Resonance In AC Circuits

1. Resonance In ACCircuits

2. 3.1 Introduction An example of resonance in the form of mechanical : oscillation Potential energy change to kinetic energy than kinetic energy will change back to potential energy. If there is no lost of energy cause by friction potential energy is equal to kinetic energy. mgh = ½ mv ² It will oscillate for a long time. Ep=mgh M M h M v Ek= ½ mv ²

3. Resonance in electrical circuit i Potential energy stored in capacitor change to magnetic energy that stored in inductor. Then magnetic energy change back to potential energy stored in capacitor. If there is no lost of energy by resistor potential energy equal to magnetic energy ½ CV ² = ½ LI ² It will oscillate for a long time. i C L Ep= ½ CV ² Em= ½ LI ²

4. The frequency response of a circuit is maximum The impedances is purely resistive. Characteristic of resonance circuit The voltage Vs and current I are in phase Power factor equal to one Circuit reactance equal zero because capacitive and inductive are equal in magnitude

5. At frequency resonance, CV² = LI² We know V = I * XL or V = I * XC =I * ωL = I * 1/ωC = I * 2חfL = I / 2חfC Substitute into 1 C (2חfLI)² = LI² f² = L C (2חfL)² f = 1 2ח√LC ½ CV ² = ½ LI ² (1) I -jXC V jXL

6. Ideal case ( no resistance) i t Practical ( energy loss due to resistance) i t

7. Main objective we analysis resonance circuits to find five resonance parameters : • Resonance frequency, ωo • Angular frequency when value of current or voltage is maximum • b) Half power frequency, ω1 and ω2 • Frequency where current (or voltage) equal Imax/√2 (or Vmax/√2 ). • c) Quality factor, Q • Ratio of its resonant frequency to its bandwidth • d) Bandwidth, BW • Difference between half power frequency

8. 3.3 Series Resonance Circuits R j XL R By KVL : V = VR + VL + VC = VR+ jVL – jVC At resonance XL = XC Hence V = VR + 0 = VR = IR * R VL VR V Vc - j XC Figure 1

9. Series Resonance Circuits R j XL |Z| VL VR XL V Vc - j XC Z = R + j XL - jXC R (XL-XC) = R + j (XL – XC) Figure 1 f where XC XL = 2πf L f0 1 XC = 2πf C

10. |V| |I| = |Z| |Z| |I| f f0

11. Resonance parameter for series circuit • Resonant frequency,ωo • The resonance condition is • ωoL = 1 / ωoC or ωo = 1 / √LC rad/s • since ωo = 2Пfo • fo = 1/ 2ח√LC Hz • b) Half power frequencies • At certain frequencies ω = ω1, ω2, the half power frequencies are obtain by setting Z = √2R • √R² + (ωL – 1/ ωC)² = √ 2R

12. Solving for ω, we obtain • ω1 = - R/2L + √(R/2L)² + 1/LC rad/s • ω2 = R/2L + √(R/2L)² + 1/LC rad/s • Or in term of resonant parameter, • ω1 = ωo [ - 1/ 2Q + √(1/ 2Q)² + 1 ] rad/s • ω2 = ωo [ 1/ 2Q + √ (1/ 2Q)² + 1 ] rad/s • Quality factor, Q • Ratio of its resonant frequency to its bandwidth.

