Driving-Point Quantities, Transfer Functions & Network Coefficients

A network coefficient is meaningful only after its input, output, sign references, and termination conditions are fixed. "Gain" without those qualifiers is not a complete circuit quantity: connecting a load can change both the output and the input operating point.

Driving-point quantities #

At one port:

$$Z_{in}(s)=\frac{V(s)}{I(s)},\qquad Y_{in}(s)=\frac{I(s)}{V(s)}.$$

They are driving-point functions because excitation and response share the same port. In DC they reduce to resistance/conductance; at sinusoidal steady state evaluate them at $s=j\omega$.

Transfer quantities #

Common definitions include

$$H_v=\frac{V_2}{V_1},\qquad H_i=\frac{I_2}{I_1},$$

$$Z_t=\frac{V_2}{I_1},\qquad Y_t=\frac{I_2}{V_1}.$$

The last two are transimpedance and transadmittance. Older course material may call off-diagonal conductance relations "mutual conductances" and simple current or voltage ratios "transfer coefficients".

Terminations are part of the function #

For a two-port, $V_2/V_1$ under open circuit is generally different from the ratio with a finite load. A complete statement looks like

$$H_v(\omega;Z_S,Z_L)=\frac{\underline V_L}{\underline V_S}.$$

The source impedance $Z_S$ and load $Z_L$ are parameters of the measurement setup, not incidental details.

Poles and zeros #

For linear time-invariant lumped networks, rational functions of $s$ expose dynamics. Zeros suppress a transfer at specific complex frequencies; poles describe natural modes and stability. A driving-point function of a passive network obeys stronger realizability constraints than an arbitrary algebraic ratio.

State whether a transfer uses open-circuit output, a specified load, or a buffered measurement. That termination belongs in the function definition because it can move both poles and gains.