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# Revision of Phasors

## A phasor is a complex number

A phasor is a way of writing down a sinusoidal signal
a
jb X
u
+ - x of complex numbers

A complex number can be

In Cartesian form: a + jb

In Polar form: XZu
a
jb X
u
2 2
b a X + =
a
b
tan
1

=
cos X a =
sin X b =
a + jb XZu
To + and - complex numbers
Complex numbers should be in Cartesian form
Add/subtract real and imaginary parts separately
Eg. What is 3 - j7 + 4Z31 ?
+ 4Z31 = 3.43 + j2
3 - j7 + 3.43 + j2 = 6.43 j5
To x and of complex numbers
If in Cartesian form:
Multiply
( )( ) ( ) ( ) ad bc j db ac jd c jb a + + = + +
Divide:

( )
( )
( )( )
( )( )
( ) ( )
2 2
d c
jd c jd c
jd c jb a
jd c
jb a
+
+ +
=
+
+
=
+
+
( )
( )
( ) ( )
2 2 2 2
d c
j
d c
bd ac
jd c
jb a
+

+
+
+
=
+
+
If in polar form:
1
Z X
2
Z Y
2 1
+ Z XY
x
=
( )
2 1
2
1

Z =
Z
Z
Y
X
Y
X
Divide:
Multiply
Representation of a sinusoidal signal
In a linear circuit, all signals at the same frequency e = 2tf
But signals will have different magnitude and have different
phase displacements to each other
Eg. The following are two signals in a circuit

10
B
80
A
10
B
80
A
If we say that A goes through zero at t =0
Then A is
And B is
t sin 10
Z0 10
80 4 t sin Z 80 4
The "magnitude" becomes the magnitude of the
complex number
The phase displacement becomes the angle of the
complex number
10
B
80
A
Actually, the magnitude of the phasor (complex number)
is, by convention, the RMS value of the signal
(peak/root 2).
The phasor for A is thus
The phasor for B is thus or
(the angle can be either degrees or radians)
4 1 83 2 . . Z or
Z0 07 7.
Z 80 83 2.
A
B
10
The phasors can be put on an Argand diagram:
0 07 7.
80 83 2.
10
A B
60
Question 1
Go from "waveform on a scope" to "phasor" to "argand
diagram". E.g. you see B on a scope
Write down its phasor and put it on the Argand diagram
A is
Z0 07 7.
B is
Z 120 10
Adding sinusoidal currents or voltages etc.
them as phasors (complex numbers). NEVER JUST ADD
THEIR MAGNITUDES.
Consider two AC currents and , both of peak 10A, flowing
in the circuit below:
Ammeter 1
Ammeter 2
Ammeter 3
?
2
I
~
1
I
~
10
1
= I

Ammeter 1
Ammeter 2
Ammeter 3
?
10
2
= I

## Q3 If ALL ammeters are set to "AC", what do

Q2. If ALL the ammeters are set to "DC", what will they
All 0A
Both 7.07A
Dont know. Not told phase of sinusoids
Question 4
Let and be as in the example above:
1
I
~
2
I
~
10
80
0 07 7
1
Z = . I
~
80 83 . 2
~
2
Z = I
I
1
= 7.07 + j0
I
2
= 2.83{cos(-80)+jsin(-80)} = 0.49 - j2.78
I
1
+I
2
= 7.56-j2.78 = 8.05Z-20. It will read 8.05A
2
I
~
1
I
~
10
Question 5

Let
1
I
~
2
I
~
and
be as follows:

0 A !!
R, C and L in AC Circuits

In steady state AC circuits, R, L and C all have an
IMPEDANCE.

This complex for inductors and capacitors:

Resistor (purely real)
Inductor (purely imaginary)
Capacitor (purely imaginary)

You will normally be given the values of
R, L, C and e = 2tf, where f = 50Hz (60Hz in the US).

