I recently made a post about radio-frequency terminology and decided to provide examples for a specific section: Bandwidth. This topic can be especially confusing when researching online because the term doesn’t make sense without being able to see the effects without having a complicated oscilloscope setup to see waveforms. This post aims to explain how a basic frequency filter works and then expanding on that filter to a more complex design without looking at the complex mathematics.

Here is my Maker.io post on making a physical circuit that takes any audio input and a user selects which audio frequency range they want to hear on the output via switches. This uses the Butterworth filters mentioned at the end: Audio Frequency Separator. Sometimes having a physical example is easier to understand instead of reading about it so that is why I linked this.

# First Order Filters: Low-Pass, High-Pass, and Band-Pass

Frequencies generated by a transponder/transceiver sometimes have no use in particular applications or are unwanted noise. This is why many devices filter out specific frequencies via different filters. The term first-order simply means mathematically there is only one exponent for any variables used.

In the physical world of electronics, it simply means an inactive filter using resistor-capacitor networks or resistor-inductor networks. The term “inactive” means there is no extra power being added to the system, so voltage can only drop due to physical loss such as heat or any other form of energy. These filters are usually used in conjunction with higher-order (active) filters to improve the overall effect of the filter.

## Resistor-Capacitor Low Pass and High-Pass Filters

I will only be covering RC networks since I don’t usually use RL networks for filtering, full designs usually have combinations depending on what the application requires. Here is what an RC low pass filter looks like:

The resistor allows slower frequencies to pass through with some loss while the capacitor may not have time to charge up enough to affect the lower frequency. It does matter what values of R and C are, but almost always the R is chosen depending on if more voltage is required or if more current is required. R is also chosen so C comes close to a typical capacitance value. Then a C value is calculated given the frequency one wants to begin “cutting-off”. The equation used for calculation is:

So let’s say one wanted a 1500Hz cutoff and wanted to use a 100 Ohm resistor, the following capacitance would be used:

I used LTSpice to simulate a frequency sweep by clicking run and choosing AC Sweep. I used 1000 data points, swept from 1Hz to 5000Hz (5kHz), and used a linear sweep to get the following graph.

I changed the X-axis by changing it to a logarithmic scale to see a better view. I measured the AC voltage which is the blue line at 14dBV. The green line shows how the filter reacts, I also have a vertical measurement at the 1500Hz location which is about where something called the -3dB roll-off point is. In other words, this is where the voltage starts to severely drop as frequency increases.

Here is what a high-pass filter looks like:

The equation for calculation stays the same, the locations of the capacitor and resistor swap, however. The capacitor behavior is often described as “blocking” the lower frequencies and “passing” higher frequencies. Electrically it is actually charging and discharging in an alternating pattern but for easier visualization, the standard is often described the other way to help beginners. This will block frequencies lower than 1500Hz and pass the ones above.

Again, I used AC sweep with 1000 data points, started at 1kHz, and ended at 10kHz. It severely drops in amplitude beneath 1500Hz and will slowly get back up to nearly 14dB (some losses on this one). If you combine these two filters, you get a band-pass filter. Usually, the high pass filter comes first, and then the low-pass filter is second (not always the case). Let’s say we want signals from 1500Hz to 5000Hz. Our cutoff capacitor would stay the same for the high pass filter while the low pass filter would need a 0.32uF capacitor. Here is the circuit:

Here is the frequency response:

Notice that this doesn’t go anywhere near the initial 14dB point, there are a lot of losses going on because it’s not an active circuit. I marked the severe cut-off points on both ends of the filter, this would really only allow about 9.1 to 9.9dBV or 2.8 to 3.1V on the output.

# Higher Order Filters

Second-order filters and beyond are way more effective and efficient because an external power source is added to the mix. Not only do you get a potential for increased power, but active filters also aid the signal so that it stays above a certain voltage threshold until the cut-off point. It also has the ability to attenuate the signal even better after the -3dB roll-off point. Here is a diagram to show what I mean:

All the colored lines are starting at 5dB, I didn’t want to overlap them in the same position. In the world of electronics, it is impossible to achieve perfect filtering as seen by the yellow line. However, it is very possible to nearly approximate this by getting steeper with higher-order filters. Graphs like these usually describe a “decibel per decade drop” to describe how steep the cutoff is. All it really means is the slope of the line after the -3dB cutoff point. Achieving a higher slope is usually the goal for a good filter. Does this mean the calculations are harder? If you want to do circuit analysis, the answer is yes, but if you simply want to build one the calculation for frequency actually stays the same. Here is what a third-order low-pass filter looks like, this is called a non-inverting Butterworth filter design:

I labeled the “sections” of the circuit so it is easier to understand. Each low-pass RC network uses the same calculation for values.

This means all the capacitors will have the same value and the corresponding resistors will have the same value (not the gain resistors). From research, it seems Ra and Rb for gain are usually the same value getting a gain of 2. Again, the other resistors used in the different low-pass networks are chosen the same way as first-order filters and then the capacitor values are calculated given the desired cut-off frequency. Just to show results, I decided to run a simulation using a cutoff of 10kHz. The resistor I usually choose is 10k Ohms since I want the voltage to stay around the same level. Based on these values, the capacitor would need to be 1.59nF. Ra and Rb I chose 1k Ohm as a value. I used an AC sweep again using a Decade sweep with 5000 points per decade, started at 1kHz, and ended at 50kHz. Here is the circuit with values:

Off-screen I do use a -5 and 5VDC source which is noted by V+ and V-.

Here is a graph comparing a regular inactive filter compared to the third-order filter after doing the AC sweep:

The blue line is the active filter and the green line is the inactive filter set at the same cutoff frequency. Notice that the blue line stays constant longer much closer to the 10kHz cutoff point (this is good). Second, note the significant differences in slope after the 10kHz point, the blue filter performs much better. A high pass filter looks exactly the same except the 10k Ohm resistors and capacitors are swapped around.

To get a band-pass filter, you simply combine both filters by feeding one into the other. The only recommended change is adding something called a unity gain buffer between stages and between output and the load. This acts as an “impedance bridge” between the output of the first op-amp and the input of the second op-amp. A unity gain buffer will increase the ability for current to pass from the input to the output more efficiently. The mathematical reason is that the first-order filter changes the impedance for the next stage. Here is a finalized circuit for a low-frequency cutoff of 10kHz and a high-frequency cutoff of 50kHz bandpass filter.

Finally, here is a simulation using an AC sweep with decade sweeping, 5000 data points per decade, starting at 8kHz and ending at 80kHz. The blue line is the base signal that is un-filtered. The red line is the inactive filter response. The green line is the active filter response.

I added a voltage divider at the end because the op-amp for the low-pass portion actually amplifies the signal again to a much higher voltage level (the two filters compound the gain). As you can see, there are two unity gain buffers to bridge between the high-pass filter and the low-pass filter as well as the output of the low-pass filter and the voltage divider.

The graph again shows how well an active filter performs compared to an inactive filter. Stability and ability to attenuate is important for filters. These aren’t the only filters available (there are far more advanced ones that do an even better job), but they all technically work the same way. The results will look very similar to the graphs I’ve shown. Things also get more complicated at Ultra High Frequencies because ideal components can actually start to behave differently due to resonant frequency.