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THE THREE METHODS FOR TRANSFER OF DC POWER TO A LOAD:
Impedance Matching and Impedance Bridging
INTRODUCTION
In the first place, it is not commonly appreciated that there are three (3) distinctly different ways to
transfer DC power to a load. The reason there are three different methods for transferring DC power is
wholly mathematical in nature. All three methods can be concisely and pictorially be related as shown
here.
If one regards all signals as variations of power delivery, then confusing discussions about how modernday amplifiers, with nearly zero output impedance, drive 8-Ohm speakers can vanish. To say, for
example, that such speakers are driven by Voltage Bridging is a somewhat incomplete story!
THE THREE METHODS OF DC POWER TRANSFER
If the source resistance 𝑅𝑆 is held fixed, while the load resistance is adjusted, then two ways for
transferring power appear. Maximum power transfer occurs if the adjustable load resistance 𝑅𝐿 is made
equal to the fixed source resistance 𝑅𝑆 . For this same fixed source𝑅𝑆 , if the adjustable load resistance
𝑅𝐿 is made greater than or equal to 10 times the fixed 𝑅𝑆 , then a minimum-current-maximum-voltage
power transfer arrangement occurs.
If the load resistance is held fixed, while the source resistance is adjusted, then only one way for
transferring power appears. Maximum power transfer again occurs, if the fixed load resistance 𝑅𝐿 is
greater than or equal to 10 times the adjustable 𝑅𝑆.
If the same value 𝑅𝐿 appears in either of the two maximum-power transfer cases, then
the second case has 4 times more power delivered to 𝑅𝐿 that does the first case! [This
is a matter of 1840 technology being superseded by 1880 technology.]
ANALYSIS OF THE THREE METHODS OF DC POWER TRANSFER
For analysis, the voltage source is replaced by its Thevenin Equivalent: an ideal voltage source 𝑉𝑆 in
series with its internal resistance 𝑅𝑆 . Further, in series with this source resistance 𝑅𝑆 is the load
resistance 𝑅𝐿 with voltage 𝑉𝐿 .
Here the author reveals a powerful way to inter-compare these three methods, by first performing some
minor algebraic analysis.
Along with the usual result,
𝑉𝐿 =
𝑅𝐿
∙𝑉
𝑅𝑆 + 𝑅𝐿 𝑆
(1)
we make two definitions:
𝑃𝑆,0 =
and
1
𝑉𝑆 2
,
𝑅𝑆
(2)
THE THREE METHODS FOR TRANSFER OF DC POWER TO A LOAD:
Impedance Matching and Impedance Bridging
𝑉𝑆 2
,
𝑅𝐿
𝑃𝐿,0 =
(3)
Equation (2) is the power delivered to a fixed source resistor 𝑅𝑆 when the load resistor 𝑅𝐿 has
zero resistance.
Equation (3) is the power delivered to a fixed load resistor 𝑅𝐿 when the source resistor 𝑅𝑆 has
zero resistance.
The power delivered to a load resistor 𝑅𝐿 is given by the expression
𝑃𝐿 = 𝑉𝐿 2 ⁄𝑅𝐿
(4)
Substitution of Equation (1) into Equation (4) yields
𝑃𝐿 = 𝑉𝑆 2 ∙
𝑅𝐿
.
(𝑅𝑆 + 𝑅𝐿 )2
(5)
Note Equation (5) has dependent variable 𝑃𝐿 as a function of two independent variables
𝑅𝑆 and 𝑅𝐿 . In our discussions, the power depends on one of the independent variables
being held constant (or fixed) while the other independent variable is varied (or adjusted).
When Equation (5) is normalized by the fixed power of Equation (2), one finds that the scaled or
normalized power obtained by adjusting the value of the load resistor 𝑅𝐿 is given by:
𝑃𝐿
1
=
,
1
𝑃𝑆,0
+2+ 𝜌
𝜌
(6)
where
𝜌 ≡
𝑅𝐿
.
𝑅𝑆
(7)
This normalization changes nothing insofar as the functionality of Equation (5) is concerned. In both
Equation (5) and Equation (6), the only variable is the load resistor 𝑅𝐿 , because both the source voltage
𝑉𝑆 and the source resistance of resistor 𝑅𝑆 are held fixed.
Note The process of normalization adds nothing but convenience in analysis.
Similarly, when the load resistor 𝑅𝐿 is held fixed, a variable source resistor 𝑅𝑆 yields
𝑃𝐿
1
= 𝜌∙
.
1
𝑃𝐿,0
𝜌 +2+ 𝜌
2
(8)
THE THREE METHODS FOR TRANSFER OF DC POWER TO A LOAD:
Impedance Matching and Impedance Bridging
The two different fixed/variable relationships given by Equations (6) and (8) are plotted in the following
figure as the blue and red curves:
Figure 1. Three Extrema for the Blue and Red Curves
Load Power/( VS2/RS) [Fixed RS ],
Load Power/(VS2/RL) [Fixed RL],
Power-Transfer Efficiency [green]
EXTREMA FOR DC POWER TRANSFER TO A LOAD
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
7
8
9
10
RL/RS
Notes 𝑉𝑆 is the (fixed) source, open-circuit voltage (or the Thevenin voltage or the Emf)
When the source resistor 𝑅𝑆 is held fixed and the load resistor 𝑅𝐿 is varied, the bluecolored curve results.
