Announcements

• Assignment 0 due now.– solutions posted later today

• Assignment 1 posted, – due Thursday Sept 22nd

• Question from last lecture: – Does VTH=INRTH

– Yes!

Lecture 5 Overview

• Alternating Current

• AC Components.

• AC circuit analysis

Alternating Current

• pure direct current = DC

• Direction of charge flow (current) always the same and constant.

• pulsating DC

• Direction of charge flow always the same but variable

• AC = Alternating Current

• Direction of Charge flow alternates

pure DC

pulsating DC

pulsating DC

AC

V

V

V

V

-V

Why use AC? The "War of the Currents"

• Late 1880's: Westinghouse backed AC, developed by Tesla, Edison backed DC (despite Tesla's advice). Edison killed an elephant (with AC) to prove his point.

• http://www.youtube.com/watch?v=RkBU3aYsf0Q

• Turning point when Westinghouse won the contract for the Chicago Worlds fair

• Westinghouse was right

• PL=I2RL: Lowest transmission loss uses High Voltages and Low Currents

• With DC, difficult to transform high voltage to more practical low voltage efficiently

• AC transformers are simple and extremely efficient - see later.

• Nowadays, distribute electricity at up to 765 kV

AC circuits: Sinusoidal waves

• Fundamental wave form

• Fourier Theorem: Can construct any other wave form (e.g. square wave) by adding sinusoids of different frequencies

• x(t)=Acos(ωt+)

• f=1/T (cycles/s)

• ω=2πf (rad/s)

• =2π(Δt/T) rad/s

• =360(Δt/T) deg/s

RMS quantities in AC circuits

• What's the best way to describe the strength of a varying AC signal?

• Average = 0; Peak=+/-

• Sometimes use peak-to-peak

• Usually use Root-mean-square (RMS)

• (DVM measures this)

rmsrmsavep

rmsp

rms VIPV

VI

I ,2

,2

i-V relationships in AC circuits: Resistors

Source vs(t)=AsinωtvR(t)= vs(t)=Asinωt

tR

A

R

tvti R

R sin)(

)(

vR(t) and iR(t) are in phase

Complex Number Review

2

2

Phasor representation

i-V relationships in AC circuits: Resistors

Source vs(t)=AsinωtvR(t)= vs(t)=Asinωt

tR

A

R

tvti R

R sin)(

)(

vR(t) and iR(t) are in phase

Complex representation: vS(t)=Asinωt=Acos(ωt-90)=real part of [VS(jω)]

where VS(jω)= A[cos(ωt-90)-jsin(ωt-90 )]=Aej (ωt-90)

Phasor representation: VS(jω) =A(ωt-90)

IS(jω)=(A/R) (ωt-90)

Impedance=complex number of Resistance Z=VS(jω)/IS(jω)=R

Generalized Ohm's Law: VS(jω)=ZIS(jω)

http://arapaho.nsuok.edu/%7Ebradfiel/p1215/fendt/phe/accircuit.htm

Capacitors

In Parallel:

V=V1=V2=V3

q=q1+q2+q3

321321 CCC

V

qqq

V

qCeq

What is a capacitor?Definition of Capacitance: C=q/VCapacitance measured in Farads (usually pico - micro)Energy stored in a Capacitor = ½CV2

(Energy is stored as an electric field)

i.e. like resistors in series

Capacitors

In Series:V=V1+V2+V3

q=q1=q2=q3

321

321 1111

CCCq

VVV

q

V

Ceq

No current flows through a capacitorIn AC circuits charge build-up/discharge mimics a current flow.A Capacitor in a DC circuit acts like a break (open circuit)

i.e. like resistors in parallel

Capacitors in AC circuitsCapacitive Load

CjCj

jj

C

j

Cj

j

tCAj

tAj

tCAdt

dqi

Cvq

tAv

C

C

CC

CC

C

1.

901)(

)(

)0()(

)90()(

)cos(

sin

C

C

I

VZ

I

V

C

• 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

http://arapaho.nsuok.edu/%7Ebradfiel/p1215/fendt/phe/accircuit.htm

jj )90sin()90cos(

"capacitive reactance"

InductorsWhat is an inductor?Definition of Inductance: vL(t)=-LdI/dtMeasured in Henrys (usually milli- micro-)Energy stored in an inductor: WL= ½ LiL

2(t)(Energy is stored as a magnetic field)

• Current through coil produces magnetic flux• Changing current results in changing magnetic flux• Changing magnetic flux induces a voltage (Faraday's Law v(t)=-dΦ/dt)

Inductors

Inductances in series add:

Inductances in parallel combine like resistors in parallel (almost never done because of magnetic coupling)

An inductor in a DC circuit behaves like a short (a wire).

Inductors in AC circuits

Lj

Lj

j

tL

Aj

tAj

tL

At

L

Ai

tL

Atdt

L

Ai

dt

diLtA

dt

diLv

tAv

L

LL

L

L

L

L

LL

S

Z

I

VZ

I

V

90)(

)(

)180()(

)90()(

)180cos()90sin(

cossin

sin

sin

L

L

• Voltage and current not in phase:• Current lags voltage by 90 degrees• Impedance of Inductor increases with increasing frequency

Inductive Load

http://arapaho.nsuok.edu/%7Ebradfiel/p1215/fendt/phe/accircuit.htm

jj )90sin()90cos(

(back emf )

from KVL

AC circuit analysis• Effective impedance: example• Procedure to solve a problem

– Identify the sinusoid and note the frequency– Convert the source(s) to complex/phasor form– Represent each circuit element by it's AC impedance– Solve the resulting phasor circuit using standard circuit solving

tools (KVL,KCL,Mesh etc.)– Convert the complex/phasor form answer to its time domain

equivalent

Example

0)()()(

)()()()(

221

211

jIZZZjIZ

jVjIZjIZZ

RLCC

SCCR

221

2

2

1

21

))((

)()(0

)(

)(CRLCCR

SRLC

RLCC

CCR

RLC

CS

ZZZZZZ

jVZZZ

ZZZZ

ZZZ

ZZZ

ZjV

jI

)(7505.01500

)(7.667.66

101500

116

jjLjZ

jjjCj

Z

L

C

4450)68375)(7.66100(

015)68375()(1

jj

jjI

687

7.8375

683tantan

22

11

baA

a

b

4450)68375)(7.66100(

015)68375()(1

jj

jjI

0157.83687015)68375( j

8.4785500

633005755044506835000456007500

4450)68375)(7.66100(

jjj

jj

Top:

Bottom:

Amps)63.01500cos(12.0)(

radians63.012.0

9.3512.08.4785500

0157.83687)(

1

1

tti

jI