# Match Point

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This exercise studies matching networks used to enable maximum power transfer in the sinusoidal steady state. Consider first the left-hand network shown below. Assume that both impedances take the form Z = R+jX with nonzero resistance R. Prove to yourself that for a given Z_1, maximum time-average power is delivered to Z_2 when Z_2 is the complex conjugate of Z_1; note that the time-average power delivered to Z_2 is the power dissipated in R_2.

Now consider the right-hand network shown below in which the Thevenin equivalent V_A\cos(\omega t) and R_A might model a receiving antenna, the load resistor R_L might model the input resistance of an amplifier, and L and C are a matching network. Using the power maximization result derived above, determine L and C such that maximum power is delivered to the amplifier R_L for a given R_A. In doing so assume R_L> R_A.

For capacitance your answer should be in terms of any of the following: (`RA`

, `RL`

, `L`

, `omega`

)

For inductance your answer should be in terms of any of the following: (`RA`

, `RL`

, `C`

, `omega`

)