Signals and Network Analysis

1. Signals and Network Analysis Target group 3rd year ECED Students 2 Lecture Hours 3 Tutorial Hours Contact hours: School of Computing and Electrical Engineering Eeng-3121

2. Signals and Network Analysis Chapter 2 Realizability Theory and Positive Real Functions

3. Realizability Theory and Positive Real Functions Realizability criteria for passive network deriving-point functions A deriving-point function (Z(s) or Y(s)) is realizable using lumped passive elements (R, L, C, M) elements if it is positive real (PR) rational function of s • a real rational function of s with real and positive coefficients • if Re[s] > 0, then Re[H(s)] > 0. Equivalent Positive-real conditions • H(s) should be quotient of polynomials with positive and real coefficients • For all real ω, Re[H(jω)] ≥ 0 • All poles and zeros of H(s) should lie in the closed Left-hand S-plane. All jω-axis poles and zeros must be simple (multiplicity of one) with positive residues.

4. Realizability Theory and Positive Real Functions Hurwitz and Strictly Hurwitz Polynomials P(s) = ansn+ an-1sn-1+… + a1s1+a0 • A polynomial with its zeros restricted to the closed Left-hand S-plane is called Hurwitz polynomial. • A polynomial with its zeros restricted to the inside if Left-hand S-plane (excluding the jω –axis) is called Strictly Hurwitz polynomial. • A polynomial that is not Hurwitz is called Non-Hurwitz polynomial • Eg. P(s) = s(s +1)(s2 + 4) Eg. P(s) = (s + 3)(s2 + 2s + 2) • Eg. P(s) = (s+2)(s2 – 4s + 5)

5. Realizability Theory and Positive Real Functions Tests for Hurwitz nature of polynomials P(s) = ansn+ an-1sn-1+… + a1s1+a0 Factorize P(s), find all zeros (roots of the polynomial) and inspect their location on the s-plane. Method 1 - Factorization Method 2 – Continued Fraction Expansion steps • Separate P(s) in to even (Pe(s)) and odd (Po(s)) parts. • Form φ(s) = Pe(s)/Po(s)orPo(s)/Pe(s). Select the higher degree to be numerator. • Expand φ(s) using CFE (Continued Fraction Expansion) If there is no premature termination, denominator of the last division is a constant. If the last denominator is not a constant, then we say there is premature termination • A polynomial is Hurwitz if all coefficients have the same sign • Otherwise a polynomial is Non-Hurwitz • A polynomial is Strictly Hurwitz if all coefficients have the same sign, and there is no premature termination.

6. Realizability Theory and Positive Real Functions Tests for Hurwitz nature of polynomials Example Test for Hurwitz nature of the following polynomial using CFE. P(s) = (s2 + 2s + 1)(s2 + s + 1)(s2+4) = s6 + 3s5 + 8s4 + 15s3 + 17s2 + 12s + 4

7. Realizability Theory and Positive Real Functions Tests for Hurwitz nature of polynomials … P(s) = ansn+ an-1sn-1+… + a1s1+a0 Method 3: Routh – Hurwitz Array steps • Construct a triangular array as follows: Special case 1: • when all coefficients in a row are zero ( Vanishing row) • The polynomial with coefficients in a row prior to the vanishing row is a factor of P(s), and it has roots on the jw-axis. • If the vanishing row is at sk-1, create a polynomial, Pk(s), using coefficients of a row at sk (prior to the vanishing row) • Pk(s) = u1sk + u2sk-2 + u3sk-4 + u4sk-6 +… • Define an auxiliary polynomial, dPk(s)/ds, and replace the vanishing row with coefficients of this polynomial (derivative of Pk(s)) • Continue the process until you reach the last row at s0.

8. Realizability Theory and Positive Real Functions Tests for Hurwitz nature of polynomials … Special case 2: • the first coefficient of a row is zero, and there is at least one non-zero coefficient in a row If a row begins with a zero and the row has some non-zero coefficients, replace the leading zero (the first zero) with ε (assume a small positive number) and find the remaining coefficients (in terms of ε). Interpretation • If all coefficients of the first column have same sign, and if there is no vanishing row, the polynomial is strictly Hurwitz. (If there are coefficients that contain ε, evaluate sign of the coefficients by taking the limit ε→ 0+). • If there is at least one sign change, the polynomial is non-Hurwitz. • If all coefficients of the first column have same sign, and there exists vanishing row (root on the jw-axis), the polynomial is Hurwitz.

9. Realizability Theory and Positive Real Functions Tests for Hurwitz nature of polynomials Example Test Hurwitz nature the following polynomial using Routh-Hurwitz array P(s) = s4 + s3 + 5s2 +3s+2

10. Realizability Theory and Positive Real Functions Sturm’s Theorem and Sturm’s Test Recall Positive-real condition • For all real ω, Re[H(jω)] ≥ 0 Consider a network function H(s) = N(s)/D(s) First, separate both polynomials in to even and odd parts. Now, multiply H(s) by D(-s)/D(-s) = He(s) + Ho(s) Even function Odd function

11. Realizability Theory and Positive Real Functions Sturm’s Theorem and Sturm’s Test … For s = j He(jω) – is purely real Ho(jω) – is purely imaginary. De(s)2 – Do(s)2 = D(s)D(-s) = D(jω)D(-jω) = |D(jω)|2 which is always positive. Ne(jω)De(jω) – No(jω)Do(jω) = P(ω2) is an even polynomial function of ω Therefore, Re[H(jω)] = He(jω) • Condition B: • For all real ω, Re[H(jω)] ≥ 0 • He(jω) ≥ 0 • where He(jω) = P(ω2)/ |D(jω)|2 • P(ω2) ≥ 0 for all real ω OR for ω2 ≥ 0 • Let x = ω2 • P(x)≥ 0 for all x ≥ 0 This implies that the graph of P(x) should not cross the x – axis in the right half of the xy – plane (positive values of x)

