RL Network Synthesis

In network synthesis, the goal is to design a physical network that realizes a given mathematical function. RL network synthesis deals specifically with constructing networks using only resistors and inductors such that the driving point impedance matches a specified rational function of s. Unlike analysis, synthesis starts from the transfer or immittance function and works backward to find the circuit.

Core Concept: RL Driving Point Impedance

A driving point impedance Z(s) is the ratio of the Laplace transform of the voltage at a terminal pair to the current entering that terminal, with all internal sources set to zero. For a network to be realizable as an RL network, the impedance function must satisfy certain necessary and sufficient conditions rooted in the properties of real physical components.

The fundamental property of an RL impedance is that its poles and zeros must lie exclusively on the negative real axis of the s-plane. Furthermore, they must alternate, meaning a pole must separate each pair of zeros and vice versa. The critical frequency nearest to the origin must be a pole (or the function value at s=0 must be finite and nonzero from the pole side), and the critical frequency farthest from the origin (as s approaches infinity) must be a zero or the function must have a finite nonzero value.

The residues of Z(s) at each of its poles must be real and positive. This condition ensures that each pole corresponds to a physically realizable element value (positive R or L). If any residue were negative, the corresponding element value would be negative, which is not physically realizable with passive elements.

Mathematical Expression and Properties

The general form of an RL driving point impedance is a ratio of polynomials in s. Both the numerator and denominator must be Hurwitz polynomials (or have roots only on the negative real axis for RL specifically). The function takes the form:

Z(s) = H (s + z1)(s + z2)...(s + zm) / (s + p1)(s + p2)...(s + pn)

where all zi and pi are positive real numbers representing zeros and poles respectively. For RL impedance, the degree difference between numerator and denominator is at most 1, and the partial fraction expansion of Z(s)/s yields only positive real residues. The slope condition requires that dZ(jw)/dw is positive, meaning the impedance is a monotonically increasing function of frequency along the imaginary axis.

The behavior at the two extremes is diagnostic. As s approaches 0, Z(0) equals R (purely resistive), and as s approaches infinity, Z(s) approaches sL (inductive), confirming the presence of inductors. The ratio Z(infinity)/Z(0) gives a sense of the inductive dominance at high frequencies.

Practical Understanding of RL Synthesis

When given a Z(s) to synthesize, the first step is to verify all the conditions: check that poles and zeros are on the negative real axis, confirm they alternate, verify residues are positive, and check the degree condition. Only after verification should any synthesis procedure begin.

The partial fraction expansion method (Foster I) expands Z(s) into a sum of simpler terms, each corresponding to a series R-L combination. The continued fraction method (Cauer I) removes elements one by one starting from the behavior at infinity, producing a ladder network. Both methods yield different circuit topologies but the same impedance function.

In practice, RL networks are less common than RC networks in filter design, but they appear in power electronics, transformer modeling, and high-frequency inductor loss modeling. Understanding RL synthesis also forms the conceptual foundation for understanding the more general positive real function theory used in advanced filter design.

Given:
Z(s) = (s + 2)(s + 6) / (s + 4)(s + 8)
     = (s^2 + 8s + 12) / (s^2 + 12s + 32)

Why this formula applies:
Both poles and zeros are on the negative real axis and alternate: zeros at -2,-6, poles at -4,-8.
Lowest critical frequency is a zero (s=-2), highest is a pole (s=-8).
Check: degree of numerator = degree of denominator = 2, so Z(inf) = 1 (finite, valid for RL).

Formula:
Z(s)/s expanded via partial fractions to get residues:
Z(s) = K0 + K1/(s+4) + K2/(s+8)

Substitution:
K1 = [(s+4) * Z(s)] at s=-4
   = (-4+2)(-4+6) / (-4+8) = (-2)(2)/(4) = -1
Note: since residue is negative, this form is not a Foster I for Z(s).
Instead expand Z(s)/s:
Z(s)/s = (s+2)(s+6) / [s(s+4)(s+8)]
K0 = s * Z(s)/s at s=0 = (2)(6)/(4)(8) = 12/32 = 0.375
K1 = (s+4)*Z(s)/s at s=-4 = (-2)(2) / (-4)(-4+8) = -4/(-4*4) = -4/-16 = 0.25
K2 = (s+8)*Z(s)/s at s=-8 = (-6)(-2) / (-8)(-8+4) = 12/32 = 0.375

Calculation:
Z(s)/s = 0.375/s + 0.25/(s+4) + 0.375/(s+8)
Z(s)   = 0.375 + 0.25s/(s+4) + 0.375s/(s+8)
Each term Ks/(s+p) = R*sL/(R+sL) where R=p*L and residue K = L
For 0.25s/(s+4): L1 = 0.25 H, R1 = 4*0.25 = 1 ohm
For 0.375s/(s+8): L2 = 0.375 H, R2 = 8*0.375 = 3 ohm
Constant term 0.375 = R0 = 0.375 ohm (series resistor)

Final Answer:
Foster Form I: R0=0.375 ohm in series with [R1=1 ohm + L1=0.25H] and [R2=3 ohm + L2=0.375H] series branches in series.

Exam Tip: For RL impedance Z(s), the lowest critical frequency (nearest to origin) is always a zero and highest is always a pole. For RC impedance Z(s), the lowest is always a pole and highest is always a zero. Do not mix these up in GATE problems.

• All poles and zeros of Z(s) lie on the negative real axis only.
• Poles and zeros strictly alternate. No two poles or two zeros are adjacent.
• The critical frequency closest to the origin is always a zero for RL impedance.
• The critical frequency farthest from the origin is always a pole for RL impedance.
• All residues in the partial fraction expansion of Z(s)/s are real and positive.
• Z(jw) is a monotonically increasing function of frequency.

Quick Revision

• RL driving point impedance has all poles and zeros on the negative real axis, alternating.
• Lowest critical frequency is a zero; highest is a pole (for RL Z(s)).
• Residues of Z(s)/s at all poles must be real and positive.
• Z(jw) increases monotonically with frequency.
• Z(0) is finite (resistive behavior at DC); Z(infinity) grows without bound (inductive behavior).
• Synthesis methods: Foster I (series RL), Foster II (parallel RL admittance), Cauer I and II (ladder forms).
• Common GATE trap: confusing RL and RC pole-zero order near origin. Remember RL lowest = zero, RC lowest = pole.