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Fundamentals of
Electric Circuits
Chapter 11
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Overview
• This chapter will cover the concept of power
in an AC circuit.
• The difference between instantaneous power
and average power will be discussed.
• The difference between resistive and reactive
power will be introduced.
• Other forms of averaged measurements will
be covered
• Apparent power and complex power will also
be covered.
2
Instantaneous Power
• This is the power at any instant in time.
• It is the rate at which an element absorbs
power
• Consider the generalized case where the
voltage and current at the terminals of a
circuit are:
v t   Vm cos t  v  i t   I m cos t  i 
• Multiplying the two together, yields:
1
1
p  t   Vm I m cos  v  i   Vm I m cos  2t   v  i 
2
2
3
Instantaneous Power II
• Note that this has two components.
– One is constant, depending on the phase
difference between the voltage and current
– The second is sinusoidal with a frequency twice
that of the voltage and current.
• A sketch of the possible instantaneous
power is below.
4
Instantaneous Power III
• Note that the figure shows times where the
power goes negative.
• This is possible with circuit elements like
inductors or capacitors which can store and
release energy.
• Note also that instantaneous power is very
hard to measure as it is constantly changing.
• The more common power measured is
average power.
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Average Power
• Average power is the instantaneous power
averaged over a period.
• It is given by:
T
1
P   p  t  dt
T0
• When evaluated, this returns the component
of instantaneous power that was constant.
• The time dependent part is a sinusoid and
thus averages to zero.
6
Average Power II
• In order to get the instantaneous power, you
need to work in the time domain.
• But for average power it is possible to work
in frequency domain.
• In this case, the average power is:
1
1
P  Re VI *  Vm I m cos v  i 
2
2
7
Resistive vs. Reactive
• Consider the case when θv= θi the voltage
and current are in phase and the circuit is
purely resistive:
1
1
1 2
P  Vm I m  I m2 R  I R
2
2
2
• When θv- θi = ±90°, the circuit absorbs no
power and is purely reactive
1
P  Vm I m cos 90  0
2
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Maximum Average Power
Transfer
• Previously we considered
how to maximize power
delivered to a resistive load.
• It was shown that maximum
power was transferred when
the load resistance equaled
the Thevenin resistance of
the supply circuit.
• This will now be extended
to AC circuits.
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Maximum Average Power
• In rectangular form, the Thevenin impedance
and load impedance are:
ZTh  RTh  jX Th
Z L  RL  jX L
• The current through the load is:
I
VTh
VTh

ZTh  Z L  RTh  jX Th    RL  jX L 
• The average power delivered to the load is:
2
P
VTh RL / 2
1 2
I RL 
2
2
2
 RTh  jX Th    RL  jX L 
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Maximum Average Power II
• We want to find the peak in this function,
which means finding the derivative and
identifying where it goes to zero.
• This must be done for both RL and XL.
• For ∂P/∂XL
VTh RL  X Th  X L 
P

2
X L
 RTh  jX Th 2   RL  jX L 2 


2
• For ∂P/∂RL
V
P
  Th
RL
2
 RTh  RL    X Th  X L   2 RL  RTh  RL 
2
 RTh  jX Th 2   RL  jX L 2 


2
2
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Maximum Average Power III
• Combining these one finds that XL=-XTh and
RL=RTh satisfy the requirements:
*
Z L  RL  jX L  RTh  jX Th  ZTh
• The load impedance must be equal to the
complex conjugate of the Thevenin
impedance.
• The maximum average power will be:
Pmax 
VTh
2
8RTh
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Effective Value
• When a time varying source is delivering
power to a resistive load, we often want to
know the effectiveness of that source on
delivering power.
• This value is the DC current that delivers the
same average power to a resistor as the
periodic current
• For a periodic current, the average power
absorbed is:
T
T
1
R
P   i 2 Rdt   i 2 dt
To
T o
13
Average Power
• For a DC current, the average power
absorbed is:
P  ieff2 R
• Equating these two and solving for the
effective DC current yields:
T
I eff
1 2

i dt
T 0
• The effective voltage is found in a similar
manner:
T
1 2
Veff 
v dt
T 0
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RMS
• The effective values for both current and
voltage take the form of the square root of
the average of the square of the periodic
signal.
• This is typically referred to as the root mean
square, or RMS value for short.
• This can be extended to any periodic
function:
T
X rms
1 2

