Poles, Zeros and the Transfer Function

To understand the effects of the capacitors, it is often useful to re-arrange equation (4.5) in the following form:  

$$\frac{v_{out}}{v_{in}}= \left(\frac{-R_C\vert\vert R_L}{R_{E... ...}C_1}}\right) \left(\frac{s-0}{s+\frac{1}{C_2(R_L+R_C)}}\right)$$ (71)

The first term in the parentheses of equation (4.6) represents the midband gain, the second term represents the high pass filter at the input, and the third term comes from the high pass filter at the output.

In electronics, it is useful to write polynomial expressions like those in equation (4.6) in the notation of what is commonly known as poles and zeros. If we define zeros as z₁=0 and z₂=0, and poles as $p_1=\frac{-1}{R_{in}C_1}$ and $p_2=\frac{-1}{C_2(R_L+R_o)}$, then equation (4.7) can again be expressed as:  

$$\frac{v_{out}}{v_{in}}= \left(\frac{-R_C\vert\vert R_L}{R_{... ...eft(\frac{s-z_1}{s-p_1}\right) \left(\frac{s-z_2}{s-p_2}\right)$$ (72)

As can be seen from equation (4.7), the zeros are the constants in each factor of the form (s-z) found in the numerator of the equation. The poles, on the other hand, are the constants in each factor of the form (s-p) found in the denominator. The overall equation for $\frac{v_{out}}{v_{in}}$ is called the transfer function. The poles and zeros are important because they indicate the angular frequencies where changes in the transfer function occur. In the notation of poles and zeros, we describe equation (4.7) as a second order transfer function with zeros z₁=0, z₂=0, and poles $p_1=\frac{-1}{R_{in}C_1}$, $p_2=\frac{-1}{C_2(R_L+R_o)}$.