Transcript V m sin ( t )

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
Introduction
Typical periodic signals normally found in electrical
engineering:
Sawtooth wave
Square wave
t
t
Triangle wave
pulse wave
t
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
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 !
Sinusoidal waveform
Let a sinusoidal signal of a voltage is given by:
v(t) = Vm sin (t)
v(t)
Vm

2
3
4
Vm – the amplitude or maximum value
 – the angular frequency (radian/second)
t – the argument of the sine function
t
Sinusoidal waveform
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/2
T
(3/2)T
2T
t
Sinusoidal waveform
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
v(t)
 T is known as the period of the waveform
Vm
• In 1 second there are 1/T cycles of waveform
• The number of cycles per second is the frequency f
T/2
T
(3/2)T
2T
1
f
T
t
The unit for f is Hertz
Sinusoidal waveform
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
Sinusoidal waveform
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

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 ( + )
Sinusoidal waveform
Some important relationships in sinusoidals
v(t)
Vm sin (t)
-Vm sin (t)
Vm
t
Sinusoidal waveform
Some important relationships in sinusoidals
v(t)
Vm sin (t)
-Vm sin (t)
Vm
t
180o
Sinusoidal waveform
Some important relationships in sinusoidals
v(t)
-Vm sin (t)
180o
Therefore, Vmsin (t  180o) = -Vmsin (t )
t
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
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:
1
Xav e 
T

t T
x( t )dt
t
e.g. for a sinusoidal voltage,
1
Vav e 
2
Vav e  0

 2 

Vm sin(t )d(t )
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)
v(t)
i(t)
R
Average power:
(absorbed)
Ieffec
Vdc
R
Average power:
(absorbed)
P

T
2
1
i R dt
T 0
P  I2effec R
Power to be equal:
2
effec
I

T
1 2
R
i R dt
T 0
 Ieffec 
1
T

T
2
i dt
0
Sinusoidal waveform
Average and effective value of a sinusoidal waveform
For a sinusoidal wave, RMS value is :
Vrms 
Vm
2
or
Irms 
Im
2
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
Phasors
How do we transform sinusoidal waveforms to phasors ??
Phasor is rooted in Euler’s identity:
e j  cos   j sin 
Real
 
sin   e 
 cos    e

j
j
Imaginary
cos  is the real part of e
j
sin  is the imaginary part of e j
Supposed v(t) = Vm cos (t + )
j( t   )

 This can be written as v(t) = Vme
Phasors
How do we transform sinusoidal waveforms to phasors ??
j( t   )

v(t) = Vme
Phasors
How do we transform sinusoidal waveforms to phasors ??
j( t   )

v(t) = Vme
j( t   )

= Vme
j  j
= Vme e 

j j
v(t) =  Vme e

Vm e j is the phasor transform of v(t)
v(t) = Vmcos (t +)
phasor transform
V  Vme j
Phasors
V  Vme j
Phasors
V  Vme j
We will use
these notations
V  Vmo
Polar forms
V  Vm cos   Vm j sin 
Rectangular forms
Some examples ….
va(t) = Vmcos (t -)
i(t) = Imcos (t +)
Va  Vm  o
I  Im o
vx(t) = Vmsin (t +)  vx(t) = Vmcos (t + - 90o)
Vx  Vm(o  90 o )
Phasors
V  Vme j
We will use
these notations
V  Vmo
Polar forms
V  Vm cos   Vm j sin 
Rectangular forms
Phasors can be graphically represented using Phasor Diagrams
Im
V  Vmo
Vm
o
Re
Phasors
V  Vme j
We will use
these notations
V  Vmo
Polar forms
V  Vm cos   Vm j sin 
Rectangular forms
Phasors can be graphically represented using Phasor Diagrams
Im
V  Vmo
Vm sin 
o
Re
Vm cos 
Phasors
V  Vme j
We will use
these notations
V  Vmo
Polar forms
V  Vm cos   Vm j sin 
Rectangular forms
Phasors can be graphically represented using Phasor Diagrams
Draw the phasor diagram for the following phasors:
V1  20125 o
V2  40100 o
V3  5  j5
Phasors
To summarize …
• va(t) = Vmcos (t -)
phasor transform
Va  Vm  o  Vm cos   Vm j sin 
• 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:
V  Vmo
inverse phasor transform
v(t) = Vmcos (t + )
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)
IR  Ii
v R  iRR
 vR = R (Im cos (t + i))
VR  RI i  Vm  v
vR and iR are in phase !
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 
IL  Im i
If iL = Im cos (t + i)
vL  L
VR  jLIm i
diL
dt
 vL = L (Im (-sin (t + i)))
 vL = L (Im cos (t + I +90o))
vL leads iL by 90o !
 LIm(i  90o )
 Vm  v
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)
Vc  Vm  v
iC  C
Ic  jCVm  v
dv c
dt
 ic = C (Vm( -sin (t + v)))
 ic = C (Vm cos (t + v +90o))
ic leads vc by 90o !
 CVm(v  90 o )
 Im i
STEADY STATE AC CIRCUIT ANALYSIS
KVL and KCL in AC circuit
Exactly the same laws (as we have seen in DC circuit) apply for AC circuit:
However, in AC circuit we will deal with complex numbers.
KVL: Sum of voltages around a closed loop = 0
v1(t) + v2(t) + v3(t) + …. + vn(t) = 0

V1 + V2 + V3 + …. + Vn = 0
KCL: Sum of currents leaving (or entering) a node = 0
i1(t) + i2(t) + i3(t) + …. + in(t) = 0

I1 + I2 + I3 + …. + In = 0
Voltage-division and Current-division rules
Series connected impedances:
Vs  I(Z
s

1
 Z2  Z3  Z 4  Z5 )
Vs
 Z1  Z 2  Z 3  Z 4  Z 5
Is
Vs  I(Z
s
eq
)
 Z eq  Z1  Z 3  Z 3  Z 4  Z 5
Voltage-division and Current-division rules
Voltage-division rule:
Voltage-division and Current-division rules
Parallel connected impedances:
Is  V(Y
s

Is  V(Y
s
1
 Y2  Y3  Y4 )
Is
 Y1  Y2  Y3  Y4
Vs
eq
)
 Yeq  Y1  Y2  Y3  Y4
Voltage-division and Current-division rules
Current-division rule: