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ADDITIONAL MAXIMUM VOLTAGE AND MAXIMUM POWER TRANSFER CONDITIONS [Technical letter] Fayez Mohammed EL-Sadik http://fayezmohammed.googlepages.com/ Department of Electrical & Electronic Engineering, University of Khartoum, Sudan ABSTRACT Results of a generalized load impedance statement that will include as special cases the conditions for maximum power transfer with conjugate impedance matching and the maximum load voltage condition are presented. These are based on an analytic solution to the problem of stability limits of radial transmission circuits with inter-nodal constraints as shown in Figure 1 where maximum received power is defined by the resulting magnitudes of sending and receiving end voltages as the angular separation between the two variables approaches the transmission angle. Figure 1 In this Letter, a presentation in terms of graphical demonstrations of the results of the generalized load impedance function is given. This leads to proofs of the well-known maximum power and maximum voltage conditions where the terminal and all other circuit voltages will come out as a result. 1. RESULTS OF A GENERALIZED LOAD IMPEDANCE STATEMENT In a previous paper [1], the impedance conditions for Brainerd’s hypothesis of maximum load voltage and Jacobi’s theorem for maximum power transfer with complex conjugate impedance matching were verified as two special cases of a continuous P-V stability boundary function that has been based on a transducer circuit model with an internal node voltage constraint. This has, in a fulcrum-cantilever type of action, imposed restrictions on the magnitude of the source voltage of any given transducer circuit as the conditions are identified through two different equilibrium states. • In this Letter, the load impedances for both conditions will be derived as special states of a generalized power-maximizing impedance function where transducer circuit source voltage is independently Sudan Engineering Society JOURNAL, January 2007, Volume 53 No.48 65 retained. As a result, a given transmission circuit can be made to deliver maximum powers less than that which can be obtained from the upper limit of conjugate matching; with the extreme lower limit of power maximums evolving in Brainerd’s hypothesis for the maximum load voltage condition. • Received power The results to be presented will show that generality of the powermaximizing load impedance statement is derived from an inherent property of concurrent load voltage maximization. This will in addition introduce a new cut-off impedance condition for the termination of voltage and hence power transfer of any given transducer circuit. 2. CONDITION FOR CONCURRENT MAXIMUM VOLTAGE AND MAXIMUM POWER • Terminal voltage ZL Figure 3 • Consider the case of a total series impedance ZS = 1.0 + j√3.0 p.u at a supply voltage ES = 1.0 /_0o p.u as shown in Figure 2. Figure 2 • • When conjugate matched, the available transfer capability of the circuit gives Pmax = 0.25 p.u and this occurs at a receiving end voltage VR = 1.0 p.u and a load impedance of magnitude ZL = 2.0 p.u Now consider the terminal voltage and power curves shown in Figure 3 where it can be seen that: 66 3. a. Maximum voltage of 1.0 p.u is synonymous with the maximum power of 0.25 p.u ; both occurring at the load impedance magnitude 2.0 p.u b. The two curves will terminate at a specific impedance value of 4.0 p.u. The curves are plotted as demonstration of the existence of a capacitive load impedance function ZL = R (ZL) - j X (ZL) that will extract powers with a maximum of circuitmatched value which is concurrent with the corresponding maximum value of terminal voltage. In addition, the simultaneous discontinuities of voltages and powers shown are indicative of the existence of a cutoff load impedance value above which no transfer of voltage and hence power will take place. CONDITIONS FOR REDUCED POWER MAXIMUMS • In its generalized form, the power-maximizing load impedance statement is in fact both voltage and power-dependent i.e., ZL = R (ZL , pv) - j X (ZL , pv), where pv is the defining parameter for those terminal conditions. Sudan Engineering Society JOURNAL, January 2007, Volume 53 No.48 • The sample set of power curves in Figure (4.a) and the associated load voltage curves in Figure (4.b) showing the influence of this parameter for the present case study reveals the natural tendency of terminal voltage increase with decreased load powers as well as the associated differences in the transfer cut-off impedance limits. Notice that no such limit is attached to the lower curve-c powers within the range of ZL considered. 0.25 a 0.2 0.15 b 0.1 0.05 may however be conducted. This entails finding the load impedance elements that will generate associated power and voltage curves while observing the occurrence of the maximums of these curves at the circuit transmission angle. 4. PROOF OF BRAINERD’S MAXIMUM LOAD VOLTAGE CONDITION • In addition to its verification of the upper power point of conjugate matching, a significant feature in support of the continuous powerand voltage--maximizing load impedance function is provided by the lower limiting zero-state of the voltage-producing power curve (a) in Figure 5; as this continuous state evolves in Brainerd’s impedance condition for maximum voltage. c 2 0 0 5 10 ZL ZL 15 1.8 1.6 Figure 4a: Load powers 1.4 1.2 2 1 1.8 0.8 c 1.6 0.6 1.4 b 1.2 0.2 1 0 0 c 0.8 0.6 0.2 0 5 10 ZLZL 15 Figure 4b: Corresponding load voltages • 0.5 1 1.5 2 2.5 3 3.5 ZL 4 Figure 5: Impedance for maximum voltage 0.4 0 (a) lower limiting zero-power states 0.4 While the complete derivation procedure of the analytic load impedance function is left for a full publication, a numerical solution for the behaviors of the voltage- and power- producing elements of ZL As stated in his paper published in 1933 [2], while the conditions for current and power maximums are commonplace, specific problems of voltage maximization are solved by differentiation and, according to the author’s knowledge, the condition for maximum voltage has evolved in a class discussion involving logical reasoning as to the nature of physical loads followed by differentiation. Sudan Engineering Society JOURNAL, January 2007, Volume 53 No.48 67 • • Bernard Miller, a senior student in the Moore School, was in particular credited for the discussion leading to the hypotheses which states that maximum voltage at the terminals of a passive linear transducer of constant source voltage is obtained when the load impedance is purely capacitive and has the magnitude ZS2/XS. This can be seen as concurring with the impedance result obtained from Figure 5 giving a corresponding maximum voltage of 2.0 p.u for the case considered. 68 References 1. EL-Sadik, F.M.: “Voltage Constraints for the Maximum Power Transfer Theorem & Brainerd’s Resonance Voltage Impedance Condition ” ; Sudan Engineering Society Journal, Volume 52, Number 45, January 2006. 2. Brainerd, J.G.: “Some General Resonance Relations and A Discussion of Thevenin’s Theorem”; Proceedings of the Institute of Radio Engineers, Volume 21, Number 7, July 1933. Sudan Engineering Society JOURNAL, January 2007, Volume 53 No.48