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:

1. N(s) is real for s real
2. Q(s) is Hurwitz polynomial
3. If N(s) has poles on (jw) axis, poles are simple and residues thereof are real and positives
4. Real N(jw) >=0 for all values of w

Properties of prf

Given a transfer function N(s) = P(s)/Q(s)

1. 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
2. The higher and lowest power of P(s) and Q(s) differ only by one. This condition prohibits multiple poles or zeros at s=0 and s=∞
3. If N(s) is a prf then its inverse is also a prf
4. The sum of positive real functions is also prf (difference of prf may not be prf)