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 emf source

rms current Effective(Integrated) values of I and V it   I max sin  t   2  f  2  T ; T is the period of oscillation Instantaneous power  it  vt  Heat dissipated= Power used in load  it  2 R  I m 2 R sin 2  t  Average power over one cycle P ave  1 T I m 2 R sin 2  t  0 T  dt  I m 2 R 1 T sin 2  t  0 T  dt  I m 2 R 2 Define P ave  I rms 2 R  I  I m 2

Alternating Alternating Current Circuits

 t) V eff and v(t) I eff and i(t) ac-R circuit

Phasor diagram for R i R (t) tt I Rm sin(  t)= Phasor Diagram Current through Load Resistance

Phasor diagram for R cont. v R (t) tt V Rm sin(  t)= Phasor Diagram PD across Load Resistance

Instantaneous current and voltage

ac-C circuit  t) v C (t) i C (t) v C (t) tt i C (t) Current in Circuit and PD across Capacitor

ac-L circuit v L (t) tt i L (t)  t) v L (t) i L (t) Current in Circuit and PD across Inductor

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) Summary

Series RLC circuit Series RLC circuit series ac-RLC circuit

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

Picture Total Potential Drop across R, L & C.

Phasor Diagram for RLC circuit I v L (t) v R (t) v C (t) 

Instantaneous PD

Phasor Diagram for RLC circuit II v L (t) v R (t) v C (t) v RLC (t) 

Instantaneous PD as projection onto y-axis v RLC (t)  v(t 1 ) v(t 2 )

Phase Angle  Phase Angle v RLC (t)   tan    V Lm  V Cm V Rm  I m X L  I m X C I m R  X L  X C R  tan  1 X L  X C R      

series ac-RLC graph

Impedance The magnitude of the Total Potential Phasor is V m  V R 2  V Lm  V Cm  2  I m 2 R 2  I m X L  I m X C  2  I m R 2  X L  X C  2  I m Z Impedance: Z  R 2  X L  X C  2

Table of definitions

Impedance and reactance

Generalized Ohm's Law. ImpedanceZ  R 2  X L  X C  2

Phase Angle between total PD across circuit and the current

Power Factor Power is only used in AC circuit in load resistance Pt   it  2 R (energy is not used in inductor or capacitor)  I m 2 sin 2  t -   R (current is always in phase with PD across total resistance) P ave =I rms 2 R  I m 2 2 R   Z         I R   I R Z   I cos      Power Factor

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

Max I ; Min Z I m    V m Z     m Z   Z     R 2  X L  X C  2  R 2  L  1  C       2  R 2   2 LC  1  C       2 Z   is a minimum when  2 LC  1  0 which occurs when  0  1 LC  X L  X C

P ave    1 2 I m 2   R  1 2 V m 2 R Z   2  1 2 V m 2 R R 2  L  1  C       2  1 2 V m 2 R R 2  L 2  2  2  0 2  2  1 2 V m 2 R  2 R 2  2  L 2  2  0 2  2 Power as a function of 

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

I max and P ave versus  ImIm P ave 

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

RC Filters I RC Filters V out V in Low Pass Filter

RC Filters V out V in RC Filters II High Pass Filter

Transformers I Step up and Step down Transformers

Transformers II V 1  N 1 d  B dt V 2  N 2 d  B Fluxes are the same V 2  N 2 N 1 V 1