Properties of LC/RC/RL Immitance

Understanding the properties of LC, RC, and RL immittance functions is essential for network synthesis. An immittance function refers collectively to either an impedance Z(s) or an admittance Y(s). The properties of these functions on the s-plane and along the imaginary axis determine whether a given rational function can be physically realized as an LC, RC, or RL network using passive elements.

Core Concept: What is an Immittance Function

The term immittance is a portmanteau of impedance and admittance. In network synthesis, the driving point immittance F(s) is the function seen looking into a terminal pair of the network. It can be either Z(s) (impedance) or Y(s) (admittance). A function is said to be realizable if there exists a passive network (using only R, L, C elements) whose driving point immittance equals that function.

A necessary condition for any passive immittance function is that it must be a positive real (PR) function. A rational function F(s) is positive real if: it is real when s is real, and the real part of F(s) is greater than or equal to zero when the real part of s is greater than or equal to zero. This condition ensures that the network is passive (cannot supply energy on its own).

Properties of LC Immittance Functions

An LC (lossless) network has no resistors, only inductors and capacitors. The driving point impedance of an LC network has very specific properties. All poles and zeros of Z(s) lie on the imaginary axis (jw-axis) of the s-plane. This happens because lossless networks store but do not dissipate energy, meaning the impedance is purely imaginary for any real frequency.

The poles and zeros on the jw-axis must alternate. Between any two consecutive poles there is exactly one zero, and between any two consecutive zeros there is exactly one pole. The critical frequencies at the two extremes (lowest and highest) must be of opposite type, meaning if the lowest is a pole the highest must be a zero, and vice versa. The slope of the reactance X(w) = Im[Z(jw)] with respect to w must always be positive, that is dX/dw is greater than 0 for all w.

All residues of Z(s) at its poles must be real and positive. The function Z(s)/s must have positive real residues at all poles. Furthermore, Z(s) is an odd rational function, meaning Z(-s) = -Z(s), which forces the numerator and denominator polynomials to contain only even or only odd powers of s alternately.

Properties of RC Impedance Functions

For an RC network (resistors and capacitors only), the driving point impedance Z(s) has its poles and zeros on the negative real axis of the s-plane. Poles and zeros alternate along this axis. The lowest critical frequency (closest to origin) must be a pole of Z(s), and the highest critical frequency (farthest from origin) must be a zero.

The residues of Z(s) at each of its poles must be real and positive. Along the real frequency axis, Z(jw) decreases monotonically as w increases, meaning dZ(jw)/dw is negative for all w. At zero frequency, Z(0) takes its maximum value (purely resistive), and as w approaches infinity, Z approaches its minimum value (also resistive, given by the sum of all series resistors that remain when capacitors are open at DC and short at high frequency).

The RC admittance Y(s) = 1/Z(s) has the same properties as RL impedance: lowest critical frequency is a zero, highest is a pole, and Y increases monotonically with frequency. This duality is important in Foster synthesis where Form II of RC impedance is obtained by working with Y(s).

Properties of RL Impedance Functions

For an RL network (resistors and inductors only), the driving point impedance Z(s) again has poles and zeros on the negative real axis. However, the order is exactly opposite to RC. The lowest critical frequency must be a zero of Z(s), and the highest critical frequency must be a pole. This means as frequency increases, Z(jw) increases monotonically.

At DC (s=0), the inductor is a short circuit and the network reduces to the series resistors. So Z(0) is a finite real number representing the DC resistance. As s approaches infinity, Z(s) approaches sL, growing without bound due to the inductive reactance. The residues of Z(s)/s at every pole must be positive and real.

Given:
F(s) = (s+3)(s+5) / [(s+1)(s+4)]

Why this formula applies:
Need to identify whether this is an LC, RC, or RL immittance.
All critical frequencies are negative real: poles at s=-1,-4 and zeros at s=-3,-5.
Check alternation: -1(pole), -3(zero), -4(pole), -5(zero) -> p-z-p-z order.
Lowest (nearest to origin) = pole at -1 -> indicates RC impedance or RL admittance.
Highest (farthest from origin) = zero at -5 -> confirms RC impedance (lowest=pole, highest=zero).

Formula:
Verify residues for RC impedance: expand Z(s) by partial fractions.
Z(s) = A + K1/(s+1) + K2/(s+4)

Substitution:
A = limit of Z(s) as s->inf = (s^2)/(s^2) = 1 (constant term -> series resistor at high freq)
K1 = (s+1)*Z(s) at s=-1 = (-1+3)(-1+5)/(-1+4) = (2)(4)/(3) = 8/3
K2 = (s+4)*Z(s) at s=-4 = (-4+3)(-4+5)/(-4+1) = (-1)(1)/(-3) = 1/3

Calculation:
Both residues K1=8/3 and K2=1/3 are positive real. Confirmed as RC impedance.
Z(s) = 1 + (8/3)/(s+1) + (1/3)/(s+4)
Each term K/(s+p) represents a parallel RC: R = K/p, C = 1/K
Term 1: R_inf = 1 ohm (series resistor, residue from infinity)
Term 2: K1=8/3 at p=1: R1 = (8/3)/1 = 8/3 ohm, C1 = 3/8 F
Term 3: K2=1/3 at p=4: R2 = (1/3)/4 = 1/12 ohm, C2 = 3 F

Final Answer with units:
RC Foster Form I: R_inf=1 ohm in series, parallel RC1 (R1=8/3 ohm, C1=3/8 F) in series,
parallel RC2 (R2=1/12 ohm, C2=3 F) in series.
Verification: all residues positive, Z decreasing with w, lowest = pole -> RC confirmed.

Exam Tip: The single most tested property distinction in GATE is: for RC impedance, the lowest critical frequency is a pole and highest is a zero (Z decreases with w). For RL impedance, the lowest is a zero and highest is a pole (Z increases with w). LC has all poles and zeros on the jw-axis only.

• LC: all poles and zeros on jw-axis, alternating. Reactance slope dX/dw always positive.
• RC impedance: poles and zeros on negative real axis, alternating. Lowest = pole, highest = zero. Z decreases with frequency.
• RL impedance: poles and zeros on negative real axis, alternating. Lowest = zero, highest = pole. Z increases with frequency.
• RC admittance has the same properties as RL impedance (duality).
• All residues must be real and positive for a realizable passive network.
• All three types are special cases of the positive real function requirement.

Quick Revision

• LC immittance: poles and zeros on jw-axis only. Odd rational function. Reactance always has positive slope.
• RC impedance Z(s): poles and zeros on negative real axis. Lowest = pole, highest = zero.
• RL impedance Z(s): poles and zeros on negative real axis. Lowest = zero, highest = pole.
• Z(jw) for RC decreases monotonically. Z(jw) for RL increases monotonically.
• Residues at all poles must be real and positive for all three types.
• RC admittance = RL impedance in properties. RL admittance = RC impedance in properties (duality).
• GATE trap: RC lowest is pole (not zero). RL lowest is zero (not pole). This reversal is the most commonly confused point.