1. STEADY STATE AC CIRCUIT ANALYSIS

2. 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

3. Typical periodic signals normally found in electrical engineering:
Introduction
Typical periodic signals normally found in electrical engineering:
Sawtooth wave
Square wave t t
Triangle wave
pulse wave t t

4. 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

5. 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.

6. 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 !

7. Sinusoidal waveform v(t) = Vm sin (t)
Let a sinusoidal signal of a voltage is given by:
v(t) = Vm sin (t)
v(t) Vm t  2 3 4
Vm – the amplitude or maximum value
 – the angular frequency (radian/second)
t – the argument of the sine function

8. Sinusoidal waveform v(t) = Vm sin (t) v(t) = Vm sin (t)
Let a sinusoidal signal of a voltage is given by:
v(t) = Vm sin (t)
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

9. Sinusoidal waveform v(t) = Vm sin (t) v(t) = Vm sin (t)
Let a sinusoidal signal of a voltage is given by:
v(t) = Vm sin (t)
The voltage can also be written as function of time:
v(t) = Vm sin (t)
In T seconds, the voltage goes through 1 cycle
T/2 T
(3/2)T 2T t
v(t) Vm
 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

10. Sinusoidal waveform v1(t) = Vm sin (t + )
A more general expression of a sinusoidal signal is
v1(t) = Vm sin (t + )
 is called the phase angle, normally written in degrees
Let a second voltage waveform is given by: v2(t) = Vm sin (t - )
v(t)
v1(t) = Vm sin (t + )
v2(t) = Vm sin (t - ) Vm   t

11. Sinusoidal waveform Vm   v(t) t v1(t) = Vm sin (t + )

12. Sinusoidal waveform Vm   v1 and v2 are said to be out of phase
v(t) Vm
v2(t) = Vm sin (t - )
v1(t) = Vm sin (t + )  
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 ( + )

13. Sinusoidal waveform Some important relationships in sinusoidals Vm
v(t)
Vm sin (t)
-Vm sin (t) Vm t

14. Sinusoidal waveform Some important relationships in sinusoidals Vm
v(t)
Vm sin (t)
-Vm sin (t) Vm t
180o

15. Sinusoidal waveform Some important relationships in sinusoidals
v(t)
-Vm sin (t) t
180o
Therefore, Vmsin (t  180o) = -Vmsin (t )

16. 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)
v(t) Vm t
250o
110o

17. 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

18. 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,

19. 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) R
v(t)
i(t)
Power to be equal:
Average power:
(absorbed)
Ieffec R
Vdc
Average power:
(absorbed)

20. Sinusoidal waveform
Average and effective value of a sinusoidal waveform
For a sinusoidal wave, RMS value is : or

21. 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

22. Phasors How do we transform sinusoidal waveforms to phasors ??
Phasor is rooted in Euler’s identity:
Real
Imaginary
cos  is the real part of
sin  is the imaginary part of
Supposed v(t) = Vm cos (t + )
 This can be written as
v(t) =

23. Phasors
How do we transform sinusoidal waveforms to phasors ??
v(t) =

24. Phasors v(t) = How do we transform sinusoidal waveforms to phasors ??
is the phasor transform of v(t)
v(t) = Vmcos (t +)
phasor transform

25. Phasors

26. Phasors Polar forms We will use these notations Rectangular forms
Some examples ….
va(t) = Vmcos (t -)
i(t) = Imcos (t +)
vx(t) = Vmsin (t +) 
vx(t) = Vmcos (t + - 90o)

27. Phasors Polar forms We will use these notations Rectangular forms
Phasors can be graphically represented using Phasor Diagrams Im Re

28. Phasors Polar forms We will use these notations Rectangular forms
Phasors can be graphically represented using Phasor Diagrams Im Re

29. 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:

30. Phasors To summarize … va(t) = Vmcos (t -)
phasor transform
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:
inverse phasor transform
v(t) = Vmcos (t + )

31. 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 R iR IR
+ vR 
+ VR 
If iR = Im cos (t + i)
 vR = R (Im cos (t + i))
vR and iR are in phase !

32. 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 L iL IL
+ 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 !

33. 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 C ic Ic
+ 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 !