Network Synthesis Lecture Notes Network analysis: You are given a network then you analyze the behavior (output) by applying certain excitation. Network Synthesis: You know the output response for certain excitation, then you are required to find appropriate Z(s) or Y(s) that will give that desired behavior. Let us consider the network equations in terms of impedance and admittance:   Process of network synthesis: let us first consider figure-1(a) Z(s) = Z1(s) + Z2(s) Rearranging, Z2(s) = Z(s)  -  Z1(s) That is we find Z2(s) by removing Z1(s) from Z(s). This removal of Z1(s) from Z(s) can be performed in following different ways: Removal of a pole at infinity: It means that the degree of P(s) is of one degree higher than the degree of Q(s). It infers that

Positive Real Function (prf) We know that the driving point impedance (Z(s) and driving point admittance (Y(s))are of the following type: N(s) = P(s) / Q(s) The function is prf if: N(s) is real for s real Q(s) is Hurwitz polynomial If N(s) has poles on (jw) axis, poles are simple and residues thereof are real and positives Real N(jw) >=0 for all values of w Properties of prf Given a transfer function N(s) = P(s)/Q(s) Bothe P(s) and Q(s) are Hurwitz, poles and zeros cannot be on the right hand of s-plane, and only simple poles with +ve real residues can only exist on jw axis The higher and lowest power of P(s) and Q(s) differ only by one. This condition prohibits multiple

Properties of the Routh Hurwitz Polynomial: A polynomial to be Hurwitz when- P(s) is real when s ir real The roots of P(s) have real parts which are zero or negative. Properties of Hurwitz polynomial P(s) = ansn + an-1sn-1 +................a1s + a0 are: Coefficient of s must be positive Both odd and even parts of Hurwitz polynomials should have roots on the imaginary axis only The continued fraction expansion of the ratio of even to odd (if even part is of higher degree) or odd to even(if the odd part is of the polynomial is of higher degree) of Hurwitz polynomial should be positive quotients If a polynomial is Hurwitz to even multiplicative factor W(s) i.e. P1(s) =W(s)P(s), if

Let the network function be written as : N(s) = P(s) / Q(s) Above equation consist of factors of the general form where pr and pn are complex frequencies. Their difference is also a complex frequency which  can be written as: Where Mnr is the magnitude of the phasor and Φnr is the phase angle of the same phasor Thus All Ms and phase angles can be calculated using the pole-zero plot on s-plane Procedure for calculation of M and Φ Plot the pole-zero of the network function Measure the distance of M1r, M2r ... of a given pole from each zero Measure the distance of Mar, Mbr ... of a given pole from each other finite pole Measure the angle Φ1r, Φ2r ... from

Stability of a system from Pole-zero concept Given a polynomial N(s) = P(s) / Q(s) By factorizing the numerator and denominator polynomials, we can easily show that the polynomial becomes zero when the ‘s’ terms in the numerator polynomial have the values s=0, -z1,-z2.......-zn, thus the roots of numerator define the zeros. Also, the roots of the denominator polynomial define the poles of the function. Zeros are marked by ‘O’ on the s-plane while a pole is indicated by a X The s-plane is shown in figure below Case-1 Unit step function F(s) =1/s F(cos(w)) = s/(s2+ω2) F(sin(w)) = w/(s2+ω2) The function has a only one pole at s=0   The function has a zero at s=0 and two poles at +/- jω There are two poles at    +/-jω In all the