# 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:

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 **jω** by **s**. The circuit then in the **Laplace Domain** becomes:

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:

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:

**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

**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:

**voltage divider rule**:

**b)** Determine the zero

**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

**b)** Determine the pole

#### Answer

**c)** Determine the -3dB frequency