1

Physics 202, Lecture 19 Today’s Topics

  AC Circuits with AC Source   AC Power Source   Phasor   Resistors, Capacitors and Inductors in AC Circuit

  RLC Series In AC Circuit   Impedance

  ΔV = ΔVmax Sin(ωt+φ0) = ΔVmax Sin(ωt)

AC Power Source

Initial phase at t=0 (usually set φ0=0)

T=2π/ω

ω: angular frequency ω=2πf

t=0

AC Circuit: Series RLC  Find out current i and voltage difference ΔVR, ΔVL, ΔVC.

Notes: •  Kirchhoff’s rules still apply ! •  A technique called phasor analysis is convenient.

i

ΔVmax Sin(ωt)

Phasor   A sinusoidal function x= Asinφ can be represented

graphically as a phasor vector with length A and angle φ (w.r.t. to horizontal)

φ

A

Asi

nφ

2

A Phasor is Like a Graph   One graphical representation is using rectangular coordinates.

  The voltage is on the vertical axis.   Time is on the horizontal axis.

 The phasor is drawn similar to polar coordinate.   The radial coordinate represents the amplitude of the

voltage.   The angular coordinate is the phase angle.   The vertical axis coordinate of the tip of the phasor

represents the instantaneous value of the voltage.   The horizontal coordinate does not represent anything.

 Alternating currents can also be represented by phasors.

(A). Resistors in an AC Circuit  ΔV(t) – i(t) R = 0 at any time

i

iR=ΔV/R=Imax sinωt, Imax=ΔVmax/R

 The current through an resistor is in phase with the voltage across it Phasor view

Function view

(B). Inductors in an AC Circuit   ΔV – L di/dt = 0

iL=Imax sin(ωt-π/2) Imax=ΔVmax/ ωL =ΔVmax/XL, XL= ωL inductive reactance  The current through an indictor is 90o behind the voltage across it.

Phasor view

Function view i

(C). Capacitors in an AC Circuit

  ΔV - q/C=0, dq/dt =i

i

iL=Imax sin(ωt+π/2) Imax=ΔVmax/ [1/(ωC) ]=ΔVmax/XC, XC= 1/(ωC) capacitive reactance  The current through a capacitor is 90o ahead of the voltage across it. Phasor view

Function view

3

Summary of Phasor Relationship

IR and ΔVR in phase IL 90o behind ΔVL ΔVL 90o ahead of IL

IC 90o ahead of ΔVC ΔVC 90o behind IC

|IR|=|ΔVR|/R |IL|=|ΔVL|/XL |IC|=|ΔVC|/XC

RLC Series In AC Circuit   The current at all point in a series circuit has the same amplitude and phase (set it be i=Imaxsinωt)  ΔvR= ImaxR sin(ωt + 0) ΔvL= ImaxXLsin(ωt + π/2)   ΔvC= ImaxXCsin(ωt - π/2)

Voltage across RLC: ΔvRLC = ΔvR + ΔvL + ΔvR = ImaxRsin(ωt) + ImaxXLsin(ωt + π/2) + ImaxXCsin(ωt - π/2) =ΔVmaxsin(ωt+φ)

i

how to get them?

Phasor Technique (again)   The phasor of ΔvRLC = vector sum of phasors for ΔvR , ΔvL , ΔvC.

Note: XL= ωL, XC=1/(ωC)

Current And Voltages in a Series RLC Circuit

ΔVmax Sin(ωt +φ)

i= Imax Sin(ωt)

ΔvR=(ΔVR)max Sin(ωt) = R i(t) ΔvL=(ΔVL)max Sin(ωt + π/2) ΔvC=(ΔVC)max Sin(ωt - π/2)

Can the voltage amplitudes across each components , (ΔVR)max , (ΔVL)max , (ΔVC)max larger than the overall voltage amplitude Δvmax ? (each one of them)

4

Impedance  For general circuit configuration:   ΔV=ΔVmaxsin(wt+φ) , ΔVmax=Imax|Z|   Z: is called Impedance.

  e.g. RLC circuit :

  In general impedance is a complex number. Z=Zeiφ. It can be shown that impedance in series and parallel

circuits follows the same rule as resistors. Z=Z1+Z2+Z3+… (in series) 1/Z = 1/Z1+ 1/Z2+1/Z3+… (in parallel) (All impedances here are complex numbers)

ΔV

i

Z

Summary of Impedances and Phases

Comparison Between Impedance and Resistance

Resistance Impedance Symbol R Z

Application Circuits with only R Circuits with R, L, C Value Type Real Complex: Z=|Z|eiφ

I - ΔV Relationship ΔV=IR ΔV=IZ, ΔVmax=Imax|Z| In Series: R=R1+R2+R3+… Z=Z1+Z2+Z3+…

In Papallel: 1/R=1/R1+1/R2+1/R3+… 1/Z=1/Z1+1/Z2+1/Z3+…