Positive Real Functions

The concept of a positive real function is the central mathematical condition that determines whether a given impedance or admittance function can be realized as a physical passive network. Every passive network, no matter how complex, must have a driving point impedance that satisfies the positive real conditions. GATE and university exams test these conditions both in theory and through verification problems.

Core Concept: What Is a Positive Real Function?

A positive real (PR) function is a rational function F(s) of the complex frequency variable s that satisfies two fundamental conditions: it must be real-valued when s is real (meaning all coefficients of the numerator and denominator polynomials are real numbers), and its real part must be non-negative whenever the real part of s is non-negative. The second condition represents the physical constraint of passivity: a passive network cannot supply more energy than it receives.

The significance of positive real functions in synthesis is that a function Z(s) can be realized as the driving point impedance of a passive network with positive R, L, C elements if and only if Z(s) is a positive real function. This statement is both necessary and sufficient, making it the most powerful single characterization in network synthesis theory.

The concept applies directly to driving point impedance Z(s) and driving point admittance Y(s) = 1/Z(s). If Z(s) is positive real, then so is Y(s), since the reciprocal of a PR function is also PR. However, transfer functions (two-port parameters like Z12, Y12, H) are generally NOT required to be positive real.

Mathematical Expression: PR Function Conditions

A rational function F(s) = P(s)/Q(s) is said to be positive real if and only if all of the following conditions hold. First, all coefficients of P(s) and Q(s) must be real and positive. Second, P(s) and Q(s) must be Hurwitz polynomials (roots in the left half s-plane). Third, any poles on the imaginary axis (s = jω) must be simple (not repeated) and the corresponding residues must be real and positive. Fourth, Re[F(jω)] must be greater than or equal to zero for all real values of ω.

The fourth condition is often verified by substituting s = jω and computing the real part of F(jω). If this real part is a non-negative function of ω for all ω from zero to infinity, the function passes this test. For LC networks (lossless), Re[Z(jω)] = 0 for all ω because there is no resistive dissipation; this is a special limiting case where the inequality becomes an equality.

A practical necessary condition (but not sufficient alone) is the degree condition: the degrees of numerator and denominator of a PR function can differ by at most 1. That is, if deg(P) = m and deg(Q) = n, then |m - n| ≤ 1. Violation of this condition immediately disqualifies a function from being positive real.

Practical Understanding

In filter design, the synthesis procedure always begins with verifying that the driving point impedance derived from the filter specification is a positive real function. If it fails any PR condition, no passive lossless or lossy network can realize it, and an active circuit (with amplifiers) would be required instead. This distinction between passive realizability and active realizability is practically important in circuit design.

The imaginary axis poles of a driving point impedance correspond to resonances in the network: a pole at s = jω0 means the impedance becomes infinite at frequency ω0. For this to represent a physical resonance, the residue at that pole must be positive. A negative residue would imply the network is supplying energy at resonance, which is physically impossible for a passive element.

Numerical Example

Checking whether a given rational function is positive real involves verifying each condition in sequence. The most common GATE question format is to test whether a given Z(s) satisfies all PR conditions. The example below demonstrates the complete verification procedure.

Given:
Z(s) = (2s² + 4) / (s³ + 2s)

Why this formula applies:
Check all four PR conditions for Z(s) = P(s)/Q(s):
P(s) = 2s² + 4, Q(s) = s³ + 2s

Formula / Conditions:
1. All real coefficients? Yes → P(s): 2, 0, 4 and Q(s): 1, 0, 2
   Note: coefficients of s¹ and s¹ in P and s⁰ in Q are absent.
   Modified Hurwitz check needed.
2. Modified Hurwitz (imaginary axis roots allowed for LC):
   Roots of Q(s) = s³ + 2s = s(s² + 2): s = 0, s = ±j√2 → all on jω axis → Modified Hurwitz.
   Roots of P(s) = 2(s² + 2): s = ±j√2 → on jω axis → Modified Hurwitz.
3. Degree check: deg(P) = 2, deg(Q) = 3, |3-2| = 1 ≤ 1 → Passes.
4. Partial Fraction Expansion to check residues:
   Z(s) = (2s² + 4) / (s(s² + 2))
   = A/s + (Bs + C)/(s² + 2)
   A = [s · Z(s)]_s=0 = 4/2 = 2 > 0 (Positive residue at s = 0)
   Residue at s = j√2: [（s²+2)·Z(s)/s]_(s²=-2) = (2(-2)+4)/(j√2) = 0/(j√2) = 0
   Since residue at jω axis poles is ≥ 0, condition passes.
5. Re[Z(jω)] = Re[(2(jω)² + 4)/((jω)³ + 2(jω))]
   Numerator: 2(-ω²) + 4 = 4 - 2ω² (real)
   Denominator: -jω³ + 2jω = j(2ω - ω³) (imaginary)
   Z(jω) = (4 - 2ω²) / (j(2ω - ω³)) → purely imaginary
   Re[Z(jω)] = 0 for all ω

Final Answer:
Z(s) = (2s² + 4)/(s³ + 2s) IS a Positive Real function.
It represents a valid LC (lossless) driving point impedance.

Exam Tip: For LC driving point impedances, Re[Z(jω)] = 0 always (lossless). For RC, Re[Z(jω)] ≥ 0. For RL, same. If the problem gives a function and asks whether it is PR, start by checking degrees differ by at most 1 and all coefficients are of the same sign. These two quick checks eliminate most wrong options in GATE.

Mechanism: Key Properties

• A function F(s) is positive real if: it has real coefficients, Re[F(s)] ≥ 0 for Re[s] ≥ 0, and imaginary axis poles are simple with positive real residues.
• The degree condition: |deg(numerator) - deg(denominator)| ≤ 1 is a necessary condition for PR functions.
• The reciprocal of a PR function is also PR. So if Z(s) is PR, then Y(s) = 1/Z(s) is also PR.
• For LC networks: Re[Z(jω)] = 0. For RC and RL networks: Re[Z(jω)] ≥ 0 with the real part being a positive function of ω.
• Poles and zeros of LC driving point impedances alternate on the imaginary axis (interlacing property), which is a direct consequence of positive realness.
• A positive real function is analytic in the right half s-plane (no poles in RHP), which physically means the network is stable.

Quick Revision

• PR function conditions: real coefficients, Re[F(s)] ≥ 0 in RHP, simple imaginary axis poles with positive residues.
• Degree condition: |deg(P) - deg(Q)| ≤ 1 must hold.
• Reciprocal of PR is PR. Sum of PR functions is PR.
• LC: Re[Z(jω)] = 0. RC/RL: Re[Z(jω)] ≥ 0.
• Trap: Not every function with positive coefficients is PR. The Re[F(jω)] ≥ 0 test must still be done.
• Trap: Repeated poles on jω axis → NOT PR (poles must be simple on the imaginary axis).
• PR condition is necessary and sufficient for passive realizability. Failure means active elements are needed.