12. Realizability Theory and Positive Real Functions Sturm’s Theorem and Sturm’s Test … Cases: P(x) = anxn + an-1xn-1 + …+ a1x + a0 • vIf all coefficients ai are positive, then P(x) ≥ 0 for x ≥ 0 (condition met) • vP(x) = a0 at x = 0 hence, ao ≥ 0 (necessary) • vP(x) → anxn as x → ∞ hence, an ≥ 0 (necessary) • vIf a0 ≥ 0, an ≥ 0 but ai < 0 for some values of i, • then P(x) ≥ 0 for x ≥ 0 provided that P(x) does not have odd-ordered zeros on the positive x-axis (except at x = 0 and x → ∞). If the multiplicity of zeros of P(x) is odd, then the graph of P(x) crosses the x-axis at those points (condition not met); otherwise, if multiplicity of zeros of P(x) is even, the graph of P(x) touches the x-axis at those points. • Therefore, even-ordered (even multiplicity) zeros (x-intercepts) are allowed, but odd-ordered zeros are not.

13. Realizability Theory and Positive Real Functions Sturm’s Theorem and Sturm’s Test … P(x) = anxn + an-1xn-1 + …+ a1x + a0 Consider the last case • vIf a0 ≥ 0, an ≥ 0 but ai < 0 for some i, • then P(x) ≥ 0 for x ≥ 0 provided that P(x) does not have odd-ordered zeros on the positive x-axis (except at x = 0 and x → ∞). There are three methods to check whether P(x) has odd ordered roots on the positive x-axis Method 1 – Factorization Factorize P(x) and observe the order of roots on the positive x-axis Method 2 - Plotting

14. Realizability Theory and Positive Real Functions Sturm’s Theorem and Sturm’s Test … Method 3 – Sturm’s test P(x) = anxn + an-1xn-1 + …+ a1x + a0 Procedures Define P0(x) = P(x) P1(x) = P’(x) To find P2(x), divide P0(x) by P1(x) to get a two term quotient and a remainder. The remainder is negative of P2(x). i.e. P0(x) / P1(x) = b1x + c1 + [-P2(x)] / P1(x) Repeat this stem for P3, P4 … Euclid Algorithm:P0(x) = P(x) P1(x) = P’(x) P0(x) = q1(x)P1 + [-P2(x)] P1(x) = q2(x)P2 + [-P3(x)] … … Pk-2(x) = q1(x)Pk-1 + [-Pk(x)] The process stops when the remainder Pk(x) becomes a constant (when k = n) or zero ( when k ≤ n premature termination)

15. Realizability Theory and Positive Real Functions Sturm’s Theorem and Sturm’s Test … Case 1: Pk(x) = constant when k = n Sturms’s Theorem The number of odd-ordered zeros which Pk(x) has in the interval a ≤ x ≤ b is equal to |Sb – Sa| where Sa and Sb are the number of sign changes in the test (P0, P1,P2,…,Pk) evaluated at x=a and x=b respectively. Here, we are interested in the presence of odd-ordered zeros on the positive x-axis, hence we take a = 0 and b → ∞.

16. Realizability Theory and Positive Real Functions Sturm’s Theorem and Sturm’s Test … Case 2: Pk(x) = 0 when k < n • This shows that Pk-1(x) is a factor of P(x) • All zeros of Pk-1 are zeros of P(x) • The multiplicity of these zeros in P(x) is one higher than their multiplicity in Pk-1(x). • The test continues by taking polynomials P0 to Pk-1. • The zero count |Sb – Sa| in this case is the sum of number of odd-ordered zeros plus multiple zeros (due to Pk-1(x) each counted once).

17. Realizability Theory and Positive Real Functions Sturm’s Theorem and Sturm’s Test … Example Test whether the following polynomials satisfy the condition P(x) ≥0 for all x ≥ 0. • p(x) = x2 – 4x + 3 • p(x) = x4 – 8x3 + 23x2 – 28x + 12

18. Realizability Theory and Positive Real Functions Testing Deriving-point functions Recall Equivalent Positive-real conditions • H(s) should be quotient of polynomials with positive and real coefficients • For all real ω, Re[H(jω)] ≥ 0 ►P(x) > 0 for all x > 0 • All poles and zeros of H(s) should lie in the closed Left-hand S-plane. All jω-axis poles and zeros must be simple (multiplicity of one) with positive residues. If a deriving point function satisfies these conditions, then it can be realized using passive elements (resistors, inductors, capacitors and transformers)

19. Testing Deriving-point functions … Realizability Theory and Positive Real Functions The following are general test procedures with all common factors removedZ(s) is considered. The same procedure can be followed for Y(s) • Inspection test for necessary conditions • all polynomial coefficients are real and positive • Degree of N(s) and D(s) differ at most by1 • Lowest degrees of N(s) and D(s) differ at most by 1 • There should be no missing terms in N(s) and D(s) unless all even or odd terms are missing. • jω – axispoles and zeros must be simple • Test for necessary and sufficient conditions • Re[Z(jω)] ≥ 0 for all real ω • Or P(x) ≥ 0 for all x ≥ 0 • b) N(s) + D(s) must be strictly Hurwitz

20. Realizability Theory and Positive Real Functions Testing Deriving-point functions … Example Test positive real nature of the following Dp impedance function.