IEEE TRANSACTIONS ON EDUCATION, VOL. 50, NO. 3, AUGUST 2007
251
“Deglorifying” the Maximum Power Transfer
Theorem and Factors in Impedance Selection
James C. McLaughlin and Kenneth L. Kaiser
Abstract—The limited usefulness of the maximum power
transfer theorem in practice is argued. Inappropriately, the
utility and value of the maximum power transfer theorem are
often elevated to be religious icons of electrical engineering.
While the theorem appears to be useful, often in real circuits
the load impedance is not set equal to the complex conjugate of
the equivalent impedance of the connecting source. When the
load impedance happens to be equal to the complex conjugate
of the source impedance, other practical reasons for this type of
impedance matching exists, other than effecting maximum power
transfer. Some reasons are discussed in a straightforward fashion.
II. EFFICIENCY
If a source is an open-circuit voltage in series with an equivalent resistance, or has the appearance of a Thévenin equivalent,
then as the load resistance increases, the power efficiency increases
Index Terms—Conjugate matching, impedance matching, maximum power transfer theorem, noise matching, voltage matching.
If
, then the efficiency approaches 100%. Of course,
if the load corresponds to an open circuit, the apparent efficiency
would be 100%, but then no power would be delivered to the
load. In many applications, especially related to power generation, distribution, and amplification, efficiency is of prime importance, not maximum power transfer. Note that the efficiency
, corresponding to maximum
is merely 50% when
power transfer to the load
I. INTRODUCTION
I
N electrical engineering circuits textbooks, a substantial discussion on the maximum power transfer theorem and its importance in “many applications” is still common. Briefly, this
theorem states that the maximum average power that can be
, is obtained
delivered to a load impedance,
when the load impedance is equal to the complex conjugate of
the equivalent impedance of the source connected to the load.
If the Thévenin or equivalent impedance of the source is
, then maximum power is delivered to a load when
. A conjugate match exists when
is equal
.
to the complex conjugate of
As many experienced engineers know, although the maximum power transfer theorem is an interesting result, its
usefulness in practice is extremely limited. A few electrical
engineering books and articles have discussed the lack of utility
of this theorem in specific applications [1]–[6], but its supposed
importance is still claimed or emphasized in several textbooks
and numerous articles [7]–[11]. Indeed, the theorem has such
a religious aura that one might be led to believe the use of
the maximum power transfer theorem will solve all problems.
In this discussion, the chief reasons why the input or output
impedance of a device is selected or adjusted to a particular
value are discussed. These reasons are not necessarily independent of each other, and frequently several of these reasons
are considered in a design. Examples that have been used to
argue incorrectly the usefulness of the maximum power transfer
theorem are included and briefly analyzed.
Manuscript received July 11, 2005; revised April 12, 2007.
The authors are with the Electrical and Computer Engineering Department,
Kettering University, Flint, MI 48504 USA (e-mail: kkaiser@kettering.edu).
Digital Object Identifier 10.1109/TE.2007.900030
When
, the efficiency is 75%, and the power delivered to the load is
Seventy-five percent of the maximum possible power is delivered to the load in this case.
Attaining high efficiency is important in many devices, especially cellular phones, laptop computers, and satellites. When a
load is connected across a battery, rarely is the load impedance
selected to be equal to the impedance (or conjugate impedance)
of the battery. Because the battery’s impedance is often low, intentionally connecting a low-impedance load across the battery
could excessively load the battery, causing its output voltage to
fall or even causing an explosion.
Not to mention that the efficiency of an actual active device
and the efficiency of its Thévenin equivalent are not necessarily
the same would be misleading. For example, a power supply
with a low effective resistance is not necessarily efficient. The
low effective resistance could be caused by feedback and not be
an actual ohmic resistance. Indeed, consider the case of modeling a network with a Thévenin equivalent circuit and also modeling the same network with a Norton equivalent circuit. Assume that the two models present to a load exactly what the
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IEEE TRANSACTIONS ON EDUCATION, VOL. 50, NO. 3, AUGUST 2007
network presents. If one were to make conclusions about the
power absorbed by each of the models as a function of load resistance, one will see different results. For a resistive equivalent impedance and load, the powers absorbed by the equivalent
Thévenin and Norton resistances are
The previous powers are identical only when
. The
validity of the applicable theorem that says that such equivalent
circuits are only equivalent as far as what they do to the outside
world is thus demonstrated.
