STEADY STATE AC CIRCUIT ANALYSIS. Previously we have analyzed circuits with time-independent sources – voltage and current that do not change with time

STEADY STATE AC CIRCUIT ANALYSISSTEADY STATE AC CIRCUIT ANALYSIS

Previously we have analyzed circuits with time-independent sources – voltage and current that do not change with time

DC circuit analysis

x(t) = x(t +nT), where n = 1,2 3, … and T is the period of the signal

Introduction

In this section we will analyze circuits containing time-dependent sources – voltage and current vary with time

One of the important classes of time-dependent signal is the periodic signals

Introduction

Typical periodic signals normally found in electrical engineering:

t

tt

Sawtooth wave Square wave

Triangle wavepulse wave

t

Introduction

In SEE 1003 we will deal with one of the most important periodic signal of all :- sinusoidal signals

Signals that has the form of sine or cosine function

t

Introduction

In SEE 1003 we will deal with one of the most important periodic signal of all :- sinusoidal signals

Circuit containing sources with sinusoidal signals (sinusoidal sources) is called an AC circuit. Our analysis will be restricted to the steady state behavior of AC circuit.

Signals that has the form of sine or cosine function

Why do we need to study sinusoidal AC circuit ?Why do we need to study sinusoidal AC circuit ?

• Dominant waveform in the electric power industries worldwide – household and industrial appliations

• ALL periodic waveforms (e.g. square, triangular, sawtooth, etc) can be represented by sinusoids

• You want to pass SEE1003 !

Sinusoidal waveformSinusoidal waveform

Let a sinusoidal signal of a voltage is given by: v(t) = Vm sin (t)

2 3 4 t

v(t)

Vm

– the angular frequency (radian/second)

t – the argument of the sine function

Vm – the amplitude or maximum value



T/2 T (3/2)T 2T t

v(t)

Vm

The voltage can also be written as function of time: v(t) = Vm sin (t)



T/2 T (3/2)T 2T t

v(t)

Vm

The voltage can also be written as function of time: v(t) = Vm sin (t)

• In T seconds, the voltage goes through 1 cycle

• In 1 second there are 1/T cycles of waveform

• The number of cycles per second is the frequency f

T

1f

The unit for f is Hertz

T is known as the period of the waveform


A more general expression of a sinusoidal signal is v1(t) = Vm sin (t + )

is called the phase angle, normally written in degrees

t

v(t)

Vm

v2(t) = Vm sin (t - )v1(t) = Vm sin (t + )

Let a second voltage waveform is given by: v2(t) = Vm sin (t - )


t

v(t)

Vm



t

v(t)

Vm


v1 is said to be leadingleading v2 by (-) or ( + )

v2 is said to be lagginglagging v1 by (-) or ( + )

alternatively,

v1 and v2 are said to be out of phaseout of phase


t

v(t)

Vm

-Vm sin (t) Vm sin (t)

Some important relationships in sinusoidals


t

v(t)

Vm

-Vm sin (t) Vm sin (t)


180180oo


t

v(t)

-Vm sin (t)


Therefore, VVmmsin (sin (t t 180 180oo) = -V) = -Vmmsin (sin (t )t )

180180oo



Vmsin (t) = Vmsin (t 360o)

Therefore, Vmsin (t + ) = Vmsin (t + 360o)

VVmmsin (sin (t + t + )) = V= Vmmsin (sin (t t (360 (360o o ))))

e.g., Vmsin (t + 250o) = Vmsin (t (360o 250o))

= Vm sin (t 110o)

t

v(t)

Vm

250o 110o



It is easier to compare two sinusoidal signals if:

• Both are expressed sine or cosine

• Both are written with positive amplitudes

• Both have the same frequency


Average and effective value of a sinusoidal waveform

An average value a periodic waveform is defined as:

Tt

tave dt)t(x

T

1X

e.g. for a sinusoidal voltage,

2

mave )t(d)tsin(V2

1V

0Vave



An effective value or Root-Mean-Square (RMS) a periodic current (or voltage) is defined as:

