Network functions

dsa In any linear, time-invariant (LTI) circuit, once you drive it with a sinusoid at angular frequency ω, all voltages and currents in the network will also be sinusoids at that same ω, but with different amplitudes and phase‐shifts. By representing them as complex phasors you can form network functions—ratios of an output phasor to an input phasor—which depend only on the circuit’s topology and component values, not on how large your input is.

Suppose you pick one “input” quantity (either a voltage or a current) and one “output” quantity. The four possibilities are:

1. Voltage‐to‐voltage transfer

$$H_V(jω);=;\frac{V_{\rm out}(jω)}{V_{\rm in}(jω)}$$

tells you the gain (magnitude change) and phase shift that the circuit applies to an input voltage.

2. Voltage‐to‐current transfer (also called a transfer admittance) $$Y_{t}(jω);=;\frac{I_{\rm out}(jω)}{V_{\rm in}(jω)}$$
3. Current‐to‐current transfer $$H_I(jω);=;\frac{I_{\rm out}(jω)}{I_{\rm in}(jω)}$$
4. Current‐to‐voltage transfer (also called a transfer impedance) $$Z_{t}(jω);=;\frac{V_{\rm out}(jω)}{I_{\rm in}(jω)}$$