A. RF Devices
A number of RF applications may exist where the use the
maximum power theorem is inappropriate. Imagine that an RF
transmitter that is able to deliver 1 kW to a 50 load has an actual internal impedance of
and is somehow connected to
. Assuming the
an antenna with an impedance of
transmitter does not fail, the maximum power possible would
be delivered to the antenna (which is likely to be greater than
1 kW), but the efficiency of this power transfer would be only
would be very great.
50%. Also, the power absorbed by
A large cooling system would have to be effected to disperse the
large amount of power dissipated within the abused transmitter.
Worse, the spectral purity of the emissions probably would be
compromised.
Many individuals believe that a 50 transmitter has an equivalent impedance of 50 . Hence, the argument proceeds that for
maximum power transfer, the load to the transmitter should also
be equal to 50 . Maximum power transfer is not the reason for
this load selection since the 50 transmitter likely has an equivalent output impedance much less than 50 . The transmitter
is probably designed to transfer its power “best” with a 50
load. Fifty ohms is the impedance that the transmitter “likes to
see [12].”
Conjugate matching is often found abused in conjunction
with active devices used at RF. The manufacturer of an active
device, such as a transistor to be used at 460 MHz, places
representative devices into a fixture that is able selectively to
present a wide range of impedances
to the device. Experimentally, the
that is best for gain or efficiency is found and
provided on data sheets. Such impedances are not the conjugate
of the actual internal impedance of the device. However, often
the provided desirable impedances are incorrectly discussed as
if conjugate matching is involved.
factor of one is frequently strived for in order to minimize
losses in the transmission line leading to the load.
Electrical energy distribution systems strive to keep voltage
changes with current changes as small as possible. Thus, the
effective impedance of conventional electrical energy distribution systems is very small, and this small effective impedance is
not considered when effecting power factor correction. In other
words, conjugate matching between a load, corrected or not,
and the impedance looking back into the connected transmission line is not considered when desiring to minimize losses in
the transmission line.
C. Transformer Considerations
When selecting a transformer or selecting the source and load
impedance for a given transformer, such as a step-down transformer to a speaker, the impedances are sometimes selected for
lowest iron and copper losses in the transformer. The impedances are often selected so that under load the iron losses are
about equal to the wire copper losses and thus tend to optimize
efficiency of the transformer. Thus, the transformer efficiency,
the ratio of the power delivered to the load to the power delivered into the primary of the transformer, might have a peak
value at some “matched” load [1]. In other cases involving multiple transformers or split transformers, the actual loads might
be selected to split the total available power. The impedances
involved more often than not have to do with optimizing efficiency and, in spite of the “matched” language often used, do
not involve conjugate matching.
III. REFLECTIONS
When an electromagnetic wave is incident on a load, whether
the load is a metallic shield or discrete resistor, a certain percentage of the wave is reflected from the load, and a certain percentage is delivered to the load. In many cases, one wishes to
reduce or minimize the reflections from the load (or source). In
high-speed circuits, reflections can be a source of oscillation and
electrical noise. For some receivers, reflections generate undesirable echoes. The overshoot generated by reflections can also
cause a device to malfunction by producing a voltage exceeding
the load’s voltage rating.
At very high frequencies (VHFs) and higher, reflections on a
transmission line between a low-noise receiver and its antenna
give rise to another issue. The losses in the transmission line increase causing a reduction in the signal delivered to the receiver
and an increase in the noise delivered since the transmission line
is “warm.” In well-designed systems, such undesirable effects
are small.
