INC 112 Basic Circuit Analysis

Week 7

Introduction to

AC Current

Meaning of AC Current

AC = Alternating current

means electric current that change up and down

When we refer to AC current, another variable, time (t) must be in our consideration.

Alternating Current (AC)

Electricity which has its voltage or current change with time.

Example: We measure voltage difference between 2 points

Time 1pm 2pm 3pm 4pm 5pm 6pm

DC: 5V 5V 5V 5V 5V 5V

AC: 5V 3V 2V -3V -1V 2V

Signals

Signal is an amount of something at different time, e.g. electric signal.

Signals are mentioned is form of1.Graph2.Equation

Graph Voltage (or current) versus time

V (volts)

t (sec)

v(t) = sin 2t

1st Form

2nd Form

V (volts)

t (sec)

DC voltage

v(t) = 5

Course requirement of the2nd half

Students must know voltage, current, power at any point in the given circuits at any time.

e.g.

What is the current at point A?What is the voltage between point B and C at 2pm?What is the current at point D at t=2ms?

Periodic Signals

Periodic signals are signal that repeat itself.

DefinitionSignal f(t) is a periodic signal is there is T such that

f(t+T) = f(t) , for all t

T is called the period, where

when f is the frequency of the signal

fT

1

Example:

v(t) = sin 2t

Period = π Frequency = 1/π

v(t+π) = sin 2(t+π) = sin (2t+2π) = sin 2t(unit: radian)

Note: sine wave signal has a form of sin ωt where ω is the angular velocity with unit radian/sec

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Sine waveSquare wave

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-5 -4 -3 -2 -1 0 1 2 3 4 5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fact:

Theorem: (continue in Fourier series, INC 212 Signals and Systems)

“Any periodic signal can be written in form of a summationof sine waves at different frequency (multiples of the frequency of the original signal)”

e.g. square wave 1 KHz can be decomposed into a sum of sine wavesof reqeuency 1 KHz, 2 KHz, 3 KHz, 4 KHz, 5 KHz, …

.....)2.14sin(5.0)2.03sin(1)5.02sin(3)3.0sin(8 tttt

Implication of Fourier Theorem

Sine wave is a basis shape of all waveform.

We will focus our study on sine wave.

Properties of Sine Wave

1. Frequency

2. Amplitude

3. Phase shift

These are 3 properties of sine waves.

Frequency

0 1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

sec

volts

fT

1

Period ≈ 6.28, Frequency = 0.1592 Hz

period

Amplitude

0 1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

sec

volts

Blue 1 voltsRed 0.8 volts

Phase Shift

0 1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

)1sin(8.0

)sin(

ty

ty

red

blue

Period=6.28

PhaseShift =

1

Red leads blue 57.3 degree (1 radian) 3.57360

28.6

1

Sine wave in function of time

Form: v(t) = Asin(ωt+φ)

AmplitudeFrequency (rad/sec)

Phase (radian)

e.g. v(t) = 3sin(8πt+π/4) volts

Amplitude3 volts Frequency

8π rad/sec or 4 Hz

Phaseπ/4 radian or 45 degree

Basic Components

• AC Voltage Source, AC Current Source

• Resistor (R)

• Inductor (L)

• Capacitor (C)

AC Voltage SourceAC Current Source

AC AC

+

-

AC +-

AC

Voltage Source Current Source

AC

+

-

10sin(2πt + π/4)

เชน

Amplitude = 10VFrequency = 1HzPhase shift = 45 degree

AC

+

-

10sin(2πt + π/4) What is the voltage at t =1 sec ?

volts

v

07.7)4/sin(10

)4/2sin(10

)4/)1(2sin(10)1(

Resistors

Same as DC circuits

Ohm’s Law is still usable

V = IR

R is constant, therefore V and I have the same shape.

AC

+

-

10sin(2πt + π/4) 2Ω

i(t)

Find i(t)

)4/2sin(5)(2

)4/2sin(10)(

2)()4/2sin(10

)()(

tti

tti

tit

Rtitv

Note: Only amplitude changes, frequency and phase still remain the same.

Power in AC circuits

R

vRiP

22

In AC circuits, voltage and current fluctuate. This makes powerat that time (instantaneous power) also fluctuate.

Therefore, the use of average power (P) is prefer.

Average power can be calculated by integrating instantaneous power within 1 period and divide it with the period.