The value of the DC current (or voltage) which, flowing through a R-ohm resistor delivers the same average power as does the periodic current (or voltage)

Ieffec

RVdc

Rv(t)

i(t)Average power:(absorbed)

dtRiT

1P

T

0

2

RIP 2effecAverage power:

(absorbed)

Power to be equal:

dtRiT

1RI

T

0

22effec

dtiT

1I

T

0

2effec



For a sinusoidal wave, RMS value is :

2

VV m

rms or2

II mrms

PhasorsPhasors

A phasorphasor: A complex number used to represent a sinusoidal waveform. It contain the information about the amplitude and phase angle of the sinusoid.

Why used phasors ?

Analysis of AC circuit will be much more easier using phasors

In steady state condition, the sinusoidal voltage or current will have the same frequency. The differences between sinusoidal waveforms are only in the magnitudes and phase angles

PhasorsPhasors

How do we transform sinusoidal waveforms to phasors ??

Phasor is rooted in Euler’s identity:

sinjcose j

jecos cos is the real part of je

jesin sin is the imaginary part of je

Real Imaginary

Supposed v(t) = Vm cos (t + )

This can be written as )t(jm eV v(t) =

PhasorsPhasors


)t(jm eV v(t) =

PhasorsPhasors


)t(jm eV v(t) =

)t(jmeV =

v(t) = jjm eeV

jjm eeV=

jmeV is the phasor transform of v(t)

v(t) = Vmcos (t +) phasor transform jmeVV

PhasorsPhasors

jmeVV

PhasorsPhasors

jmeVV

omV V

Polar forms

sinjVcosV mmV Rectangular forms

We will use these notations

va(t) = Vmcos (t -)o

mV aV

Some examples ….

i(t) = Imcos (t +)o

mI I

vx(t) = Vmsin (t +) vx(t) = Vmcos (t + - 90o) )90(V oom xV

PhasorsPhasors

jmeVV

omV V

Polar forms



Phasors can be graphically represented using Phasor Diagrams

omV V

mV

o

Im

Re

PhasorsPhasors

jmeVV

omV V

Polar forms




omV V

cosVm

o

sinVm

Im

Re

PhasorsPhasors

jmeVV

omV V

Polar forms




Draw the phasor diagram for the following phasors:

o125201Vo100402V 5j5 3V

PhasorsPhasors

To summarize …

• va(t) = Vmcos (t -) phasor transform omV aV sinjVcosV mm

• It is also possible to do the inverse phasor transform:

omV V inverse phasor transform v(t) = Vmcos (t + )

• If v1(t), v2(t), v3(t), v4(t), ….vn(t) are sinusoidals of the same frequency and

V = V1 + V2 + V3 +V4 + …+Vn

v(t) = v1(t) + v2(t) + v3(t) + v4(t) + ….+vn(t) , in phasors this can be written as:

Phasor Relationships for R, L and CPhasor Relationships for R, L and C

The relationships between V and I for R, L and C are needed in order for us to do the AC circuit analysis

+ vR

iR

RiRRv

If iR = Im cos (t + i)

vR = R (Im cos (t + i))

RR

IR

+ VR

iR I I

vmiR VRI V

vvRR and i and iRR are in phase ! are in phase !



+ vL

iL

dt

diL LLv

If iL = Im cos (t + i)

vL = L (Im (-sin (t + i)))

LL

IL

+ VL

imL I I

imR LIj V

vvLL leads i leads iLL by 90 by 90oo ! !

vL = L (Im cos (t + I +90o))

)90(LI oim

vmV



+ vc

ic

dt

dvC cCi

If vc = Vm cos (t + v)

ic = C (Vm( -sin (t + v)))

CC

Ic

+ Vc

vmc V V

vmc CVj I

iicc leads v leads vcc by 90 by 90oo ! !

ic = C (Vm cos (t + v +90o))

)90(CV ovm

imI

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STEADY STATE AC CIRCUIT ANALYSIS. Previously we have analyzed circuits with time-independent sources – voltage and current that do not change with time