Complex Notation for AC Quantities:
Complex Impedance
Aims:
To appreciate:
•Use of complex quantities.
•Role of impedance and j.
•Influence of power factor.
•Appreciate operation of low, high and band pass filters
To be able:
•To analyse some basic circuits.
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Lecture 12
Revision of Complex Number Arithmetic
what is j?
three ways to express a complex number
j = −1
Z = R + jX
j = −1
Z = R 2 + X 2 (modulus) : tan φ =
2
j 3 = − −1 = − j
R is real part; X is imaginary part
X
(argument)
R
Z = Z e jφ
1
=−j
j
Z = Z (cos φ + j sin φ )
complex arithmetic
Z1 + Z 2 = ( R1 + R2 ) + j ( X 1 + X 2 )
Z1Z 2 = Z1 Z 2 e
j (φ1 +φ2 )
Z = R − jX = Z e − jφ (complex conjugate)
Z
Z1
= 1 e j (φ1 −φ2 )
Z2 Z2
jZ = Z e
⎛ π⎞
j⎜φ + ⎟
⎝ 2⎠
complex conjugate
*
ZZ * = R 2 + X 2
⎛
π⎞
j⎜φ − ⎟
Z
= Z e ⎝ 2⎠
j
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Lecture 12
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Phasors as Complex Numbers
If we plot phasors on an Argand
diagram we can use complex
number representation:
Imaginary
V
Vsin ωt
ωt
Vcos ωt
Real
V ωt ≡ V (cos ωt + j sin ωt )
Which means that we can use the
powerful tools of complex algebra
to manipulate AC quantities.
Note that we use j for √-1 and not i
This is to avoid confusion with i as a
symbol for currents.
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Complex AC quantities
In general, all AC quantities are complex numbers containing amplitude and phase:
Voltage
Current
Impedance
V = VRE + jVIM = V e jφ
I = I RE + jI IM = I e jφ
Z = R + jX = Z e jφ
The complex quantities obey all the laws and techniques that we
have derived for DC networks:
• Kirchhoff’s Current Law
• Kirchhoff’s Voltage Law
• Ohm’s Law V=IZ
• Impedances in series: Z = Z1+Z1
• Impedances in parallel: Y = Y1+Y2
The physical significance of the real and imaginary parts of current and voltage:
• Real currents and voltages are associated with energy dissipation (power
averaged over one cycle is positive).
Measurable
• Imaginary parts are associated with energy storage (power averaged over
one cycle is zero – charging and discharging). Not measurable
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Lecture 12
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Look Back at Inductive Reactance:
Let
dI
L
I L = I o e jωt and we know that VL = L dt
What is VL ?
The reciprocal of impedance is ADMITTANCE, symbol Y, units Siemens
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Lecture 12
Look Back at Capacitative Reactance:
Let
dV
C
VC = Vo e jωt and we know that I C = C dt
What is IC ?
The reciprocal of impedance is ADMITTANCE, symbol Y, units Siemens
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Lecture 12
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Power in AC Circuits
Power in an AC circuit is given by W=VI
V = V0 e jωt
I = I 0 e j (ωt +φ )
W = V0 I 0 e j (2ωt +φ )
This reduces to:
(
W = V0 I 0 cos φ e + j 2ωt + sin φ e − j 2ωt
)
Reactive power
Resistive power
If we average over one cycle (from t=0 to t=2π/ω):
e+j2ωt averages to ½ and e-j2ωt averages to 0, so
1
W = V0 I 0 cos φ
2
or W = VRMS I RMS cos φ
The cosφ term is called the power factor
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Lecture 12
Power Factor:
W = VRMS I RMS cos φ
This tells us that when the current and voltage are π/2 out of phase
(e.g.in a pure L or pure C), the power dissipated is zero.
Power factor is a big issue for electrical engineers.
• Many industrial loads have a high inductance in
series with the resistance (e.g. heating coils for
large tanks)
• This can affect the power factor and reduce the
power dissipated in the resistor.
• In many cases a capacitor is used to correct the
power factor
50 Hz
Power factor correction
R
L
Power ratings of industrial equipment are often quoted in
“kVA” – kilo-volt-amp – rather than kW to indicate that
the power factor may not be 1
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Lecture 12
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RC Network with Complex Numbers
I
V
R
VR
VC
C
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Lecture 12
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RCL Network in Series
L
C
V
R
VR
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5
RCL Network in Series
3. Get the current
R
C
L
I=
4. Get the voltage across the resistor
1. Complex impedance is given by
Z = R + jX
VR = IR =
1
where X = ω L −
ωC
⎛ 1 − ω 2 LC ⎞
Z = R2 + ⎜
⎟
⎝ ωC ⎠
RVA j (ωt −φ )
e
Z
VR
R − jφ
=
e
V
Z
2. Convert to exponential form for multiplication:
VR
=
V
2
X ω LC − 1
=
R
ωCR
Z = Z e jφ
2
tan φ =
VAe jωt VA j (ωt −φ )
=
e
Z e jφ
Z
1
⎛ 1 − ω LC ⎞
1+ ⎜
⎟
⎝ ωCR ⎠
2
2
e − jφ
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Lecture 12
RCL Network in Series
VR
=
VAe jωt
L
C
VA
R
VR
1
⎛ 1 − ω LC ⎞
1+ ⎜
⎟
⎝ ωCR ⎠
2
2
e − jφ
when ω=0, denominator →∞, VR →0
when ω →∞, denominator →∞, VR →0
when ω = ω0, denominator = 1, VR=VA
ω0 =
1
LC
VR
ω0
This is a series resonant circuit.
At resonance XC=-XL so the
reactance in the circuit is zero
ω
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Lecture 12
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