I last studied electronics about twelve years ago. \u00a0Ideas from it return to mind now and then. \u00a0Researching for my Nov 10 article on auto batteries,<\/a> I read a remark that whatever the battery’s resistance was, the starter should match it to receive maximum possible power. I recognized the idea as a case of the maximum power transfer theorem:<\/p>\n Whatever the resistance of the surrounding circuit, the load resistor should match it in order to receive maximum power.<\/em><\/p>\n Here’s a proof using calculus:<\/p>\n Let’s imagine a series circuit with peripheral resistance $R$ and load resistance aR, where a≥0. Since the two resistances are in series, Rtotal<\/sub>=R+aR.<\/p>\n Now, since V=IR, we have I=V\/R. In particular,<\/p>\n I=V\/(R+aR)<\/p>\n Furthermore, the power dissipated by a resistor is given by P=I^2R. Therefore, the power in the load resistor of our circuit is<\/p>\n P=(V\/(R+aR))^2(aR)=aRV^2\/(R+aR)^2<\/p>\n To find the value of a that gives the maximum value of P, we take the derivative dP\/da, set it equal to zero, and solve for a.<\/p>\n To take the derivative dP\/da, we use the quotient rule:<\/p>\n dP\/da=((R + aR)^2(RV^2) – aRV^2(2(R + aR)R))\/(R + aR)^4<\/p>\n Set dP\/da to zero and solve for a:<\/p>\n 0=((R + aR)^2(RV^2) – aRV^2(2(R + aR)R))\/(R + aR)^4<\/p>\n Multiply both sides by (R + aR)^4<\/p>\n 0=(R + aR)^2(RV^2) – 2aV^2R^2(R + aR)<\/p>\n Factor R + aR:<\/p>\n 0=(R +aR)[(R + aR)RV^2 – 2aV^2R^2]<\/p>\n Divide out R + aR from both sides:<\/p>\n 0=(R + aR)RV^2 – 2aV^2R^2<\/p>\n Factor out RV^2:<\/p>\n 0=RV^2[(R + aR) – 2aR]<\/p>\n Divide out RV^2:<\/p>\n 0=R + aR – 2aR<\/p>\n Factor out R:<\/p>\n 0=R(1 + a -2a)<\/p>\n Divide out R:<\/p>\n 0=1 + a -2a<\/p>\n Simplify:<\/p>\n 0=1 -a<\/p>\n Finally, <\/p>\n a=1<\/p>\n Let’s recall that, in our circuit, the peripheral resistance is R, while the load resistance is aR. We now find that for maximum power, a=1. It follows that the load resistance should be 1R=R, the same as the peripheral resistance, for maximum power.<\/p>\n The maximum power transfer theorem, while many never encounter it, is a fundamental part of everyday life for many others. Anticipating what we may need to know in the future is often a challenge….<\/p>\n