Poles and Zeroes

Poles and Zeroes, what could they be? I'll give you a left of field hint, poles are related to the poles which hold up circus tents.

Okay, from a less abstract frame of reference they are very useful metrics when it comes to describing an electronic system in the frequency domain.

Describing Electronic Systems in the Frequency Domain

So far we have primarily looked at analyzing the behavior of circuits within the time domain, i.e. how does the voltage or current change with time. While this is certainly useful, it is also highly beneficial to describe how the amplitude and phase change with frequency. It is this aspect which we will now dive more deeply into.

We have touched on it briefly by plotting the frequency response of filters, however, is there a way we can look at a circuit and determine an equation which relates the output quantity, such as a voltage, to an input quantity such as the supply voltage to develop intuition of how the output relates to the input? The answer is yes and it is known as the Transfer Function.

Let's consider the classic RC circuit to build up the Transfer Function:

Now the question is, can we formulate a frequency domain representation of this circuit? Yes we can, and we have already encountered one representation which is the phasor form which we can re-annotate the circuit as below:

RC circuit in the Phasor Domain

Note that the above is great for representing sinusoidal signals, however, can we generalize the concept of impedance to any periodic signal such as a square wave or triangle wave? The answer is to look to Laplace.

Recall from mathematics that the Laplace Transform is an integral transform that converts a function of a real variable t, like a sine, square or triangle wave in the time domain, to a complex variable s, in the complex frequency domain. So how can we apply this to sinusoidal signals as a starting point? Well it turns out we simply replace by s. The circuit then in the Laplace Domain becomes:

RC Circuit in the Laplace Domain

It is at this point we can derive a Transfer function by determining the voltage across the capacitor in relation to the input voltage. We can determine the output voltage by applying the voltage divider rule to the above circuit as follows:

We can then determine H(s) by dividing by vs:

Let's now substitute in the impedance values and replace vo/vs with H(s):

We now have an equation which neatly describes the input-output relationship expressed in the Laplace Domain which we can also convert back to the Phasor Domain (replace s with jω). This now brings us to the concept of poles. How would the transfer function look when plotted on a curve? Let's draw that now assuming C=R=1:

If we set s to -1 then we will approach infinity from the right side and approach -infinity from the left side. If we visualize this in 3D space and say the vertical axis is the z-axis then one would imagine the graph going up into the sky as we approach -1 from the right hand side. One could imagine this as a pole in a tent going towards the ceiling and this is where the name comes from.

If we now replace s with j𝝅fC we have:

Which has the following spectrum:

Frequency spectrum of the RC circuit

Recall the concept of the -3dB point in terms of frequency from our discussion of filters is equal to 1/2piRC and 1/RC for frequency in terms of ω.

This is where we can connect the -3dB point to poles. If we equate the denominator of our s-domain equation to 0, what would the solution be?

Look at that, it's equal to the cutoff frequency in terms of ω. This then means that the poles exist at the cutoff frequency. This is a very powerful construct, it allows us to very elegantly quickly determine the cutoff frequency for any electronic system.

To close out this discussion we have the related concept of zero's which is the value at which the numerator equates to zero and going back to our visual analogy this would be the bottom of the tent, where the value approaches 0.

We can now generalise the transfer function to:

Where the poles and zero's are given by the following respectively:

There is one final point here worth also mentioning, it turns out that at the point of the pole the amplitude will rolloff at a rate of -20dB/decade and for zeroes will increase at a rate of +20dB/decade.

This then allows us to quickly sketch the frequency response of any electronic system by just knowing the location of the poles and zeroes!

Examples:

1. a) Determine the transfer function for the circuit shown below:

Let's begin by determining the output voltage across the resistor via the voltage divider rule:
From here we can substitute in the impedances in terms of s:

b) Determine the pole

Recall that the pole is determined by equating the denominator of the Transfer Function to zero, so let's do that :)

c) Determine the -3dB frequency

2. a) Determine the transfer function for the circuit shown below:

Once again we start by using our old mate the voltage divider rule:

b) Determine the zero

Recall that the zero is determined by equating the numerator of the Transfer Function to zero, so let's do that :)

c) Determine the pole

Questions:

1. a) Determine the transfer function for the circuit shown below:

Answer

H(s) = 1/(1+2*10^-3s)

b) Determine the pole

Answer

s = -1/2*10^-3

c) Determine the -3dB frequency

Answer

f = 79.577Hz

Ready for the next module? Return to the course home.