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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 π π 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 π πΏ is made equal to the fixed source resistance π π . For this same fixed sourceπ π , if the adjustable load resistance π πΏ is made greater than or equal to 10 times the fixed π π , 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 π πΏ is greater than or equal to 10 times the adjustable π π. If the same value π πΏ appears in either of the two maximum-power transfer cases, then the second case has 4 times more power delivered to π πΏ 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 [This is a matter of 1840 technology being superseded by 1880 technology.]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 ππ in series with its internal resistance π π . Further, in series with this source resistance π π is the load resistance π πΏ with voltage ππΏ . 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, ππΏ = π πΏ βπ π π + π πΏ π (1) we make two definitions: ππ,0 = and 1 ππ 2 , π π (2) THE THREE METHODS FOR TRANSFER OF DC POWER TO A LOAD: Impedance Matching and Impedance Bridging ππ 2 , π πΏ ππΏ,0 = (3) Equation (2) is the power delivered to a fixed source resistor π π when the load resistor π πΏ has zero resistance. Equation (3) is the power delivered to a fixed load resistor π πΏ when the source resistor π π has zero resistance. The power delivered to a load resistor π πΏ is given by the expression ππΏ = ππΏ 2 βπ πΏ (4) Substitution of Equation (1) into Equation (4) yields ππΏ = ππ 2 β π πΏ . (π π + π πΏ )2 (5) Note Equation (5) has dependent variable ππΏ as a function of two independent variables π π and π πΏ . 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 π πΏ is given by: ππΏ 1 = , 1 ππ,0 +2+ π π (6) where π β‘ π πΏ . π π (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 π πΏ , because both the source voltage ππ and the source resistance of resistor π π are held fixed. Note The process of normalization adds nothing but convenience in analysis. Similarly, when the load resistor π πΏ is held fixed, a variable source resistor π π yields ππΏ 1 = πβ . 1 ππΏ,0 π +2+ π 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 ππ is the (fixed) source, open-circuit voltage (or the Thevenin voltage or the Emf) When the source resistor π π is held fixed and the load resistor π πΏ is varied, the bluecolored curve results. When the load resistor π πΏ is held fixed and the source resistor π π 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 5-10Kβ¦. Speakers were typically having an 8 β¦ impedance. Given that the source impedance was fixed, it was necessary to adjust the impedance of the speaker via βreflectedβ impedance presented by an output transformer. Impedance Matching occurred when the abscissa π = π πΏ βπ π 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 ππ β ππΏ , when π πΏ β₯ 10 β π π . A curious green curve further sits in Figure 1. The power-transfer efficiency π, ratio of load power ππΏ to the total power (ππ + ππΏ ) supplied by the source, can be expressed as 3 THE THREE METHODS FOR TRANSFER OF DC POWER TO A LOAD: Impedance Matching and Impedance Bridging π = 1 1 1 +π . (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 π πΏ exceeds 10 times the value of π π . According to Figure 1, π β₯ 10 corresponds to power-transfer efficiency π β 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 π πΏ exceeds 10 times the value of π π . This is an instance of voltage ππΏ being nearly equal to the value ππ , but the current is being minimized. Power may also be expressed as the product of voltage and current; so now the voltage ππΏ approaches the constant source value ππ whereas the loop current approaches the asymptotic value of zero at infinite π. 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 β when π πΏ = π π maximizes power transfer; and the blue curve is also Impedance (Voltage) Bridging curve that minimizes power transfer β when π πΏ β₯ 10 β π π . The red curve is the Impedance (Voltage) Bridging Curve that maximizes power transfer β when π πΏ β₯ 10 β π π . APPLICATIONS It can be said that audio equipment manufacturers adjust (or designed) preamp input impedances to be about 2 β 3 πβ¦ 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 β¦. 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 πβ¦ and a magnetic cartridge has output impedance of about 50 πβ¦. To deal with these exceptional cases, one uses a DI Box to convert the high output impedance of the source to a lower value. 4 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 βnewerβ 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. 5