13. or Q = VC V = [ I ] x XC [ I ] x R = 1 ; Q = XC ωC R R = 1 2ח fr CR Q = VL VS = [ I ] x XL [ I ] x R = ωL ; Q = XL R R = 2 חfr L R

14. Bandwidth, BW • BW = ω2 – ω1 • = ωo [√ 1+ (1/ 2Q)² + 1/ 2Q ] - ωo [√ 1+ (1/ 2Q)² - 1/ 2Q ] • = ωo [ 1/ 2Q + 1/2Q ] • = ωo [2/ 2Q] • = ωo / Q • Q = ωoL /R = 1/ ωoCR • thus, • BW = R / L = ωo / Q • or, BW = ωo²CR

15. + v - i C L R ωo Ideal Parallel RLC circuit 3.4 Parallel Resonance Circuits • Resonance can be divided into 2: • Ideal parallel circuit • Practical parallel circuit • At least 3 important information that is needed to analyze to get the resonances parameter: • In resonance frequency, ωo the imaginary parts of admittance,Y must be equal to zero. • When in lower cut-off frequency, ω1 and in higher cut-off frequency, ω2 the magnitude of admittance,Y must be equal to √2/R.

16. -jXc jXL R YT YT = Ideal Parallel RLC circuit Resonance parameter for ideal RLC parallel circuit Whereas G(ω) is the real part called the conductance and B(ω) is the imaginary parts called the susceptance.

17. Resonant frequency,ωo Angular resonance frequency is when B(ω)=0. • Lower cut-off angular frequency, ω1 Produced when the imaginary parts = (-1/R)

18. c) Higher cut-off angular frequency, ω2. Produced when the imaginary parts = (1/R) d) Quality Factor, Q e) Bandwidth, BW

19. Duality Concept Parallel circuit Series circuit + v - R j XL i C L R VL VR V Vc - j XC Ideal Parallel RLC circuit Figure 1 Y = Y1 + Y2 + Y3 Z = Z1 + Z2 + Z3 Y= Z = R + j XL - jXC Y Z = R + j (ωL – )

20. Duality Concept Parallel circuit Series circuit + v - R j XL i C L R VL VR V Vc - j XC Ideal Parallel RLC circuit Figure 1 Z = R + j (ωL – ) Y R L C C L

21. Duality Concept Parallel circuit Series circuit + v - R j XL i C L R VL VR V Vc - j XC Ideal Parallel RLC circuit Figure 1 ω1 = - R/2L + √((R/2L)² + 1/LC) rad/s ω2 = R/2L + √((R/2L)² + 1/LC) rad/s ω1 = - 1/2RC + √((1/2RC)² + 1/LC) rad/s ω2 = 1/2RC + √((1/2RC)² + 1/LC) rad/s

22. Resonance parameter for practical RLC parallel circuit I I1 IC + V - R1 C i L IC Practical Parallel RLC circuit |I1|cosθ Z1 = R1 + jXL = |Z1|/θ |I1|sinθ θ I1

23. Resonance parameter for practical RLC parallel circuit I I1 IC + V - R1 C i L IC Practical Parallel RLC circuit |I1|cosθ Resonance occur when |I1|sinθ = IC |I1|sinθ θ I1

24. Resonance occur when |I1|sinθ = IC |I1|sinθ = IC L R2 + (2πfrL)2 = |V| XL |V| C x = |Z1| |Z1| XC L - R2 (2πfrL)2 = C 1 XL = |Z1|2 XC 2πfrL = 2πfrC R2 + (2πfrL)2

25. Q factor IC |I1|sinθ Q = current magnification = = I |I1|cosθ tanθ = XL = R 2πfrL IC = R |I1|cosθ |I1|sinθ θ I1

26. i Ideal Parallel RLC circuit Resonance parameter for practical RLC parallel circuit + V - R1 C L Second approach to analyze this circuit is by changing the series RL to parallel RL circuit. The purpose of this transformation is to make it much more easier to get the resonance parameter.

27. Rl L RL in series Rl jXp RL in parallel

28. and and By matching equation ZT and YT above, we can get: Or By defining the quality factor, Rp and Xp can be write as:

29. Resonance parameter • Angular resonance frequency, ωo • Lower cut-off angular frequency, ω1 • Produced when the imaginary parts = (1/R)

30. Produced when the imaginary parts = (1/R) • Higher cut-off angular frequency, ω2. • Quality Factor, Q • Bandwidth, BW