Foster Forms

The Foster forms are among the most fundamental methods for synthesizing LC, RC, and RL networks. They were introduced by Ronald M. Foster and are based on the partial fraction expansion of the driving point immittance function. Foster synthesis is a direct method that yields two canonical forms, each giving a distinct circuit topology with the same immittance function.

Core Concept: Partial Fraction Expansion

The Foster synthesis method exploits the fact that any realizable driving point immittance function can be expanded into a sum of simpler rational functions using partial fractions. Each term in this expansion corresponds to a physically realizable element or a pair of elements. This one-to-one correspondence between algebraic terms and circuit elements is what makes the Foster method so elegant and systematic.

For Foster Form I, the driving point impedance Z(s) is expanded into partial fractions. Each term in the expansion maps directly to a series-connected element or branch. For an LC network, the general form of Z(s) is:

Z(s) = K_inf * s + K0/s + sum of [2Ki*s / (s^2 + wi^2)] for each finite nonzero pole pair

Here the K_inf*s term gives a series inductor, K0/s gives a series capacitor, and each 2Ki*s/(s^2+wi^2) term gives a series combination of an inductor and capacitor tuned to resonance at wi. The residues K must all be real and positive for the network to be realizable.

For Foster Form II, the driving point admittance Y(s) = 1/Z(s) is expanded instead. Each term in Y(s) corresponds to a parallel-connected branch. A term K0/s in Y(s) gives a shunt inductor, K_inf*s gives a shunt capacitor, and 2Ki*s/(s^2+wi^2) gives a parallel LC tank connected from the node to ground.

Mathematical Expression

For an LC network, the driving point impedance is an odd rational function meaning Z(s) = odd polynomial / even polynomial or vice versa. The general partial fraction expansion for Foster Form I is:

Z(s) = A*s + B/s + sum [2Ki*s / (s^2 + wi^2)]

Each residue is computed as Ki = lim[(s^2 + wi^2)/2 * Z(s)] as s approaches j*wi. The element values are extracted as: for the series LC branch at pole wi, the inductance Li = Ki and the capacitance Ci = 1/(Ki * wi^2). The constant term A gives the inductor value directly and B gives the capacitor value as C = 1/B (since B/s = 1/(s*C) means C = 1/B).

Practical Understanding

Foster forms are canonical, meaning for a given immittance function, the Foster I and Foster II realizations are unique. There is only one Foster I circuit and one Foster II circuit for each valid immittance function. This is unlike Cauer forms, where multiple valid ladders can exist depending on the order of element removal.

In practical filter design, the Foster form gives a canonic realization where the total number of reactive elements equals the degree of the immittance polynomial. No redundant elements are used. This minimum element count property is important in situations where component count, cost, or size must be minimized.

For RC networks, the Foster I form expands Z(s) of an RC network. The poles of Z(s) lie on the negative real axis, and each partial fraction term K/(s+p) corresponds to a parallel RC combination connected in series with others. For Foster II of an RC network, Y(s) is expanded and each term corresponds to a series RC combination connected in parallel.

Given:
Z(s) = 2(s^2 + 1)(s^2 + 9) / [s(s^2 + 4)]

Why this formula applies:
Poles of Z(s) are at s=0, s=+j2, s=-j2 (i.e., s^2=-4).
Zeros are at s=+j1, s=-j1, s=+j3, s=-j3.
This is a valid LC impedance (odd rational function, alternate poles-zeros on jw axis).
Foster Form I: expand Z(s) by partial fractions.

Formula:
Z(s) = A*s + K0/s + K1*2s/(s^2+4)

Substitution:
As s->inf: Z(s)/s -> 2s^4/s^4 -> A = 2H (inductor)
K0 = s*Z(s) at s=0 = 2*1*9 / 4 = 18/4 = 4.5 => C0 = 1/4.5 = 2/9 F
K1 = [(s^2+4)/2 * Z(s)] at s^2=-4
   = 2(-4+1)(-4+9) / [2*(-4)] * (1/2) ... let us compute directly:
   Residue at pole j2: K1 = lim[(s^2+4)/2 * Z(s)] as s^2->-4
   = 2(-4+1)(-4+9) / [(-4) * 2 / 2] = 2(-3)(5)/(-8) = -30/-8 = 3.75 H
   C1 = 1/(K1*4) = 1/15 F

Calculation:
Z(s) = 2s + 4.5/s + 2*3.75*s/(s^2+4)
     = 2s + 4.5/s + 7.5s/(s^2+4)

Element check: 3 reactive elements needed (degree 3 denominator plus 1 = 4 elements for degree-4 numerator/denominator after simplification)

Final Answer with units:
Foster Form I realization:
- Series inductor: L_inf = 2 H
- Series capacitor: C0 = 2/9 F (from B/s term)
- Series branch: L1 = 3.75 H and C1 = 1/15 F in series (resonates at w=2 rad/s)

Exam Tip: In Foster Form I, expand Z(s). In Foster Form II, expand Y(s) = 1/Z(s). The residues at every pole must be positive for realizability. If any residue is negative, the given function is not realizable by Foster method with passive elements.

• Foster Form I expands Z(s) by partial fractions. Each term gives a series-connected element or branch.
• Foster Form II expands Y(s) = 1/Z(s) by partial fractions. Each term gives a parallel shunt branch.
• For LC networks, all residues must be positive real. For RC or RL, the condition changes per network type.
• The K_inf*s term gives inductor (Foster I) or capacitor (Foster II).
• The K0/s term gives capacitor (Foster I) or inductor (Foster II).
• Foster realizations are unique and canonic (minimum number of reactive elements).

Quick Revision

• Foster Form I: expand Z(s) by partial fractions. Series connection of resulting elements.
• Foster Form II: expand Y(s) = 1/Z(s) by partial fractions. Parallel (shunt) connection of resulting elements.
• Element values from Ks/(s^2+w^2): L=K, C=1/(Kw^2).
• All residues must be positive real for the function to be realizable by passive elements.
• Foster realizations are unique (canonic). Only one Foster I and one Foster II circuit exists for each valid immittance.
• Common GATE trap: confusing which form expands Z(s) vs Y(s). Foster I always uses Z(s).
• Both forms use the same number of reactive elements equal to the degree of the immittance function.