A measure of the reflections of the voltage and electric field
waves on a line is expressed by the reflection coefficient defined
as
B. Power Factor Correction
In power factor correction, elimination or reduction of the
reactive component of a device, such as a motor, is desirable.
The reactive component of the device’s impedance is canceled
or reduced by adding the appropriate size reactive component of
opposite sign. For example, a power-factor-correcting capacitor
is added across an inductive motor. When the power factor is
one, the current and voltage are in phase since the net reactance
is zero, and the magnitude of the current is minimum. A power
where
is the impedance of the load and
the impedance
of the medium in which the incident wave is traveling. For
transmission lines,
is the characteristic or surge impedance.
For plane waves,
is the intrinsic (and some cases wave)
impedance of the medium. The previous expression leads to the
conclusion that to minimize reflections, the impedance of the
load should be similar or equal to the impedance of the line (or
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MCLAUGHLIN AND KAISER: MAXIMUM POWER TRANSFER THEOREM
medium) leading to the load. The reflection coefficient is zero
when
, which is referred to as an impedance match.
An impedance match is not a conjugate match (unless the load
is purely resistive) [1], [13].
For high-frequency modeling, parameters are probably the
simplest set of parameters to manipulate and measure. Unlike
many other parameters that can be used to model or describe a
network, measurement of parameters does not require that the
input and output of the network be short circuited and open circuited. With parameters, the source and load impedances are
usually set equal to a resistance of 50 . The classic definitions
for each of the parameters are obtained by setting the or
parameters equal to zero, which is accomplished by providing
a source or load impedance of 50 and does not involve short
circuiting (or open circuiting) the input and output ports. Here,
the matching is not performed for maximum power transfer but
to obtain more easily the parameters. The performance of filters, and their parameters, is often measured using a network
analyzer that uses a source and load impedance of 50 , and
maximum power transfer is not involved.
When tapping onto a transmission line, frequently having the
input impedance of the tapping device be equal to the line’s
characteristic impedance is not desirable. Although the final or
last load along the line might be impedance matched to the line,
the tapping loads often have an impedance much greater than the
line’s impedance. Of course, the distance between the tap and
the high-impedance tapping device must be electrically short in
order to have a high-impedance tap. Again, maximum power
transfer is not involved.
IV. LOADING AND VOLTAGE REGULATING
Output stages of most amplifiers, and some buffered transducers such as accelerometers, have a low output impedance
(i.e., a low Thévenin impedance at its output), such as 1 . The
output stages are called “drivers” in this discussion because they
drive succeeding stages. Rarely is the input impedance of an amplifier that is connected to a driver selected to be low and equal to
the driver’s output impedance. Usually, the input impedance of
. Connecting a driver
a driven amplifier is high, such as 100
with a low output impedance to an amplifier or receiver with
a high input impedance is occasionally referred to as “voltage
matching [14].” The largest signal voltage is transferred to the
input of an amplifier when its input impedance is much greater
than the driver’s output impedance. When an electrically short
transmission line has a load that is not a very large or a very
small ratio of , the effects of reflections are negligible, and
voltage matching is commonly used, which is far from conjugate matching [1].
An amplifier with a low input impedance could excessively
load down the driver connected to the amplifier. In other words,
the input current demand by the amplifier could exceed what the
driver can deliver. Unacceptable distortion could be one consequence of this loading. Sometimes transformers are used between the driver and its load to transform the load impedance
to a level acceptable to the driver [15]. This impedance transforming is not used for maximum power purposes.
Related to the concept of loading is voltage regulation associated with power supplies. Often the load impedance is selected
to be much greater than the Thévenin impedance of the power
253
supply. Changes in the load resistance are then less noticeable.
If the load resistance changes by , then the change in the load
voltage is
This change in voltage decreases with increasing
.
