuit At resonance frequency fr Z = R and current is Im Im At half power frequencies f1 and f2 the current is √ 2 Z= √ 2R p Z = R + jXL − JXC = R2 + (XL − XC )2 p R2 + (XL − XC )2 = √ 2R R2 + (XL − XC )2 = 2R2 (XL − XC )2 = R2 XL − XC ω2 = R L = R = R = 0 = 0 1 R a = 1, b = − , c = − L LC q s R 2 4 ± + R R 2 1 L LC = ± + 2 2L 2L LC Frequency is always positive R ω2 = + 2L s R 2L 2 + 1 LC In terms of frequency f2 = R At frequency ω1 the circuit impedance XC > XL XC − XL 1 − ω1 L ω1 C 1 − ω12 LC ω1 C Rω1 C − 1 + ω12 LC R 1 ω12 + ω1 − L LC = R f2 = 1 R + 2π 2L s R 2L 2 + 1 LC = R = R = R = 0 = 0 Dr. Manjunatha P Professor Dept of E&CE, JNN College of Engineering, Shivamogga 2

1.1. SERIES RESONANCE Chapter 1. Resonance Relation between ωr , ω1 and ω2 ω1 × ω2 = = = R + 2L s R 2L 2 + 1 −R × + LC 2L Resonance by varying circuit inductance s R 2L 2 + 1 LC Consider a series RLC circuit as shown in Figure 1.5 is become resonant by varying inductance of the circuit. 1 LC + vi (t ) r ωr = 1 LC ωr Let L1 is the inductance at ω 1 = ω1 .ω2 LC √ = ω1 .ω2 p fr = f1 f2 XC − XL = R 1 − ωL1 = R ωC L1 = Relation between Bandwidth Quality factor Bandwidth is B = ω2 − ω1 = = = R i(t ) - Figure 1.5: Resonance by varying inductance ωr2 = B L C 1 ω2C − R ω + R ω Let L2 is the inductance at ω ω2 − ω1 s s 2 2 R R −R R 1 1 − + + + + 2L 2L LC 2L 2L LC R radians L XL − XC 1 ωL2 − ωC = R = R L2 = 1 R R . = Hz 2π L 2πL r 1 ωr = LC s B 2 B ω1 = − + + ωr2 2 2 s B B 2 ω2 = + + ωr2 2 2 B= 1 ω2C Resonance by varying circuit capacitance Consider a series RLC circuit as shown in Figure ?? is become resonant by varying capacitance of the circuit. L C + vi (t ) R i(t ) - Figure 1.6: Resonance by varying capacitance Let C1 is the capacitance at ω1 Quality factor Q The quality factor Q is defined as the ratio of resonant frequency to the bandwidth q r 1 ωr 1 L LC Q= = R = B R C L XC − XL = R ⇒ 1 ω1 C 1 = R + ω1 L C1 = I Im Im 2 1 − ω1 L = R ω1 C 1 ω12 L 1 + ω1 R Let C2 is the capacitance at ω2 Q1 Q2 Q1 > Q 2 XL − XC ω 1 = ω2 L − R ω2 C2 ω1 ωr ω2 Figure 1.4: Plot of frequency verses Q = R ⇒ ω2 L − C2 = Dr. Manjunatha P Professor Dept of E&CE, JNN College of Engineering, Shivamogga ω22 L 1 =R ω2 C2 1 − ω2 R 3

1.1. SERIES RESONANCE Chapter 1. Resonance Table 1.1: Important Formulae Parameter At resonance Resonance Formula Z = R, XL = XC Current Ir = E R 1 ωr = √LC fr = 2π√1LC q ω1 −R 1 R 2 ω1 = 2L + + LC f1 = 2π 2L q ω2 R R 2 1 ω2 = 2L + + LC f2 = 2π 2L q ω1 −B B 2 ω1 = 2 + + ωr2 f1 = 2π 2 q ω2 B B 2 ω2 = 2 + + ωr2 f2 = 2π 2 Half power frequency Half power frequency B = ω2 − ω1 = Bandwidth Quality factor R L Radians R 2πL Hz B = f2 − f1 = q ωr L 1 Q = R = R CL √ √ ωr = ω1 ω2 OR fr = f1 f2 across VLr = VCr = IXLr ωr ω1 ω2 Voltage capacitor/inductor Value of inductor at f1 , f2 Value of capacitor at f1 , f2 L1 = 1 ω2 C − Rω , L2 = 1 ω2 C + R ω C1 = 1 ω2 L − Rω , C2 = 1 ω2 L + R ω Selectivity: is property of circuit in which the circuit is allowed to select a band of frequencies between f1 and f2 . Dr. Manjunatha P Professor Dept of E&CE, JNN College of Engineering, Shivamogga 4

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