LC Network Synthesis

LC network synthesis deals with realizing a given driving point impedance or admittance using only inductors and capacitors, with no resistors. Such networks are lossless and purely reactive. The four canonical synthesis forms, namely Foster Form I, Foster Form II, Cauer Form I, and Cauer Form II, provide systematic procedures to extract element values from a given LC driving point function. These are among the most frequently tested topics in GATE Network Analysis.

Core Concept: LC Driving Point Impedance Properties

An LC driving point impedance Z(s) is the impedance seen at the input terminals of a network containing only inductors and capacitors. Since the network is lossless, all poles and zeros of Z(s) lie on the imaginary axis of the s-plane. Furthermore, poles and zeros alternate along the imaginary axis: this is the interlacing property, which is a direct consequence of the positive real and lossless conditions combined. The frequency response X(ω) = Im[Z(jω)] is a monotonically increasing function of ω.

The general form of an LC driving point impedance is Z(s) = H · [s(s² + ω1²)(s² + ω3²)...] / [(s² + ω2²)(s² + ω4²)...] or with the roles of numerator and denominator interchanged, depending on whether there is a pole or zero at the origin and at infinity. The scale factor H must be positive. The alternating pole-zero pattern on the jω axis defines the resonant and anti-resonant frequencies of the network.

Mathematical Expression: Foster and Cauer Methods

The Foster Form I method synthesizes Z(s) by expanding it as a partial fraction in terms of its poles. Since the poles of an LC impedance are on the imaginary axis, the partial fraction form is: Z(s) = k0/s + k∞·s + sum of (2ki·s)/(s² + ωi²). Here k0 is the residue at the pole at origin (giving a capacitor 1/k0), k∞ gives an inductor k∞, and each 2ki·s/(s² + ωi²) term gives a parallel LC combination. This is a series connection of these elements, giving a series ladder from partial fraction expansion of Z(s).

The Foster Form II method works with the admittance Y(s) = 1/Z(s) and expands it as a partial fraction. The extracted elements become a parallel connection: y0/s gives a shunt inductor 1/y0, y∞·s gives a shunt capacitor y∞, and each resonant term gives a series LC in the shunt branch. This is a parallel ladder realization.

The Cauer Form I method uses continued fraction expansion of Z(s) around s = ∞ (highest power first). Each division step extracts one element: series inductors and shunt capacitors alternate, forming a ladder network. The Cauer Form II uses continued fraction expansion of Z(s) around s = 0 (lowest power first). Each step extracts series capacitors and shunt inductors alternately. The Cauer forms produce compact ladder networks directly from the continued fraction quotients.

Practical Understanding

Foster realizations are based on partial fractions and are straightforward to compute when the pole locations are known. Each partial fraction term maps to one resonant circuit element pair. The topology is clear but may result in circuits that are harder to tune in practice because elements are not in simple ladder form.

Cauer realizations produce ladder networks that are preferred in practical filter design because they are more amenable to cascading and have better sensitivity properties with respect to element tolerances. In modern filter design, the Cauer Form I ladder (series inductors, shunt capacitors) is the standard topology for low-pass LC filters. The element values obtained from Cauer synthesis correspond directly to Butterworth or Chebyshev prototype values tabulated in filter design handbooks.

Numerical Example

The Foster I synthesis procedure starts from the partial fraction expansion of Z(s). Each term maps to a specific circuit element or parallel LC pair. The example below demonstrates finding element values for a given LC driving point impedance using Foster Form I.

Given:
Z(s) = (s² + 4) / (s³ + s)

Why this formula applies:
Z(s) is an LC driving point impedance (all imaginary axis poles and zeros, real positive scale).
Foster I: partial fraction expansion of Z(s).

Formula:
Z(s) = k0/s + sum of 2ki·s/(s²+ωi²)

Substitution:
Factor denominator: s(s² + 1)
Z(s) = (s² + 4) / [s(s² + 1)]
       = A/s + (Bs + C)/(s² + 1)

Calculation:
A = [s · Z(s)]_(s=0) = 4/1 = 4
(s² + 4) = A(s² + 1) + (Bs + C)s
         = 4s² + 4 + Bs² + Cs
Compare: s²: 1 = 4 + B → B = -3
         s¹: 0 = C → C = 0
         s⁰: 4 = 4 ✓

So: Z(s) = 4/s + (-3s)/(s² + 1)
But residue k must be positive for PR.
Since B = -3 < 0, this means Z(s) is NOT a valid PR LC impedance!

--- Corrected example with valid function ---
Z(s) = (s⁴ + 5s² + 4) / (s³ + 4s)

Factor: Numerator = (s²+1)(s²+4), Denominator = s(s²+4)
Simplify: Z(s) = (s²+1)/s = s + 1/s

Foster I:
Z(s) = 1/s + s
Term 1/s → Capacitor: C = 1 F (since k0 = 1, C = 1/k0)
Term s → Inductor: L = 1 H

Final Answer:
Foster I realization: Capacitor C = 1 F in series with Inductor L = 1 H
This is a series LC ladder with Z(s) = s + 1/s.

Exam Tip: In GATE, always simplify the LC driving point impedance by canceling common factors between numerator and denominator before starting Foster or Cauer synthesis. An uncanceled common factor means the function has a removable singularity, which will produce incorrect element values if not simplified first.

Mechanism: Key Properties of LC Synthesis Forms

• LC driving point functions have all poles and zeros on the imaginary axis, alternating (interlacing property). The function is purely imaginary for s = jω.
• Foster I: partial fraction of Z(s). Series connection of C (from pole at origin), L (from pole at infinity), and parallel LC pairs (from imaginary axis poles).
• Foster II: partial fraction of Y(s). Parallel connection of L (from pole at origin in Y), C (from pole at infinity in Y), and series LC pairs.
• Cauer I: continued fraction of Z(s) at s → ∞. Quotients alternate as series inductors and shunt capacitors.
• Cauer II: continued fraction of Z(s) at s → 0. Quotients alternate as series capacitors and shunt inductors.
• All four forms realize the same driving point function but with different network topologies and element arrangements.

Quick Revision

• LC impedance: all poles and zeros on jω axis, alternating. Re[Z(jω)] = 0. Scale factor H must be positive.
• Foster I: partial fraction of Z(s) → series ladder.
• Foster II: partial fraction of Y(s) → parallel ladder.
• Cauer I: continued fraction of Z(s) at s → ∞ → L series, C shunt alternating ladder.
• Cauer II: continued fraction of Z(s) at s → 0 → C series, L shunt alternating ladder.
• Trap: Always cancel common factors in numerator and denominator before synthesis. A common factor indicates redundant poles or zeros.
• Trap: All residues in partial fraction expansion must be real and positive for a valid LC (PR) function. Negative residues indicate the function is not realizable as passive LC.