**RC Driving Point Impedance function:**

As the name indicates, the RC networks consist of only R and C components. There is no inductor in RC networks. The RC Driving Point Impedance function is denoted as Z_{RC}(s). The properties of RC Driving Point Impedance function and the properties of driving point admittance function of RL network are identical. Thus the properties Z_{RC}(s) and Y_{RL}(s) are same. There are no complex poles in RC network function. The poles and zeros alternate each other and are located in left half of s plane. To understand the properties of RC network function consider a driving point impedance function of RC network as,

The poles are at s = 0, â€“ 2

The zeros are at s = â€“1, â€“ 4

The pole-zero plot is shown in the Fig. 7.11.

**Properties of RC Driving Point Impedance Function:**

Referring to the pole zero plot of Z_{RC}(s) function considered, the various properties of RC Driving Point Impedance function can be stated as,

**The poles and zeros are simple. There are no multiple poles and zeros.****The poles and zeros are located on negative real axis.****The poles and zeros interlace (alternate) each other on the negative real axis.****We know that the poles and zeros are called critical frequencies of the The critical frequency nearest to the origin is always a pole. This may be located at the origin.****The critical frequency at a greatest distance away from the origin is always a zero, which may be located at âˆž also.****The partial fraction expansion of Z**_{RC}(s) gives the residues which are always real and positive.**There is no pole located at infinity.****The slope of the graph of Z(Ïƒ) against Ïƒ is always negative.****There is no zero at the origin.****The value of Z**_{RC}(s) at s=0 is always greater than the value ofÂ Z_{RC}(s) at s = âˆž.

It can, be seen that for a Z_{RC} considered Z_{RC}(0) = âˆž while Z_{RC}(âˆž) = 1.

Consider a simple RC network as shown in the Fig. 7.12.

Let us plot Z_{RC}(Ïƒ) against Ïƒ where s = Ïƒ.

To find the slope ofÂ Z_{RC}(Ïƒ) against Ïƒ find d Z_{RC}(Ïƒ)/dÏƒ

Thus the slope ofÂ Z_{RC}(Ïƒ) against Ïƒ is always negative for any value of Ïƒ.

The graph of Z(Ïƒ) against Ïƒ for the RC network function is shown in the Fig. 7.13.

Ï‰_{1},Ï‰_{2},Ï‰_{3} and Ï‰_{4} are the critical frequencies. At the critical frequencies like Ï‰_{3}, the Z(Ïƒ) changes its sign suddenly such that the slope always remains negative. The value of Z (âˆž) is constant so graph runs parallel to the Ïƒ axis, finally.

The nature of Z (Ïƒ) againstÂ Ïƒ graph when there is no pole at the origin is shown in the Fig. 7.14.

All the properties of driving point admittance function of RL networkÂ [Y_{RL}(s)] are exactly identical to the properties of driving point impedance function of RC network [Z_{RC}(s)].

**Realization of Impedance Function of RC Network:**

As mentioned earlier, the realization of Z_{RC}(s) function can be achieved using Foster I, Foster II, cauer I or cauer II form. Remember that the number of elements are not equal to highest power of s in overall Z(s) for RC networks.