Admin: Assignment 6 is posted. Due Monday. Collect your exam at the end of class Erratum: US mains is 60Hz (I said 50Hz last lecture!)

An example: j

An example: j θ = tan -1 (2/1) = 63.4° A=sqrt( ) = 2.24 Phasor form = ° = 2.24 e j63.4° = 2.24(cos[63.4°] + sin[63.4°]) Real part = 1 = A cosθ Imaginary part = 2 = A sinθ * Re Im

When adding or subtracting complex numbers, the rectangular form is simplest: c 1 = a 1 + jb 1 c 2 = a 2 + jb 2 c 1 + c 2 = (a 1 + a 2 ) + j(b 1 + b 2 ) When multiplying or dividing, the polar or phasor forms are simplest: θ=tan -1 (b/a) Example: (1 + 2 j) + (-1 + 4j) = 0 + 6j Example: (1 + 2 j) 2 = ° × ° = ° =5cos(126.8°) + j5sin(126.8°)= j

i-V relationships in AC circuits: Resistors Source v s (t)=Asinωt v R (t)= v s (t)=Asinωt v R (t) and i R (t) are in phase Complex representation: v S (t)=Asinωt=Acos(ωt-90)=real part of [V S (j ω) ] where V S (j ω) is the complex number A[cos(ωt-90)+jsin(ωt-90 )]=Ae j (ωt-90) Phasor representation: V S (jω) =A  ( ωt -90) I S (jω)=(A/R)  ( ωt -90) Impedance=complex number of Resistance Z=V S (jω)/I S (jω)=R Generalized Ohm's Law: V S (jω)=ZI S (jω) Play animation Note – no phase dependence for a resistor

Capacitors in AC circuits No charge flows through a capacitor A Capacitor in a DC circuit acts like a break (an open circuit) But in AC circuits charge build-up and discharge mimics a current. V -V

Capacitors in AC circuits Capacitive Load Voltage and current not in phase: Current leads voltage by 90 degrees (Physical - current must conduct charge to capacitor plates in order to raise the voltage) Impedance of Capacitor decreases with increasing frequency "capacitive reactance” X C =1/ωC Play animation "capacitive impedance”

Copyright © 2009 Pearson Education, Inc AC Circuits with AC Source Example 30-10: Capacitor reactance. What is the rms current in the circuit shown if C = 1.0 μF and V rms = 120 V? Calculate (a) for f = 60 Hz and then (b) for f = 6.0 x 10 5 Hz. X C =1/ωC Note: we can simply use reactance (X C ) if dealing with phase independent quantities like rms. But the answer is not valid at any given instant in time: for that we need the impedance (Z C )