Voltage Constraints for the Maximum Power Transfer Theorem & Brainerd’s Resonance Voltage Impedance Condition * Fayez Mohammed EL-Sadik ABSTRACT It is shown that Brainerd’s resonance voltage impedance relation and the well-known maximum power transfer impedance conditions of conjugate matching are two states of unstable equilibrium with the terminal voltages of a matched linear transducer circuit directly related to that of Brainerd’s. The relationship is based on stability indices derived from internal impedance and voltage constraints with the result that additional singularity states that may apply to radial power and communication links and for which no reference can be found in the literature are observed. These include the conditions for maximum source voltage; voltage profiles with inter-nodal constraints and the source and load voltage magnitudes evolving in states of zero load powers and a maximum of the maximums of transmitted powers. Keywords: maximum power, maximum voltage, radial power links, equilibrium states 1. INTRODUCTION The theorems for the maximum load voltage and maximum power transfer of systems that can be represented by the circuit of Figure 1 were formulated on the basis of fixed magnitudes of either source voltage E or terminal voltage V according to the application for which the circuit is intended. In electronic engineering where E is normally fixed, the two theorems evolve, as a result, in the impedance variations of a passive load circuit with the occurrence of maximum power of E2/4R, when ZL=Z*sand, according to Brainerd [1], maximum voltage when: Figure 1: Series Representation of a Voltage ZL = -jZs2/Xs. Notice that the two conditions share the occurrence of a common δ between E and the resulting V that is identical to the transmission angle γ = tan-1 Xs/Rs, with no reference made as to any constraints on the magnitude of E. On the other hand, in power systems engineering where V, e.g., can be specified such as to be found in radial power links, the following well-known expression for the receiving end power derived with Vreference gives δc = γ = tan-1 Xs/Rs, as the first maximum with no upper limit to this maximum set by any other magnitude of E thereafter: VE V2 P= sin(δ + α ) − sin α Zs Zs , α = tan Link / Terminated Transducer Circuit * −1 (1) Rs Xs . Because of the high powers involved, the lack of a clear analytic solution to the capability limits of power systems components has Department of Electrical Engineering, University of Khartoum, Sudan; fmsadik@yahoo.com Sudan Engineering Society JOURNAL, January 2006, Volume 52 No.45 57 Figure 2: Transmission Circuit with Inter-Voltage and Sectional Impedances Constrains prompted criteria for practical stability limits, e.g., thermal limits of transmission lines and the rotors of synchronous machines. In this paper, evidence of the existence of a current equilibrium state imposing constraints on the source and load voltages of a linear transducer circuit under maximum power transfer conditions is presented. The nonlinear relationship connecting the two variables is derived from an analytic solution to the problem of stability limits of a radial power link with resistance and inter-voltage constraints, Figure 2); giving two indices one of which will define an equilibrium state of a maximum no-load voltage Vmax that corresponds to Brainerd’s load impedance induction for this maximum. As a result it is shown that the conditions ZL=-jZ2s/Xs for maximum voltage and ZL=Z*s, for maximum load power have unique sets of unstable operational voltages that are directly related to that of Brainerd’s. In addition, and as a consequence of the resulting voltage stability boundary curves, other equilibrium states that may relate to radial power and communication links and for which no reference can be found in linear circuit theory are observed. These include the conditions for maximum source voltage, voltage profiles under inter-nodal constraints and the source and load voltage magnitudes evolving in states of zero load powers and a maximum of the maximums of transmitted powers. 2. RESULTS OF THE POWERMAXIMIZING RELATION The load impedances for maximum voltage and maximum power of a terminated transducer circuit are obtainable as two special cases of the following generalized current 58 equilibrium relation where β, Vmax are stability indices derived from internal impedance and voltage constraints; I eq = β Z 2 Vmax −V 2 + j 2 s Xs V Z s2 (2) The relationship is power maximizing in that the resulting source voltage E = V + IZ occurs at δ = δc for all β, Vmax & V. The corresponding receiving-end maximum of powers are obtained from the in-phase component as: Pmax = βV Z 2 s 2 Vmax −V 2 with the corresponding component given by: QPmax = (3) leading reactive Xs 2 V Z s2 (4) The statement for equilibrium current Ieq is based on the results of an analytic solution to the stability problem of a radial power link with resistance and inter-voltage constraints for which, as concluded by Kimbark some four decades ago [2], no algebraic equations have been derived. While solution details and practical implications require the space of a full publication, validity of the stated indices is verified herein for the present purposes of introducing the general shape of the resulting V-Pmax and E-Pmax stability boundary curves that will embody Brainerd’s load impedance induction for Vmax together with the necessary and sufficient conditions of conjugate matching. This uses for a case study a line with unity resistance and the sectional impedances defined by the internal voltage magnitude constraints shown in Figure 3; Sudan Engineering Society JOURNAL, August 2005, Volume 51 No.44 giving a total impedance Zs=1.0 + jXs . The general solution algorithm for this twonode system evolves in a constraint relationship connecting the two variables Va and Vb at the maximum power transfer condition of δ = δc. A state of no-load terminal voltage is then identified as a