V. SIGNAL-TO-NOISE RATIO
In some applications, one needs to increase or decrease the
impedance of the driver (which might be an antenna) by the use
network. This
of a transformer or an inductor–capacitor
transformation or adjustment of the apparent driver’s output
impedance might be to satisfy a signal-to-noise requirement of
the receiver or amplifier connected to the driver. Some amplifier data sheets provide information relating the signal-to-noise
ratio [(SNR) or noise figure] to the impedance seen by the amplifier’s input. Often, an optimal expected input impedance is
required for the amplifier to produce the greatest SNR. Experience at VHFs and higher suggests that the optimal apparent
source impedance (the driver’s apparent impedance) for best
SNR does not correspond to maximum power gain for the signal
and is not directly related to the input impedance of the amplifier. Adjustment of the input impedance seen by an amplifier
to optimize the SNR is called noise matching [16], [17]. With
modern, low-noise active devices, the difference between noise
matching and power matching is small. SNR can also be affected
by reflections at an input. Reflections will result when other than
an impedance match is effected.
VI. BALANCING AND DISCHARGING
For some devices such as instrumentation amplifiers, similar source impedances over a range of values are needed for
proper balancing. Source impedances are sometimes adjusted,
not to provide a better match for maximum power transfer, but
to obtain a better balance and better common-mode rejection.
For real differential amplifiers the common-mode rejection ratio
is not infinite, and imbalances will affect the amplification of
common-mode signals. Since no system is perfectly balanced,
some common-mode signals are present and will be amplified
by a real differential amplifier. Often, the output impedance of
drivers is low, and the stated input impedance of differential receivers is high. The input bias currents of the active differential
device will be affected by the source impedances leading to the
inputs of the differential device.
Sometimes resistors are seen across devices for electrical discharge purposes even though one might claim they are present
for matching purposes. For example, in some situations a direct
current (dc) path to ground is required at the input(s) of a device
for proper operation. The input impedance of many amplifiers is
usually very high, and a portion of this input impedance is a result of nonzero input capacitance. Even high-impedance inputs
draw some current. The dc component of this current will place
a charge on input capacitances and could eventually charge the
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IEEE TRANSACTIONS ON EDUCATION, VOL. 50, NO. 3, AUGUST 2007
capacitor to some unacceptable level. A path, often to a local reference, is required to discharge the input capacitance. Resistors
from the inputs to the local reference are used for discharging
the input capacitance.
Cathode-ray tubes and microwave ovens are two examples of
products that often contain large capacitors. Resistors are placed
across (in parallel with) high energy-storage capacitors to dissipate the charge on them after power has been turned-off to the
product. The actual value selected for the “bleeder” resistance
has nothing to do with maximum power transfer but effects the
rate of discharge desired, via the time constant(s). “Soft” ground
resistors are also used to dissipate electrostatic charge buildup
[1]. On windy days prior to a storm, for example, static charge
can buildup on antennas. This charge can transfer to equipment
connected to the antenna. To help dissipate this charge, large resistances are used unless arc discharge devices are employed.
For example, sometimes a resistor is connected between the
neutral side of the primary of the input transformer of a device
and the grounded chassis. Soft ground resistors are also used on
wrist and other types of electrostatic discharge grounding straps
to help limit the current through the user in case of an accidental
fault to a high-voltage conductor.
VII. FILTER ISSUES
There are several situations where filter terminations might be
adjusted to give the appearance that conjugate matching is being
performed. First, when designing ladder filters, the termination
impedances to the input and output of the filter are sometimes
selected to be equal to the conjugate of the impedance looking
into the input and output of the filter, respectively [18]. However,
even if such termination impedances are actually selected in
practice, one would not select them for maximum power transfer
but to minimize the sensitivity of the filter’s output power to
component variation.
Second, resistors, referred to as swamping resistors, are used
across tuned circuits to broaden their response (i.e., decrease
their and increase their losses) not to perform some matching
function. For example, placing a resistor across a parallel
circuit will lower the circuit’s and increase its bandwidth. For
double-tuned transformers, adding a resistor in shunt with the
transformer will also affect the response of the transformer.