x dt

T0
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RMS II
• If the parameter of interest is a constant, the
RMS value will be that constant.
• The RMS value is applicable to any periodic
function, regardless of its shape.
• However, for a sinusoidal waveform, the RNS
value is related to the amplitude as follows:
Vrms
Vm
2
• RMS power can be determined from either
RMS current or voltage:
PI
2
rms
2
Vrms
R
R
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Apparent Power
• The product of RMS voltage and
current will be called apparent power.
• Note that this is modulated by the
phase difference:
cos v  i 
• This will be referred to as the power
factor.
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Apparent Power II
• Apparent power is called that simply
because that is what the power in the circuit
would seem to be.
• It is measured in VA, to distinguish it from
the average or real power.
• The angle θv-θi is called the power factor
angle.
• This is equal to the angle of the load
impedance is V is the voltage across the load
and I is the current through it.
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Power Factor
• The power factor can range from zero to
unity.
• In a case where the power factor angle is
positive, the current leads the voltage, and
vice versa for negative angles.
• Purely reactive loads will have a power factor
of zero.
• Power factors affect the way utilities bill for
electricity.
19
Complex Power
• The term complex power refers to the total
effect of parallel loads on power.
• The complex power, S, absorbed by an AC
load is:
1
S  VI *
2
• This may also be expressed in terms of the
RMS values and load impedance as:
2
V
2
*
S  I rms
Z  rms*  Vrms I rms
Z
20
Complex Power II
• The complex power expressed in rectangular
form is:
2
S  I rms
 R  jX   P  jQ
• Where:
2
P  Re  S   I rms
R
2
Q  Im  S   I rms
X
• P is the average or real power
• Q depends on the load’s reactance and is
called reactive (or quadrature) power
21
Real and Reactive Power
• The real power is the only useful power. It is
measured in watts.
• The reactive power is a measure of the
energy exchange between the source anf the
reactive load.
• It is measured in units of volt-ampere
reactive (VAR)
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Summarizing Power
Complex Power  S  P  jQ  Vrms  I rms  *
 Vrms I rms   v  i 
Apparent Power  S  S  Vrms I rms  P 2  Q 2
Real Power  P  Re  S   S cos  v  i 
Reactive Power  Q  Im  S   S sin  v  i 
P
Power Factor   cos  v  i 
S
23
Conservation of AC Power
• Regardless of how circuit elements are
connected, the total complex power
delivered is equal to the total complex power
absorbed by the elements.
S  S1  S2  S3 
 SN
• The same is true of real and reactive power,
but not of apparent power.
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Power Factor Correction
• Most domestic and industrial loads, such as
washing machines, air conditioners, and
induction motors are inductive.
• They have a low, lagging power factor.
• The load cannot be changed, but the power
factor can be increased without altering the
voltage or current to the original load.
• This is referred to as power factor correction.
25
Adding a Capacitor
• To mitigate the inductive aspect of
the load, a capacitor is added in
parallel with the load.
• Looking at the phasor diagram,
showing before and after adding
the capacitor, the power factor has
improved.
26
Adding a Capacitor II
• With the same supplied voltage, the current
draw is less by adding the capacitor.
• Since power companies charge more for
larger currents because it leads to larger
power losses.
• Overall, the power factor correction benefits
the power company and the consumer.
• By choosing a suitable size for the capacitor,
the power factor can be made to be unity.
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Adding a Capacitor III
• The capacitor needed in order to shift the
power factor angle from θ1 to θ2 is:
C
P  tan 1  tan  2 
QC

2
2
Vrms
Vrms
• Note that the real power dissipated in the
load is not affected by the shunt capacitor.
• Although it is not as common, if a load is
capacitive in nature, the same treatment with
an inductor can be used.
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Wattmeter
• Power consumption in a AC system
can be measured using a
Wattmeter.
• The meter consists of two coils; the
current and voltage coils.
• The current coil is designed with
low impedance and is connected in
series with the load.
• The voltage coil is designed with
very large impedance and is
connected in parallel with the load.
29
Wattmeter II
• The induced magnetic field from both causes
a deflection in the current coil.
• Ideally, the configuration does not alter the
load and affect the power measured.
• The physical inertia of the moving coil
results in the output being equal to the
average power.
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