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IMS2022


ARTICLE | SEPTEMBER 14, 2022




FILTER BASICS 9: AN INTRODUCTION TO POLES AND ZEROS

Source: Knowles Precision Devices DLI

To help customers with filter selection, we generally provide a lot of
information on what our filters can do. But in this new Filter Basics Series, we
are taking a step back to cover some background information on how filters do
what they do.

Regardless of the technology behind the filter, there are several key concepts
that all filters share that we will dive into throughout this series. By
providing this detailed fundamental filter information, we hope to help you
simplify your future filtering decisions. 

In part 9, we go in-depth on the background information of how poles and zeros
impact a transfer function to show you how you can use this information to
improve your filter’s performance.

In Part 7 of our Filter Basics series, we discussed the different ways you can
look at Q factor, one of which is to consider the Pole Q factor (often used with
more complex systems). We also explained in that post that filters have a
transfer function 𝐻(𝑠) which tells us what an output signal will look like for
a given input signal. Note that filter transfer functions are expressed in terms
of the complex variable ‘s.’ 

Poles and zeros are properties of the transfer function, and in general,
solutions that make the function tend to zero are called, well, zeros, and the
roots that make the function tend towards its maximum function are called poles.
Let’s look at how this works using a simple RC first order lowpass filter, like
the one we looked at in Part 2 (Figure 1).

 The transfer function for this filter written in terms of the complex
frequency s, is as follows:



Thus, when s (frequency) = 0, the transfer function is 1 and we say the filter
has a DC gain of 1. At s = -1/RC the transfer function will tend to infinity, so
we say we have a single ‘pole’ at frequency s = -1/RC. 

Now, knowing there is a ‘pole’ at s = -1/RC really does not help us understand
how the filter performs versus frequency ω, not yet anyway. To determine this,
we are going to look at a more general transfer function for a first order
filter:



Then to understand the frequency response we replace s with where is the
imaginary number “i”:



When  =  the transfer function tends to infinity, and we say we have a pole.

Next, if we plot the pole at   in the complex plane of the ‘pole zero’ plot and
mark it with an X, you get the graph shown in Figure 2. To see how the transfer
function behaves at different values for frequency we can move the frequency
value up and down the imaginary (vertical) axis for different values of :

 

Our transfer function will perform in the following manner – as the distance
from the pole at to the frequency we are interested in grows, the signal will
decrease since we are dividing by the size of that green vector .

Some additional general notes about this transfer function: 

 * At  – We are as close to the pole as we can get if we stay on the imaginary
   axis and our transfer function Y will be at a maximum
 * At  – We are as far away from the pole as we can get, and our transfer
   function Y will be at a minimum
 * At  – Our amplitude will be down by a ratio of compared to its maximum, so in
   dB this is -3dB and we can say that is our cutoff frequency

Therefore, in this simple case, our pole at  gave us a cutoff frequency at .

Similarly, our RC filter above with a pole at  gives us a cutoff frequency of .
This makes sense and we probably already know this is the cutoff frequency of an
RC filter, but getting there via a roundabout route through a pole zero plot can
help us understand how poles impact filter behavior. 

Using Pole and Zero Information to Enhance Your Filter Designs

Through this single pole example, we can make the following general observation
about poles:

 * The closer your frequency of interest puts you on the complex plane relative
   to a pole, the filter’s transfer function will increase
 * The further you are away from a pole and the filter’s transfer function will
   decrease
 * Zeros have the opposite effect – the closer your frequency puts you to a
   pole, the filter transfer function will decrease and vice-versa.

As an RF designer, if you have an in-depth understanding of how poles and zeros
work, you can take advantage of this information in your filter designs and
improve your filter’s response. For example, you can place zeros near
frequencies you want to reject and poles near frequencies you want to pass. 

In Part 10 of our Filter Basics series, we will discuss all things resonators
including resonators as microwave devices, coaxial ceramic resonators, and
dielectric resonators.








KNOWLES PRECISION DEVICES DLI

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MORE FROM KNOWLES PRECISION DEVICES DLI

 * FILTER BASICS PART 7: DIFFERENT APPROACHES TO Q FACTOR
   
   In part 7 of this series from Knowles Precision Devices, a deep dive is
   performed on the different ways tocan think about Q factor for the components
   going into your filter or your filter as a whole....

 * LUMPED ELEMENT FILTERS: A COMPACT, LOW-FREQUENCY FILTERING OPTION
   
   We explore the basics of lumped element filter design,its characteristics,
   and how we can push filter design limits to develop a wide variety of
   high-performance low-frequency filtering options.

 * FILTER BASICS SERIES PART 2: DESIGNING BASIC FILTER CIRCUITS
   
   In part twoof this series, Knowles covers how RF designers can use the
   different frequency dependencies of capacitors and inductors to manipulate
   impedance and create various filter responses.

 * FILTER BASICS PART 3: FIVE KEY FILTER SPECIFICATIONS TO UNDERSTAND
   
   In part 3 of this series, Knowles aims to help simplify filter selection by
   providing an overview and reference point for five of the most commonly
   discussed filter technology specifications.

 * UNDERSTANDING THE BASICS OF CERAMIC COAXIAL RESONATOR FILTERS
   
   In general, a resonator is an essential component for constructing a bandpass
   filter since the resonator is what will allow specified frequencies, or bands
   of frequencies,to pass through the filter.

 * FILTER BASICS 8: AN IN-DEPTH LOOK AT BANDWIDTH

 * FILTER BASICS PART 5: LUMPED ELEMENT AND DISTRIBUTED ELEMENT FILTER
   CONSTRUCTION

 * FILTER BASICS 10: RESONATORS AS MICROWAVE DEVICES

 * FILTER BASICS PART 4: KEY FILTER TYPES AND TECHNOLOGIES

 * PLANAR FILTER TECHNOLOGY FOR MILLIMETER WAVE APPLICATIONS

 * FAQ: WHAT IS Q FACTOR?

 * 05.18.22 -- 5 KEY FILTER SPECIFICATIONS TO UNDERSTAND

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