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

General Terminologies of Two-Port networks: Network: A network is an interconnection of passive elements and dependent sources. There are no active or independent sources inside the box. The network is treated to be part of the square box shown in the figure. Terminal: A terminal is the endpoint of the conductor which is connected to the node of the network which is brought out of the designated square box representing the network Port:  port is a pair of terminal that is used to connect to the sources of energy i.e. independent sources and or the excitation. A network may be one-port in which there is only one driving force or multi-port network to which multiple sources of energy are connected. We consider a 2-port