1. 1 Resonant Circuits SEE 1023 Circuit Theory Frequency Response

2. 2 Series RLC Circuit When  varies, the impedance of the circuit will vary. Then, the current and the real power will also vary. We would like to study the frequency response of these quantities. VsVs R L C I + V R - + V L - +VC-+VC-   (varied)

3. 3 Series RLC Circuit Impedance as a function of frequency: Current as a function of frequency: Power as a function of frequency: Reactance as a function of frequency:

4. 4 Series RLC Circuit Excitation (Input) Response (Output) Series RLC Circuit Constant input voltage: Vs Variable Source angular frequency:  Main response: current Other responses: Power, Impedance, reactance, etc.

5. 5 Series RLC Circuit in PSpice It is too hard to study the frequency response of these quantities manually. VsVs R L C I + V R - + V L - +VC-+VC-   (varied) It is too easy to study the frequency response of these quantities PSpicely. 0 1 2 3

6. 6 Series RLC Circuit in PSpice Series resonant Circuit Vs 1 0 AC 10V R1 1 2 10 L1 2 3 100mH C1 3 0 10uF.AC LIN 1001 100Hz 220Hz.Probe.end Start FREQ. End FREQ. Total PTS. To Display graph

7. 7 In the Probe windows Trace Expression M(V(1)/I(R1)) Magnitude of Z Response P(V(1)/I(R1))Phase of Z R(V(1)/I(R1))Real part of Z IMG(V(1)/I(R1)) Imaginary part of Z

8. 8 In the Probe windows Trace Expression M(I(R1)) Magnitude of I Response P(I(R1))Phase of I R(I(R1))Real part of I IMG(I(R1)) Imaginary part of I

9. 9 In the Probe windows Trace Expression V(1,2) Magnitude of V R Response V(2,3) Magnitude of V L V(3) Magnitude of V C I(R1)*I(R1)*10 Real power, P

10. 10 Run Pspice File Frequency Response of The Current

11. 11 (Variation of the current with frequency) Frequency Response of The Current At Resonance, the current is maximum

12. 12 Basic Questions What is the minimum value of  Z  ? What is the maximum value of  I  ? What is the maximum value of P?  Z  = R

13. 13 Basic Questions The magnitude of  I  ? When the power P = P o /2, what is The magnitude of  Z  ? The magnitude of  X  ? The angular frequency?  1 lower half power frequency  2 higher half power frequency at  1 at  2

14. 14 Resonant Condition By definition the resonant angular frequency,  o, for the RLC series circuit occurs at the peak of the current response. Under this condition:  The real power is maximum  The magnitude of impedance is minimum  The circuit is purely resistive  The imaginary part of the impedance is zero  The pf = 1  The current is in phase with the voltage source

15. 15 Lower half-power angular frequency,  1, condition By definition lower half-power angular frequency,  1, occurs when the power is P o /2 and the angular frequency is below the resonant angular frequency.  The real power is P o /2  The current is I o /  2  The magnitude of impedance is  2R  X = -R  The circuit is predominantly capacitive  The pf = cos(45  ) leading

16. 16 By definition lower half-power angular frequency,  2, occurs when the power is P o /2 and the angular frequency is above the resonant angular frequency.  The real power is P o /2  The current is I o /  2  The magnitude of impedance is  2R  X = +R  The circuit is predominantly inductive  The pf = cos(45  ) lagging Lower half-power angular frequency,  2, condition

17. 17 The Voltage Phasor Diagram at  o For R: I is in phase with V R For L:I lags V L by 90  For C:I leads V C by 90  For series circuit, use I as the reference. V R = V S I VLVL VCVC at  o The circuit is purely resistive.

18. 18 The Voltage Phasor Diagram at  1 For R: I is in phase with V R For L:I lags V L by 90  For C:I leads V C by 90  For series circuit, use I as a reference. VSVS I VLVL V L +V C at  1 VRVR VCVC The circuit is predominantly capacitive.

19. 19 The Voltage Phasor Diagram at  2 For R: I is in phase with V R For L:I lags V L by 90  For C:I leads V C by 90  For series circuit, use I as the reference. VSVS I VLVL V L +V C at  2 VRVR VCVC The circuit is predominantly inductive.

20. 20 Learning Sheet 3 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,

21. 21 Learning Sheet 3 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, Note: Lower cut-off angular frequency is also popularly known as lower half-power angular frequency. The same is true for the upper.

22. 22 Learning Sheet 3 We know that, Lower cut-off angular frequency, Upper cut-off angular frequency, Are the half-power frequencies symmetrical around  o ? Generally No. The resonant frequency is the geometric mean of the half-power frequencies. But, If Q  10, the half-power frequencies can be approximately considered as symmetrical around  o. Then and

23. 23 Example: Series RLC Resonant Circuit V s = 10 Vrms, R = 10 , L = 100 mH, C = 10  F VsVs R L C I + V R - + V L - +VC-+VC-   (varied)

24. 24 Find: (ii) The magnitude of the current at  o (iii) The real power P at  o (iv) The expression for i(t) at  o (v) The expression for v L (t) and v C (t) at  o (i)The impedance of the circuit at  o (vii) The current at  1 in polar form (viii) The real power P at  1 (ix) The expression for i(t) at  1 (x) The expression for v C (t), v L (t) and v C (t)+v L (t) at  1 (vi) The impedance of the circuit at  1 in polar form

25. 25 (xii) The current at  2 in polar form (xiii) The real power P at  2 (xiv) The expression for i(t) at  2 (xv) The expressions for v L (t), v C (t) and v L (t)+v C (t) at  2 (xi) The impedance of the circuit at  2 in polar form (xvi) Draw the voltage phasor diagram at  o (xvii) Draw the voltage phasor diagram at  1 (xviii) Draw the voltage phasor diagram at  2 (ixx) Draw the waveforms of v C (t), v L (t) and v C (t)+v L (t) at  o (xx) Draw the waveforms of v C (t), v L (t) and v C (t)+v L (t) at  1 (xxi) Draw the waveforms of v L (t), v C (t) and v L (t)+v C (t) at  2

26. 26 (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