Third, all filters, including suppressors, operate by absorbing,
shunting, and/or reflecting. A transient voltage suppressor functions by introducing a low impedance in shunt with the line or
a high impedance in series with the line. Although the maximum voltage, current, power, and energy ratings of the device
are considered in the selection of a suppressor, maximum power
transfer is not.
VIII. LINEARITY AND DYNAMIC RANGE
The output (and input) impedance of many nonlinear devices
can vary with voltage swing [19]–[21]. For large voltage swings,
a single, effective output or input impedance for a nonlinear device might be difficult to define. The impedance that a nonlinear
device might like to see may correspond to some sort of “average” value. This average impedance may have little to do with
a conjugate or an impedance match.
Imagine that the dynamic output resistance of a driver actually varies from 2 to 100
while its effective or average
stated value is given as 10 . If, for instance, a 10 resistor
is placed in parallel with the output, the net impedance variation will be reduced to about 1.7–9.1 . (The resulting smaller
voltage variation might be more acceptable to the next stage
of the circuit.) This resistor is not added for maximum power
transfer (or impedance matching) but to reduce the percentage
change in the output resistance, which will reduce the output
gain, reduce the variation in delivered voltage, and possibly reduce undesirable oscillations and nonlinearities. This resistor is
referred to as a “swamping” resistor. The most obvious cost of
this shunt resistor is the additional power loss associated with its
use. Swamping resistors have been used to improve the linearity
of class-B, high-power, vacuum-tube amplifiers [22]. Without
the use of swamping resistors, the source driving the power amplifier could be subjected to impedance variations that result in
signal distortion or even injury to the source. Particularly during
modulation peaks that are within the capability of the power
amplifier, without the swamping resistors the resultant extreme
values of input impedance can cause distortion of the signal even
when using a driving source with a small output impedance.
The dynamic range of a signal can also be affected by the
value selected for an impedance. For example, if a transmission
line is terminated with a high-impedance load and impedance
matched at the source rather than at the load, one-half of the
driver’s voltage will appear along the line until the reflected
signal from the (high-) impedance load appears [1]. This
matching source impedance obviously affects the dynamic
range for a period of time.
IX. ADDITIONAL EXAMPLES
The optimization of source and load impedances associated
with Gilbert cell mixers is an example of the desirability of
nonconjugate matching. If the optimizing of conversion gain
is the only objective, references indicate the use of conjugate
matching at the input and output of a mixer [23], [24]. However,
such conjugate matching will have a detrimental effect on noise
figure, dynamic range, and intermodulation performance [25].
Rare is the mixer application where a trade of some conversion
gain for enhanced noise figure or dynamic range is not desirable.
If a transmitter with a small equivalent resistance is connected
to an antenna, in many cases selecting an antenna with a small
equivalent resistance would be unreasonable for the purpose of
obtaining an impedance match or a conjugate match (unless
other constraints, such as physical size, require the use of an
electrically small antenna, which is likely to have a small resistance). If the antenna could be electrically large with a resultant moderate radiation resistance as compared to the antenna’s
ohmic losses, then the antenna system is likely to be efficient.
Thus, a moderate to large input resistance for the antenna is a
frequent sought-after goal without consideration of maximum
power transfer.
Electrically-short antennas are frequently high- devices
with a large reactive impedance. When such antennas are
used for transmitting, connecting a reactive load across, or in
series with, the antenna’s impedance with the external load’s
reactance being of an opposite sign to that of the antenna’s
reactance is common. This reactive load is not added for maximum power transfer but is added to cancel the reactance of the
antenna’s impedance. A large reactance is likely to be unaccept-
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MCLAUGHLIN AND KAISER: MAXIMUM POWER TRANSFER THEOREM
able to the transmitter. Or, the reactance of the antenna, when
transformed back to the transmitter through a transmission
line, might be unacceptable. Modifying the antenna’s input
impedance to be purely resistive and equal to the transmission
line’s characteristic impedance will allow the transmitter to see
relatively constant impedance over a wide frequency range. In
this particular case, a conjugate match happens to exist at the
transmitting antenna. If maximum power transfer is argued as
the purpose for adding reactance to the antenna, the case of a
lossy or partially reactive characteristic line impedance can be
presented that requires an impedance match.
