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STEADY STATE AC CIRCUIT ANALYSIS

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STEADY STATE AC CIRCUIT ANALYSIS

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STEADY STATE AC CIRCUIT ANALYSIS

Introduction

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

DC circuit analysis

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

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

t

t

t

t

Introduction

Typical periodic signals normally found in electrical engineering:

Sawtooth wave

Square wave

Triangle wave

pulse 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

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

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.

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 !

v(t)

t

2

3

4

Sinusoidal waveform

Let a sinusoidal signal of a voltage is given by:

v(t) = Vm sin (t)

Vm

Vm – the amplitude or maximum value

– the angular frequency (radian/second)

t – the argument of the sine function

The voltage can also be written as function of time:

v(t) = Vm sin (t)

v(t)

t

T/2

T

(3/2)T

2T

Sinusoidal waveform

Let a sinusoidal signal of a voltage is given by:

v(t) = Vm sin (t)

Vm

The voltage can also be written as function of time:

v(t) = Vm sin (t)

v(t)

Vm

t

T/2

T

(3/2)T

2T

Sinusoidal waveform

Let a sinusoidal signal of a voltage is given by:

v(t) = Vm sin (t)

- In T seconds, the voltage goes through 1 cycle

T is known as the period of the waveform

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

- The number of cycles per second is the frequency f

The unit for f is Hertz

A more general expression of a sinusoidal signal is

v1(t) = Vm sin (t + )

v(t)

Vm

t

Sinusoidal waveform

is called the phase angle, normally written in degrees

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

v1(t) = Vm sin (t + )

v2(t) = Vm sin (t - )

v(t)

v1(t) = Vm sin (t + )

v2(t) = Vm sin (t - )

Vm

t

Sinusoidal waveform

v(t)

v1(t) = Vm sin (t + )

v2(t) = Vm sin (t - )

Vm

t

Sinusoidal waveform

v1 and v2 are said to be out of phase

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

alternatively,

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

v(t)

Vm

t

Sinusoidal waveform

Some important relationships in sinusoidals

Vm sin (t)

-Vm sin (t)

v(t)

Vm

180o

t

Sinusoidal waveform

Some important relationships in sinusoidals

Vm sin (t)

-Vm sin (t)

v(t)

180o

t

Sinusoidal waveform

Some important relationships in sinusoidals

-Vm sin (t)

Therefore, Vmsin (t 180o) = -Vmsin (t )

v(t)

t

Vm

Sinusoidal waveform

Some important relationships in sinusoidals

Vmsin (t) = Vmsin (t 360o)

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

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

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

= Vm sin (t 110o)

250o

110o

Sinusoidal waveform

Some important relationships in sinusoidals

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

Sinusoidal waveform

Average and effective value of a sinusoidal waveform

An average value a periodic waveform is defined as:

e.g. for a sinusoidal voltage,

Ieffec

Vdc

R

i(t)

v(t)

R

Sinusoidal waveform

Average and effective value of a sinusoidal waveform

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)

Power to be equal:

Average power:

(absorbed)

Average power:

(absorbed)

Sinusoidal waveform

Average and effective value of a sinusoidal waveform

For a sinusoidal wave, RMS value is :

or

Phasors

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

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

Why used phasors ?

Analysis of AC circuit will be much more easier using phasors

cos is the real part of

sin is the imaginary part of

Imaginary

Real

v(t) =

This can be written as

Phasors

How do we transform sinusoidal waveforms to phasors ??

Phasor is rooted in Euler’s identity:

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

v(t) =

Phasors

How do we transform sinusoidal waveforms to phasors ??

=

=

v(t) =

phasor transform

v(t) = Vmcos (t +)

v(t) =

Phasors

How do we transform sinusoidal waveforms to phasors ??

is the phasor transform of v(t)

Phasors

va(t) = Vmcos (t -)

i(t) = Imcos (t +)

vx(t) = Vmcos (t + - 90o)

vx(t) = Vmsin (t +)

Phasors

Polar forms

We will use these notations

Rectangular forms

Some examples ….

Im

Re

Phasors

Polar forms

We will use these notations

Rectangular forms

Phasors can be graphically represented using Phasor Diagrams

Im

Re

Phasors

Polar forms

We will use these notations

Rectangular forms

Phasors can be graphically represented using Phasor Diagrams

Phasors

Polar forms

We will use these notations

Rectangular forms

Phasors can be graphically represented using Phasor Diagrams

Draw the phasor diagram for the following phasors:

phasor transform

- va(t) = Vmcos (t -)

inverse phasor transform

v(t) = Vmcos (t + )

Phasors

To summarize …

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

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

V = V1 + V2 + V3 +V4 + …+Vn

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

R

iR

IR

Phasor 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

+ vR

If iR = Im cos (t + i)

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

vR and iR are in phase !

L

iL

IL

Phasor 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

+ VL

+ vL

If iL = Im cos (t + i)

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

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

vL leads iL by 90o !

C

ic

Ic

Phasor 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

+ Vc

+ vc

If vc = Vm cos (t + v)

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

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

ic leads vc by 90o !