Poles and Zeros of a Transfer Function
can you explain in simpler terms with explanation, please


If you have a very simple low pass filter made from a resistor (R) and a capacitor (C), you can calculate the transfer function (TF) as being: -

$$\dfrac{1}{1+sCR}$$

Then, if you re-arranged that TF you could get: -

$$\dfrac{\frac{1}{CR}}{s + \frac{1}{CR}}$$

Now the important thing to realize is that if s = \$\frac{-1}{CR}\$ the whole TF has a value of infinity. This is the position of the pole.

Take a few minutes to think about that because it is fundamental to understanding how the bode plot and pole-zero diagram are related mathematically.

For this particular simple example, that pole sits purely on the real axis of the s-plane where \$\sigma = \frac{-1}{CR}\$. This isn't the vertical \$j\omega\$ axis. The \$j\omega\$ axis is where the bode plot exists.

Considering the pole; at any point distant from that pole, the amplitude is not infinity and, at any point along the \$j\omega\$ axis you can predict the amplitude of the bode plot by using the distance from the plot and taking the reciprocal. However, that reciprocal has to factor in the natural frequency of the circuit. For this simple circuit, the natural frequency is 1/CR.

So, using a simple example at the origin of the s-plane (0, 0), the TF amplitude is the reciprocal of 1/CR divided by the scaling factor (1/CR) and in this pretty trivial example this works out at 1.

So the TF has an amplitude of 1 at DC. I say "DC" because the value of the \$j\omega\$ axis is 0 at the origin and this means 0 Hz or "DC".

If you were to move up the \$j\omega\$ axis by an amount equal to 1/CR, the distance from the pole becomes \$\sqrt2\$ times bigger and hence the TF amplitude becomes \$\frac{1}{\sqrt2}\$. This is normally called the 3 dB point because, in decibels the amplitude has fallen by approximately 3 dB. It's also called the half power point.


  What does the height of the cone indicate?


The height of the cone is infinity and, as such doesn't really tell you anything useful.


  And what do the different color rings in the cone indicate?


The coloured rings are arbitrary and don't tell you anything useful.