)sin()( tAtv Assume v(t) in form Change variable of integration to θ

sinAv We get Then, find instantaneous power

R

A

R

vp

222 sin integrate from 0 to 2π

R

A

R

A

dA

dR

A

dR

AP

24

2sin

2

1

2

2

2cos1

2sin

2

sin2

1

22

0

2

2

0

22

0

22

2

0

22

Compare with power from DC voltage source

DC

AC

+

-

Asin(ωt+Ф) R

i(t)

AC

+

-A R

i(t)

R

AP

2

2

R

AP

2

Root Mean Square Value (RMS)

In DC circuitsR

VRIP

22

In AC, we define Vrms and Irms for convenient in calculating power

R

VRIP rms

rms

22 Note: Vrms and Irms are constant,

independent of time

For sine wave Asin(ωt+φ)2

_A

valuerms

V (volts)

t (sec)

311V

V peak (Vp) = 311 VV peak-to-peak (Vp-p) = 622VV rms = 220V

3 ways to tell voltage

0

Inductors

i(t)

+ v(t) -

Inductance has a unit of Henry (H)

Inductors have V-I relationship as follows

dt

tdiLtv

)()( This equation compares to

Ohm’s law for inductors.

AC

+

-

Asin(ωt)

i(t)

L Find i(t)

)2

(sin)cos(

cossin

sin1

)(1

)(

)()(

tL

At

L

A

t

L

Atdt

L

A

tdtAL

dttvL

ti

dt

tdiLtvfrom

AC

+

-

Asin(ωt) R

i(t)

AC

+

-

Asin(ωt)

i(t)

L

)2

(sin)(

t

L

Ati )(sin)( t

R

Ati

ωL is called impedance (equivalent resistance)

Phase shift -90

Phasor Diagram of an inductor

v

i

Power = (vi cosθ)/2 = 0

Phasor Diagram of a resistor

v

i

Power = (vi cosθ)/2 = vi/2

Note: No power consumed in inductorsi lags v

DC Characteristics

i(t)

L1V1Ω

i(t)

1V1Ω

When stable, L acts as an electric wire.

dt

tdiLtv

)()(

When i(t) is constant, v(t) = 0

Capacitors

i(t)

+ v(t) -

Capacitance has a unit of farad (f)

dt

tdvCti

)()(

Capacitors have V-I relationship as follows

This equation compares to Ohm’s law for capacitors.

AC

+

-

Asin(ωt)

i(t)

C Find i(t)

)2

sin(1

)(cos

)sin()()(

t

C

A

tCAdt

tAdC

dt

tdvCti

Impedance (equivalent resistance)

Phase shift +90

Phasor Diagram of a capacitor

v

i

Power = (vi cosθ)/2 = 0

Phasor Diagram of a resistor

v

i

Power = (vi cosθ)/2 = vi/2

Note: No power consumed in capacitorsi leads v

DC Characteristics

i(t)

1V1Ω

C

i(t)

1V1Ω

When stable, C acts as open circuit.

When v(t) is constant, i(t) = 0

dt

tdvCti

)()(

Combination of Inductors

L1 L2 L1+L2

L1

L2

L1L2/(L1+L2)

21 LLLtotal

21

111

LLLtotal

Combination of Capacitors

C1 C2 C1C2/(C1+C2)

C1

C2

C1+C2

21 CCCtotal

21

111

CCCtotal

Linearity

Inductors and capacitors are linear components

dt

tdvCti

)()(

dt

tdiLtv

)()(

If i(t) goes up 2 times, v(t) will also goes up 2 timesaccording to the above equations

Transient Responseand Forced response

Purpose of the second half

• Know voltage or current at any given time

• Know how L/C resist changes in current/voltage.

• Know the concept of transient and forced response

Characteristic of R, L, C

• Resistor resist current flow

• Inductor resists change of current

• Capacitor resists change of voltage

L and C have “dynamic”

I

1V1Ω

I

2V1Ω

I = 1A I = 2A

Voltage source change from 1V to 2V immediatelyDoes the current change immediately too?

Voltage

Current

time

time

1V

2V

1A

2A

AC voltage

I

L1V1Ω

I

L2V1Ω

I = 1A I = 2A

Voltage source change from 1V to 2V immediatelyDoes the current change immediately too?

Voltage

Current

time

time

1V

2V

1A

2A

Forced Response

Transient Response + Forced Response

AC voltage

Unit Step Input and Switches

Voltage

time0V

1V

This kind of source is frequently used in circuit analysis.

Step input = change suddenly from x volts to y voltsUnit-step input = change suddenly from 0 volts to 1 volt at t=0

AC

u(t)

This kind of input is normal because it come from on-off switches.

PSPICE Example

• All R circuit, change R value

• RL circuit, change L

• RC circuit, change C

I am holding a ball with a rope attached, what is the movement of the ball ifI move my hand to another point?

Movements

1. Oscillation

2. Forced position change

Pendulum Example

• Transient Response or Natural Response (e.g. oscillation, position change temporarily)

Fade over timeResist changes

• Forced Response (e.g. position change permanently)

Follows inputIndependent of time passed

Forced response Natural responseat different time

Mechanical systems are similar to electrical system

i(t)

LR

ACconnect

i(t)

StableChanging

Transient Analysis Phasor Analysis

Transient Response

• RL Circuit

• RC Circuit

• RLC Circuit

First-order differential equation

Second-order differential equation