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Lesson 11

Lesson 11. Lesson 11. AC Circuits. AC Ciruits Power Maximum and Instantaneous voltage drops and current Phasor Diagrams Phase angle RLC Circuits Resonance frequency High and Low pass filters Step up and Step down Transformers. AC Generator. AC Generator. AC emf source.

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Lesson 11

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  1. Lesson 11 Lesson 11 AC Circuits • AC Ciruits • Power • Maximum and Instantaneous voltage drops and current • Phasor Diagrams • Phase angle • RLC Circuits • Resonance frequency • High and Low pass filters • Step up and Step down Transformers

  2. AC Generator AC Generator

  3. AC emf source

  4. rms current Effective ( Integrated ) values of I and V ( ) ( ) = i t I sin t w max p 2 w = p = 2 f ; T is the period of oscillation T ( ) ( ) = Instantaneous power i t v t ( ) ( ) 2 = = 2 2 w Heat dissipated = Power used in load i t R I R sin t m Average power over one cycle ò ò T T 2 1 1 I R ( ) ( ) = 2 2 w = 2 2 w = m P I R sin t dt I R sin t dt ave m m T T 2 0 0 Define I = 2 Þ = m P I R I ave rms rms 2

  5. Alternating Current Circuits

  6. ac-R circuit Veff and v(t) Ieff and i(t) t)

  7. Phasor diagram for R Phasor Diagram iR(t) IRmsin(t)= iR(t) t Current through Load Resistance

  8. Phasor diagram for R cont. Phasor Diagram vR(t) vR(t) VRmsin(t)= t PD across Load Resistance

  9. Instantaneous current and voltage

  10. ac-C circuit vC(t) iC(t) t) iC(t) vC(t) Current in Circuit and PD across Capacitor t

  11. ac-L circuit vL(t) iL(t) t) vL(t) t Current in Circuit and PD across Inductor iL(t)

  12. Summary The phase angle between the current and the voltage: In the resistor is 0 rad In the capacitor is - rad ( Current Ahead) In the inductor is +rad (Current Behind)

  13. series ac-RLC circuit Series RLC circuit

  14. Instantaneous current Current through all elements is the same Thus the instantaneous PD’s must be out of phase

  15. Picture Total Potential Drop across R, L & C.

  16. Phasor Diagram for RLC circuit I  vL(t) vR(t) vC(t)

  17. Instantaneous PD Why

  18. Phasor Diagram for RLC circuit II  vL(t) vRLC(t) vR(t) vC(t)

  19. Instantaneous PD as projection onto y-axis  vRLC(t) v(t2) v(t1)

  20. Phase Angle  vRLC(t)  - V V ( ) f = Lm Cm tan V Rm - - I X I X X X = = m L m C L C I R R m Phase Angle æ ö - X X - 1 f = L C ç ÷ tan è ø R

  21. series ac-RLC graph

  22. Impedance The magnitude of the Total Potential Phasor is ( ) 2 = 2 + - V V V V m R Lm Cm ( ) ( ) 2 2 = 2 2 + - = 2 + - I R I X I X I R X X m m L m C m L C = I Z m ( ) 2 = 2 + - Impedance : Z R X X L C

  23. Table of definitions

  24. Impedance and reactance

  25. Generalized Ohm's Law. ( ) 2 = 2 + - Impedance Z R X X L C

  26. Phase Angle between total PD across circuit and the current

  27. Power Factor Power is only used in AC circuit in load resistance ( ) ( ) 2 = P t i t R (energy is not used in inductor or capacitor) ( ) = 2 2 w f t I sin - R m (current is always in phase with PD across total resistance) 2 I 2 = P = I R R m ave rms 2 ß e æ ö R e e ç ÷ ( ) = = f rms I R I I cos ç ÷ rms rms rms rms rms Z Z è ø ( ) f = cos Power Factor

  28. Angular frequency dependence Power and current depend on angular frequency of circuit

  29. e Max I ; Min Z V ( ) w = = m m I ( ) ( ) w w m Z Z ( ) ( ) 2 w = + - 2 Z R X X L C æ ö 2 æ ö w - 2 2 1 LC 1 = 2 + w - = 2 + ç ÷ ç ÷ R L R è ø w w è ø C C ( ) w w - = 2 Z is a minimum when LC 1 0 which occurs when 1 w = w = 0 LC Û = X X L C

  30. Power as a function of  2 V R 1 1 ( ) ( ) w = w = 2 m P I R ( ) ave m 2 w 2 2 Z 2 2 1 V R 1 V R = = m m æ ö 2 2 L 2 1 2 ( ) 2 + w - + w - w 2 2 2 2 ç ÷ R L R è ø w 2 0 w C w 2 2 1 V R = m ( ) 2 2 2 w 2 + 2 w 2 - w 2 R L 0

  31. Resonance Circuit uses most power / current when it is in RESONANCE with applied frequency

  32. Imax and Pave versus  Pave Im  

  33. Quality of circuit • Width of Power curve is a measure of the QUALITY of the circuit • Small width - High Quality • Sharpness of response to external frequency

  34. RC Filters I RC Filters Low Pass Filter Vout Vin

  35. Low Pass Filter

  36. RC Filters II RC Filters High Pass Filter Vout Vin

  37. High Pass Filter

  38. Transformers I Step up and Step down Transformers

  39. Transformers II F d = - B V N 1 1 dt F d = - B V N 2 2 dt Fluxes are the same N = V 2 V 2 1 N 1

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