limiting condition for the existence of Va for any specified magnitude of Vb. This equilibrium state is taken as the voltage stability index Vmax which, together with β in Ieq given by (2), are determined by Vb and the sectional impedances constraints. For the present study case, taking Vb = 1.0 p.u, the two indices relate to those in Figure 3 as follows: Vmax = Z s2 β (5) β = Z − Xs 2 s For any given Xs, therefore, Vmax and β are identified to yield the simultaneous conditions δ = δc and Vb = 1.0 p.u. for all V ≤ Vmax . This gives the desired stability boundary curves a sample of which is shown plotted in Figure 4 for the case Zs=1.0+j3.0 p.u. where A, B, C, D, F, G & H represent the operational voltages and powers of seven impedance equilibrium states. While two of the states (A, B) will identify with the impedance conditions for maximum voltage and maximum power of a terminated transducer circuit, the others relate to newly- introduced impedance and voltage states under which neither a terminated transducer circuit nor a radial voltage link will be able to sustain stable equilibrium as can be seen from the following. 3. EQUILIBRIUM VOLTAGES AND IMPEDANCE STATES The conditions δ = tan-1Xs and Vb = 1.0 p.u. are satisfied for all Xs with the substitution in Ieq of the terminal voltage magnitudes V shown in table I using β, Vmax for the case Zs=1.0 + jXs considered in the configuration of Figure 3). The corresponding source voltage magnitudes E together with the equilibrium impedance ZL for each state will come out as shown in Table 1. The following identifications can be made for the seven states of unique equilibrium sets of source and load voltages which are in direct proportionality to Vmax and with their terminal impedances line parameterdictates while circuit internal voltage constraint for the configuration used is being observed. State (A): Equilibrium No-Load Voltage State: This can be seen as evolving in Brainerd’s impedance induction for load voltage resonance except that the source is constrained to the magnitude show0n. Figure 3: Case Study Configuration; ZS = 1.0 + j XS Sudan Engineering Society JOURNAL, January 2006, Volume 52 No.45 59 Figure 4: Case Study Equilibrium Voltage Stability Boundaries Zs=1.0+j3.0 p.u. State (B): Voltages of Conjugate-Matching: While the impedance derived for this state concurs with that of a matched linear transducer circuit, no reference can be found for the associated source voltage magnitude as representing a unique state of unstable equilibrium. State (C): A Condition of Maximum of the Maximums of Received Powers: This condition can be deduced from the maximum power expression (3) to give in conjunction with the corresponding E an upper limit to the maximums of received powers. The condition is synonymous with conjugate matching for values of Xs that will satisfy β = 1.0 State (D) : A Maximum Source Voltage Condition: This newly introduced condition for linear transducer circuits is the dual of Brainerd’s maximum load voltage condition with the terminal voltage V duly constrained to the magnitude shown. As for the case of upper maximums of powers stated in (C), the 60 condition is also synonymous with conjugate matching for values of Xs that will satisfy β = 1.0 States (F) & (G): Conditions for Zero Load Voltages and Powers: In addition to the no-load state due to Vmax and the associated E derived in (A), a source voltage indicated by state (F) with the magnitude shown evolves in two conditions one of which gives rise to a zero-load voltage and hence zero load power. The other state (G) that is associated with the terminal impedance and voltage relations shown can be seen as reducing to the equilibrium no-load state (A) for values of β = 1.0 State (H) : Voltage Profile Predictions With Inter-Nodal Constraints: In power systems engineering where the voltage profile under steady-state stability limits with inter-voltage constraints is of primary concern, equation (3) readily solves for the condition in terms of V for any specified maximum power with the corresponding E coming out as a result. Alternatively, the voltages and power limits at Sudan Engineering Society JOURNAL, August 2005, Volume 51 No.44 Table 1: Equilibrium Voltages and Impedance States State V A Vmax β B β 2 +1 D β 2 +1 − jZ s2 β 2 +1 Vmax Zs β 0 G 2β Vmax β 2 +1 Zs β Zs β ( Z s − 1) + β 2 2 Vmax a given profile can be obtained; e.g., a flat voltage profile indicated by the crossover state (H) in Figure 4 is governed by the impedance and voltage relations shown. 4. CONCLUSION Evidence of the existence of a constraint relationship connecting E & V under maximum power transfer conditions of systems described by E = V + IZ has been presented. The analysis uses internal voltages and sectional impedance constraints resulting in a well-known power system configuration problem for which an analytic solution in terms of steady state stability limits has long been sought. The sample of results given describes P-V & P-E stability curves for radial links that will embody known as well as newly introduced impedance conditions for linear terminated transducer circuits. The complete Xs Z *s = 1 − jX s Vmax ⎛ β +1 ⎞ ⎜ ⎟ ⎜ 2 Z ⎟Vmax s ⎠ ⎝ F H 1 Vmax Zs 2β Zs β 2 +1 2 1 ZL Vmax Vmax C E Z s2 ( β − jX s ) β 2 + X s2 Z s2 ( β 2 − jX s ) 4 2 β + Xs Vmax Vmax -- Vmax 2Z s2 β 2 − 1 + j2 X s β ( Z s − 1) + β 2 2 Vmax Zs Xs −j 2 2( Z s − 1) solution algorithm for the power limits of radial systems with resistance and multi nodal constraints will afford better insight into design optimization problems as to be seen in a future full publication. REFERENCES 1. 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. 2. Kimbark, E.W., “Power System Stability; Vol. III”, John Wiley and Sons, Second Printing , 1962. 3. Fayez Mohamed El-Sadik, “Towards the prediction of Power System Voltage Collapse”; Sudan Engineering Society Journal, Volume 47, No.38, July 2000. Sudan Engineering Society JOURNAL, January 2006, Volume 52 No.45 61