The quantity e=2tf is called the angular frequency of the

R X
R
=
L j X
L
=
C j / X
C
1 = C / j =

Why should L go to jeL in AC circuits?
Consider Faraday's Law for a coil:
dt
t di
L R t i t v
) (
) ( ) ( + =
Let the current be t sin I

) t ( i =
dt
t sin I

d
L t sin I

R ) t ( v

+ =
t cos I

L t sin I

R ) t ( v + =
( ) + + = 90 t sin I

L t sin I

R ) t ( v
( ) ( ) j I
~
L j I
~
R V
~
+ + + = 0 0 1
I
~
L j I
~
R V
~
+ = I
~
X I
~
R V
~
L
+ =
change to phasors:

Worked Example R1
A 50Hz voltage of 220V rms is applied to
a series combination of R = 10O and L = 38mH.
(a) What is the rms value of the current flowing?
(b) What is the rms voltage across R and L?

(a) Impedance of R = 10 O
Impedance of L =
Combined series Impedance
v
R
L
= 10 + j12
L
jX R Z + =
jX
L
R
Z
Impedance Phasors:
2t x 50 x 0.038 = 12 O
I
~
X R V
~
L
+ =
( ) 12 10
0 220
j X R
V
~
I
~
L
+
Z
=
+
=
( ) Z = Z =
Z
Z
= 2 . 50 1 . 14 2 . 50 0
6 . 15
220
2 . 50 6 . 15
0 220 ~
I
v
R
L
i
Note that a reactive impedance (with
+j component), causes the current
to lag the voltage (i.e. current has a
-j component)
A useful mnemonic is C I V I L:
In a Capacitive circuit I leads V;
whereas V leads I in an inductive circuit (L)
L
V
L
R
V
(b) What is the rms voltage across R and L?
I
~
L j I
~
R V
~
V
~
V
~
L R
+ = + =
2 . 50 141 2 . 50 1 . 14 10
~ ~
Z = Z = = I R V
R

Voltage and Current
Phasors:
V
-j

I
IR
-50.2
L
V
L
R
V

Z Z = Z = = 2 . 50 169 90 1 2 . 50 1 . 14 12
~ ~
j I L j V
L
e
Z = 8 . 39 169
~
L
V
-j

V
Voltage and Current
Phasors:
jIX
L
I
IR
Note that although
This is because V
R
and V
L
are sinusoids and NOT in phase
L R
V
~
V
~
V
~
+ =
, 220 = 141 + 169 !!
Note: 141
2
+ 169
2
= 220
2
Ammeter 1
Ammeter 2
Ammeter 3
?
Worked example R2
1
I
~
2
I
~
t sin ) t ( i 3
1
=
0
2
3
1
Z = I
~
t cos ) t ( i 4
2
= 90
2
4
2
+ Z = I
~
Let
Let

t cos t sin ) t ( i ) t ( i 4 3
2 1
+ = +
( )
2 1
+ = + t sin X ) t ( i ) t ( i
Result will be sinusoid. Must find
magnitude and phase
t cos t sin ) t ( i ) t ( i 4 3
2 1
+ = +

2 1
+ = + t sin X ) t ( i ) t ( i
t cos sin X cos t sin X ) t ( i ) t ( i
2 1
+ = +
3 = cos X
4 = sin X Hence: and
2 2 2 2 2 2
4 3 + = + sin X cos X 5 X 25
2
= = , X
3
4

=
cos X
sin X
3
4

1
= tan
Not a good way of adding sinusoids!!
( ) + = + 53 5
2 1
t sin ) t ( i ) t ( i
A / I
~
2 5
3
=
Much better to use Phasors:
0
2
3
1
Z = I
~
90
2
4
2
+ Z = I
~
3
4
2
5
1
3

Z = tan I
~
And the answer is obtained almost immediately
0
2
3
1
Z = I
~
90
2
4
2
+ Z = I
~
Have
and
Revision of Phasors
A phasor is a complex number
A phasor is a way of writing down a sinusoidal signal
Animations on WebCT
a
jb X
u
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