Impedance matching

For a source with internal electrical impedance $Z_s$, conjugate matching $Z_L=Z_s^{\ast }$ maximizes delivered mean power when the source is fixed. For a purely resistive source and equal load electrical resistance $R$, the load sees half the open-circuit voltage. A Johnson-Nyquist noise source with open-circuit mean-square voltage $4k_{B}TRd\nu$ therefore delivers $k_{B}Td\nu$ to the matched load, rather than $4k_{B}Td\nu$.

A Thévenin equivalent voltage source $V_s$ with positive source resistance $R_s$ supplies load power $P_L=V_s^2{R}_L/(R_s+R_L{)}^2$ to a real resistive load. Differentiating gives its maximum at $R_L=R_s$. With reactive impedances and fixed source impedance, conjugate matching maximizes the average delivered power. Maximum delivered power is distinct from maximum efficiency; at a real resistive match half of the source's internally generated power is dissipated in $R_s$.

A lossless intermediate section of positive real characteristic impedance $Z_M$ matches unequal real exterior characteristic impedances $Z_L,Z_R$ when $Z_M^2=Z_L{Z}_R$ and its propagation wave phase is an odd multiple of $\pi /2$. The length is an odd quarter of the internal wavelength. This principle also applies to the characteristic impedance of a stretched string.

For a uniform intermediate string of length $l$, put $\delta =k_{M}l$. Continuity of displacement and transverse force gives the displacement reflection coefficientAt real positive mechanical impedances, the denominator never vanishes. Its numerator vanishes only if $(Z_L-Z_R)\cos \delta =0$ and $(Z_M^2-Z_L{Z}_R)\sin \delta =0$. Thus unequal exterior strings require the quarter-wave condition. Equal exterior strings with a different middle string instead transmit perfectly at integer half-wavelengths; if all three mechanical impedances agree, every frequency is matched.