The maximum power transfer theorem might be used in the
case of certain receiving systems. In the case of a short rod antenna to be used at medium frequency or the lower part of HF
(where the effective noise factor is almost always determined
outside of the receiver), the equivalent circuit of the antenna is
comprising a small resistance and a large
an impedance
proportional
capacitive reactance in series with a voltage
to the magnitude of the incident electric field and the length of
of 0.5
the antenna. Suppose that a 1 m long antenna has
in series with 10 pF and that the first receiver stage has an input
impedance of 1800 . Clearly, to minimize the necessary overall
gain of the receiver’s active stages, one would consider using
conjugate matching. At 1 MHz, a series ideal inductor of just
over 2.5 mH and an ideal transformer with a 1:60 turns ratio is
to the first stage. The dilemmas are apneeded to deliver
parent. Even with an inductor having a of 200, about 80 of
ohmic loss resistance is added in series, the ideal turns ratio is
is delivered to the first stage, and
about 4.7, only a bit over
(worst of all) a change in frequency would be disastrous. With
real transformers, the situation is even worse. While one could
use conjugate matching in this situation, redesigning the first
stage to have as large an input impedance as possible (voltage
matching) is a better choice.
Although the values selected for a source or load impedance
might appear as an attempt to match, sometimes the impedances
are also adjusted to change the degree of inductive and capacitive crosstalk between circuits. In some special cases, the inductive coupling can be reduced by increasing the impedances
while the capacitive coupling can be reduced by decreasing the
impedances [1].
X. SUMMARY
This paper has demonstrated that multiple factors must be
considered when selecting or adjusting source and load impedances other than maximizing power transfer. Depending on the
application, this paper has illustrated that factors such as efficiency, reflections, dynamic range, SNR, and loading should
be considered. A religious aura emanates from the maximum
power transfer theorem. The authors have presented the heresy
of contending that the maximum power transfer theorem’s importance is mostly limited to the academic arena, while working
engineers will solve problems by attending to the needs of the
particular task.
255
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[4] W. Bruene, “The elusive conjugate match,” Commun. Quart., pp.
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[5] W. Bruene, “RF power amplifiers and the conjugate match,” QST, pp.
31–32, Nov. 1991.
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Switched Capacitor. Englewood Cliffs, NJ: Prentice-Hall, 1981.
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James C. McLaughlin received the B.S. degree in mathematics from the University of Michigan, Ann Arbor, the M.Sc. degree in electrical engineering from
The Ohio State University, Columbus, and the J.D. degree from the Thomas M.
Cooley Law School, Lansing, MI.
He is currently a Professor in the Electrical and Computer Engineering
Department at Kettering University (formerly General Motors Institute), Flint,
MI. He previously worked at the National Radio Astronomy Observatory
and studied at Manchester University, Manchester, U.K.. His areas of interest
are antennas including antenna structures, radio propagation, electronics and
electromagnetic compatibility/radio frequency interference.
Prof. McLaughlin is a Professional Engineer in Michigan and a Patent Attorney.
Kenneth L. Kaiser received the B.S.E.E., M.S.E.E., and Ph.D. degrees from
Purdue University, West Lafayette, IN.
He is currently a Professor in the Electrical and Computer Engineering Department at Kettering University (formerly General Motors Institute), Flint, MI.
While gaining a theoretical background in a number of fields in electrical engineering, he has obtained additional inspiration and practical experience working
in several nonacademic positions. His areas of research focus on topics of personal and industrial interest including effective teaching methods and writing a
book on electromagnetic compatibility.
Dr. Kaiser is a Professional Engineer in Michigan.
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