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 1

AC Generator AC Generator 2

AC emf source 3

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

ac-R circuit Veff and v(t) Ieff and i(t) t) 5

Phasor diagram for R Phasor Diagram iR(t) IRmsin(t)= iR(t) t Load Resistance 6

Phasor diagram for R cont. vR(t) VRmsin(t)= vR(t) t Load Resistance 7

Instantaneous current and voltage 8

ac-C circuit vC(t) iC(t) t) iC(t) vC(t) t 9

Instantaneous current and voltage 10

ac-L circuit vL(t) iL(t) t) vL(t) t iL(t) 11

Instantaneous current and voltage 12

Phase angle between current Summary Phase angle between current and voltage resistor is 0 rad capacitor is - rad ( Current Ahead) inductor is + rad (Current Behind) 13

Inductive and Capacitive Reactance 14

Instantaneous current and voltage in R,L, C circuits 15

series ac-RLC circuit Series RLC circuit 16

Instantaneous current Current through all elements is the same Thus the instantaneous PD’s must be out of phase 17

Total Potential Drop across R, L & C. Picture Total Potential Drop across R, L & C. 18

Instantaneous current and voltage in RLC circuit 19

Phasor Diagram for RLC circuit I  vL(t) vR(t) vC(t) 20

Instantaneous PD 21

Phasor Diagram for RLC circuit II vL(t) vR(t) vC(t) vRLC(t)  22

Instantaneous PD as projection onto y-axis  v(t2) vRLC(t) v(t1) 23

Phase Angle  vRLC(t)  Phase Angle ( ) V - V tan f = V - - I X I Lm Cm V Rm - - I X I X X X = m L m C = L C I R R Phase Angle m æ ö X - X f = tan - 1 ç L C ÷ è ø R 24

series ac-RLC graph 25

Impedance 26

Table of definitions 46

Impedance and reactance 28

Generalized Ohm's Law. 29

Power Factor

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

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

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

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

Imax and Pave versus  Im Pave   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

RC Filters RC Filters I Vout Vin

Low Pass Filter

RC Filters RC Filters II Vout Vin

High Pass Filter

Step up and Step down Transformers Transformers I 44

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