When the load resistor 𝑅𝐿 is held fixed and the source resistor 𝑅𝑆 is varied, the redcolored curve results.
An example of the blue curve once appeared in virtually every American home. In the pre-1960 days,
before transistors really hit the marketplace, an output tube had a fixed (source) impedance of
something like 5-10KΩ. Speakers were typically having an 8 Ω impedance. Given that the source
impedance was fixed, it was necessary to adjust the impedance of the speaker via “reflected”
impedance presented by an output transformer. Impedance Matching occurred when the abscissa 𝜌 =
𝑅𝐿 ⁄𝑅𝑆 of the blue curve equaled unity.
Actually, the adjusted impedance was not exactly matched to the output tube, to minimize
distortion.
A ready example of the red-colored curve occurs in AC power distribution systems. Therein, it is
desirable to minimize the source (generator) resistance, while the load is held (somewhat) fixed. This is
one of the two kinds of Impedance (Voltage) Bridging. Equation (1) that tells us 𝑉𝑆 ≈ 𝑉𝐿 , when 𝑅𝐿 ≥
10 ∙ 𝑅𝑆 .
A curious green curve further sits in Figure 1. The power-transfer efficiency 𝜖, ratio of load power 𝑃𝐿 to
the total power (𝑃𝑆 + 𝑃𝐿 ) supplied by the source, can be expressed as
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THE THREE METHODS FOR TRANSFER OF DC POWER TO A LOAD:
Impedance Matching and Impedance Bridging
𝜀 =
1
1
1 +𝜌
.
(9)
Note that the maximum of the blue curve (Impedance Matching) had a 50% efficiency, whereas the red
curve approaches a 100% efficiency.
Strikingly, we find that the green curve is the sum of the blue and red curves! [The reader can satisfy
himself of this claim by adding Equations (6) and (8).] Not only did the peculiar normalization achieve
dimensionless expressions for the two load power results, but this normalization scheme revealed this
heretofore latent identity.
RELATION OF IMPEDANCE (or VOLTAGE) BRIDGING TO BOTH THE RED AND BLUE CURVES
The objective in Impedance (Voltage) Bridging is to match, nearly, a load (destination) voltage to a
source voltage. By inspection of Equation (1), this goal is roughly met if the resistance of 𝑅𝐿 exceeds 10
times the value of 𝑅𝑆 . According to Figure 1, 𝜌 ≥ 10 corresponds to power-transfer efficiency 𝜖 ≈ 1
for both the blue curve and the red curve.
We already touched upon the topic of Impedance (Voltage) Bridging in the context of the red curve. In
that case of AC-power distribution, the power-transfer approached a maximum. The maximum value is
the asymptotic value at infinity.
Yet, there is another class of cases to consider for Impedance (Voltage) Bridging. Inspection of Figure 1
shows that the blue curve approaches a minimum when 𝑅𝐿 exceeds 10 times the value of 𝑅𝑆 . This is an
instance of voltage 𝑉𝐿 being nearly equal to the value 𝑉𝑆 , but the current is being minimized. Power may
also be expressed as the product of voltage and current; so now the voltage 𝑉𝐿 approaches the constant
source value 𝑉𝑆 whereas the loop current approaches the asymptotic value of zero at infinite 𝜌. Thus
the power is minimized in this other version of Impedance (Voltage) Bridging.
SUMMARY
The following associations are visually made, according to Figure 1: the blue curve is the Impedance
Matching curve – when 𝑅𝐿 = 𝑅𝑆 maximizes power transfer; and the blue curve is also Impedance
(Voltage) Bridging curve that minimizes power transfer – when 𝑅𝐿 ≥ 10 ∙ 𝑅𝑆 . The red curve is the
Impedance (Voltage) Bridging Curve that maximizes power transfer – when 𝑅𝐿 ≥ 10 ∙ 𝑅𝑆 .
APPLICATIONS
It can be said that audio equipment manufacturers adjust (or designed) preamp input impedances to be
about 2 − 3 𝑘Ω these days, to maintain low (thermal) noise in the input resistor. To be consistent with
Equation (1) most modern sources have fixed (or designed) output impedances of about 150 Ω.
Accordingly, minimal current is drawn by the preamplifier (or load resistor). Exceptions to this low
output impedance are found with guitar pickups, nearly-extinct magnetic phono cartridges and high-Z
microphones. An electric guitar has output impedance of about 10 𝑘Ω and a magnetic cartridge has
output impedance of about 50 𝑘Ω. To deal with these exceptional cases, one uses a DI Box to convert
the high output impedance of the source to a lower value.
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THE THREE METHODS FOR TRANSFER OF DC POWER TO A LOAD:
Impedance Matching and Impedance Bridging
Although Impedance Bridging has become common place in the audio and recording studio, Impedance
Matching remains important with vintage sound equipment using vacuum tubes and when avoiding
transmission line reflections.
TEACHING PRINCIPLE
There was never a need to separate the “newer” topic of Impedance (Voltage) Bridging from the more
basic, older discussions of power transfer, as all transferred signals (from source to load) involve a
transfer of power.
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