1 SEE 1023 Circuit Theory Parallel RLC Resonant Circuit

2 Excitation (Input) Response (Output) Parallel RLC Circuit Constant input current: I s Variable Source angular frequency:  Main response: Voltage Other responses: Power, Admittance, susceptance, etc. The parallel Resonant is the dual of the series resonant.

3 Parallel RLC Resonant Circuit SeriesParallel Voltage, VCurrent, I Impedance, ZAdmittance, Y Reactance, XSusceptance, B Resistance, RConductance, G Short-circuitOpen-circuit Capacitor, CInductor, L Capacitor, C Duality between series and parallel

4 Parallel RLC Resonant Circuit RL ISIS   varied) +V-+V- IRIR ILIL ICIC

5 (Variation of the voltage with frequency) Frequency Response of The Voltage At Resonance, the voltage is maximum

6 (Variation of the Admittance with frequency) Frequency Response of The Admittance At Resonance, the admittance is minimum

7 (Variation of the susceptance with frequency) Frequency Response of The Susceptance At Resonance, the susceptance is zero

8 (Variation of the power with frequency) Frequency Response of The Power At Resonance, the power is maximum

9 Three main points on the power plot. Frequency Response of The Power (P o,  o ) (0.5P o,  1 ) (0.5P o,  2 ) At resonance, the power is maximum

10 Three main points on the power plot: Analysis of Parallel RLC resonant circuit  o : angular resonant frequency  1 and  2 : half-power angular frequencies  1 : lower half-power angular frequency  2 : upper half-power angular frequency Point-0: Point-1: Point-2: (P o,  o ) (0.5P o,  1 ) (0.5P o,  2 )

11 The Current Phasor Diagram at  o V ICIC ILIL at  o The circuit is purely resistive. I R = I s For R: I R is in phase with V For L:I L lags V by 90  For C:I C leads V by 90  For parallel circuit, use V as the reference because V is common For all three elements.

12 The Current Phasor Diagram at  2 I L +I C +I R =I S IRIR ICIC I L +I C at  2 V ILIL The circuit is predominantly capacitive. For R: I R is in phase with V For L:I L lags V by 90  For C:I C leads V by 90  For parallel circuit, use V as the reference because V is common For all three elements.

13 The Current Phasor Diagram at  1 I R +I L +I C = I S IRIR ICIC I L +I C at  1 V ILIL The circuit is predominantly inductive. For R: I R is in phase with V For L:I L lags V by 90  For C:I C leads V by 90  For parallel circuit, use V as the reference because V is common For all three elements.

14 Learning Sheet 3a Five Resonant Parameters: 1. Resonant Angular frequency, 2. Lower cut-off angular frequency, 4. Bandwidth of the resonant circuit, 3. Upper cut-off angular frequency, 5. Quality factor of the resonant circuit,

15 Learning Sheet 3a Five Resonant Parameters: 1. Resonant Angular frequency, 2. Lower cut-off angular frequency, 4. Bandwidth of the resonant circuit, 3. Upper cut-off angular frequency, 5. Quality factor of the resonant circuit,

16 Example: Parallel RLC Resonant Circuit I s = 0.1 Arms, R = 1k , L = 100 mH, C = 10  F RL ISIS   varied) +V-+V- IRIR ILIL ICIC

17 Find: (ii) The magnitude of the voltage at  o (iii) The real power P at  o (iv) The expression for i(t) at  o (v) The expression for i L (t) and i C (t) at  o (i)The admittance of the circuit at  o (vii) The voltage at  1 in polar form (viii) The real power P at  1 (ix) The expression for v(t) at  1 (x) The expression for i C (t), i L (t) and i C (t)+i L (t) at  1 (vi) The admittance of the circuit at  1 in polar form

18 (xii) The voltage at  2 in polar form (xiii) The real power P at  2 (xiv) The expression for v(t) at  2 (xv) The expressions for i L (t), i C (t) and i L (t)+i C (t) at  2 (xi) The admittance of the circuit at  2 in polar form (xvi) Draw the current phasor diagram at  o (xvii) Draw the current phasor diagram at  1 (xviii) Draw the current phasor diagram at  2 (ixx) Draw the waveforms of i C (t), i L (t) and i C (t)+i L (t) at  o (xx) Draw the waveforms of i C (t), i L (t) and i C (t)+i L (t) at  1 (xxi) Draw the waveforms of i L (t), i C (t) and i L (t)+i C (t) at  2

19 (xxii) The resonant frequency, f o (xxiii) The lower cut-off frequency, f 1 (xxiv) The upper cut-off frequency, f 2 (xxv) The bandwidth, BW in Hertz (xxvi) The Quality factor, Q