At the most basic level, filters are necessary in RF devices so that unwanted frequencies do not pass through the circuit and cause interference. While filtering can become quite complex as operating frequencies increase, it can be made much less daunting by having an understanding of how basic filters are built using standard electrical components such as resistors (Rs), capacitors (Cs), and inductors (Ls).
Before we jump into types of filters though, let’s start by looking at how a voltage divider functions. A voltage divider is a passive linear circuit that produces an output voltage (Vo) that is a fraction of the input voltage (Vi). As shown in Figure 1, the voltage divider changes the Vi down to the Vo based on the values of the resistors used, which are R1 and R2 below.
If we start with Ohms Law, which as a reminder, states that electric current is proportional to voltage and inversely proportional to resistance since resistance hinders the flow of current in the circuit, we can derive the following formula for how a voltage divider works:
As you can see from the formula above, as R1 grows (or as R2 shrinks) the output voltage will drop.
Swapping Rs and Cs to Turn Our Voltage Divider into a Filter
Now that we know how our simple voltage divider works, let’s look at what happens if we trade out some Rs for Cs. Impedance of a capacitor will change with frequency and that capacitance can be calculated as follows:
Z = Impedance
Omega = Angular Frequency
i is the imaginary number since we are using complex numbers here
C = Capacitance
Thus, for a capacitor, impedance decreases with frequency. So, if we swap R2 for a C as shown in Figure 2, we will have a low-pass RC filter, which is a filter circuit that passes frequency signals below a certain cutoff frequency and blocks frequency signals higher than that point.
RC Low Pass Filter
In the RC low-pass filter, the path to ground goes through a capacitor, which means impedance will decrease with increasing frequency. Therefore, in this circuit, the ratio of the Vi down to the Vo will depend on the values of R and C and the frequency of the signal. With high frequencies, impedance is low and energy is sent to ground as Vi is divided down. Low frequencies see a higher impedance and energy is sent to the output.
If we do this the other way around and replace R1 with a C, the path to output goes through the capacitor and we get the opposite effect.
RC High Pass Filter
In this configuration, low frequencies see impedance that is higher than R, so the low-frequency signals go to ground while the high-frequency signals see an impedance lower than R, which means we get a high-frequency output.
Swapping Cs and Ls to Create Your Desired Filter
Inductors have the opposite response to frequency than capacitors. This can be calculated as follows:
L = Inductance
This means that impedance increases with frequency in an inductor. To create a low-pass filter using an R and L, we can start with an RC high-pass filter as shown in Figure 4 and swap the C for an L. We can also take the same type of approach to design a high-pass filter if we start with a RC low-pass filter as shown in Figure 2 and replace the C with an L. Both examples are illustrated in Figure 6.
As it turns out, RC filters don’t provide the best performance in terms of roll off, which is the slope over frequency from passing a signal to blocking it. To improve this, RF designers can combine Cs and Ls, balancing their opposite responses to frequency to build different low-pass, high-pass, and band pass LC filter responses as shown in Figure 7.
Manipulating Impedance to Achieve a Desired Frequency Response in Your Filter
Given that Rs, Cs, and Ls offer different variations on impedance, we can think of designing a filter response as manipulating different impedances to achieve a desired frequency response. Thinking back to the DC voltage divider we started with at the beginning of this post, we can build on this to create a high pass circuit by adding networks of impedance (Zs) using various combinations of RCs and LCs as discussed throughout this post. Figure 8 below shows a high pass circuit where Z1 decreases with frequency and Z2 increases with frequency. In general, by replacing individual Rs, Cs, and Ls with sections of a circuit that have very specific impedance vs frequency behavior, we can create more complex high-performance filters.
By breaking down the different ways to use the frequency dependencies of Cs and Ls, you can see how it is possible to get a variety of different filter responses and form simple filters that can serve as the building blocks for more complex filtering needs.
Five key filter specifications to understand and check can be followed in related article here.