How it works
A driving-point impedance Z(s) must be a Positive Real (PR) function to be physically realisable: it must have real part ≥ 0 for Re(s) ≥ 0, real for real s, and conjugate symmetry Z*(s*) = Z(s). LC (lossless) driving-point functions are purely imaginary on the jω axis, with poles and zeros alternating on it. Foster's First Form realises Z(s) as a partial fraction expansion: a series combination of L, C, and parallel LC tanks. Foster's Second Form realises Y(s) = 1/Z(s) similarly. Cauer's First Form extracts a continued-fraction expansion around ω → ∞, producing an LC ladder starting with a series inductor. Cauer's Second Form expands around ω = 0, producing a ladder starting with a shunt capacitor.
Key points to remember
The alternating pole-zero property is a necessary and sufficient condition for an LC driving-point function — poles and zeros must strictly alternate on the jω axis with no repeated values. For RC driving-point impedance, all poles and zeros are on the negative real axis, with a zero closer to the origin than any pole. RL driving-point functions have a pole closer to origin than any zero. In a Butterworth LP prototype, all poles lie on the unit circle in the left-half s-plane at angles π/2n + kπ/n. Normalised element values for Butterworth and Chebyshev ladder networks appear in standard tables; denormalisation scales L and C for desired cutoff frequency and impedance level.
Exam tip
The examiner always asks you to test whether a given Z(s) is a positive real function and then realise it in Foster's First Form — verify all three PR conditions explicitly and show the partial fraction expansion step by step before drawing the circuit.