# BJT Active low-pass filter frequency calculation

The left BJT of this circuit acts as a low-pass filter.

According to Simple transistors Low Pass filter, where I found this schematic, its cut-off frequency is 150 Hz.

How do we calculate this frequency?

simulate this circuit – Schematic created using CircuitLab

Ah, it's "one of those sites", with their fantastic made-up circuits.

The circuit is reproduced below.

simulate this circuit – Schematic created using CircuitLab

The response from input to output of stage 1 with stage 2 attached and stage 2 respectively is plotted below.

The very first stage acts like a bandpass filter, the 2nd stage provides some additional low frequency roll-off.

So no, the circuit does not quite do what the description claims. I shall cite that:

The first stage is base on NPN bipolar junction transistor (BJT) C2383 which cuts off high audio frequencies from the input to low (bass). Then, the second stage which is based on NPN BJT C2655 amplifies these low frequencies generated by the first stage.

It doesn't work that way, sorry. You'll not find 150Hz anywhere in the design of this circuit unless you actually send 150Hz to the input :)

Looking at the schematic, the 1st and 2nd stage are supposed to have a voltage gain of about 5. The 1st stage is a high pass filter, with the following response - if we disconnect stage 2:

Just adding C3 turns this into a 2nd order bandpass filter.

The 2nd stage's C4 and C5 then add some additional low-frequency roll-off, as shown on the first response graph (orange response curve).

Further from the site:

produces low (bass) audio frequencies from the range of 20Hz to 150Hz.

Yes, it somewhat does it, but the "range" described is slightly fictional, as I have no idea what criterion was used to divine those two frequencies.

I've indicated the 20Hz and 150Hz response points with the cursors on the plot below.

The "20" and "150" numbers were just made up.

The circuit is probably copied from somewhere, since the transistor choice is odd to say the least. Both transistors can be the same, and they can be pretty much any reasonable NPN type - that's the whole point of such simple designs, they are not meant to be highly dependent on the highly variable transistor specs. Sure, you could design it for selected transistors with exact gains needed, but that's only called for in designs that are running close to the limits of the technology. For a general purpose bandpass filter for such a low frequency range, a good design is one that will work with almost anything you put in it as far as transistors go. It should also not be terribly dependent on the exact values of electrolytic capacitors, since those vary a lot in practice.

Now if someone asked me to breadboard an actual 20-150Hz filter, I'd probably just use a design tool to have something to start with. The equations can be looked up in any filter design book and are mathematically very elegant. Op-amp based implementations of those are rudimentary and perform well.

The most basic design tool would give back a passive LC filter. It can be made more practical for audio frequencies using active components. A 3-section pi bandpass could be two L||C shunts to ground with a series L-C section between them. The inductors would be fairly large - unpractically so - and they could be approximated with op-amp gyrator or even a single BJT gyrator. The gyrator would convert the impedance of a small capacitor (on the order of 1μF) into the impedance of a large inductor (on the order of 1H). BJT inductors need bias, so two more gyrators could act as a "big inductors" for two DC bias networks.

So, very roughly, it could look like:

simulate this circuit

This passive circuit does not require the bias network consisting of LB1-LB3 - if you remove those three inductors, it has the same frequency response. The inductors are shown just to demonstrate that "very large" inductors can be used to push some DC current through the sections, should the sections need it - e.g. if we had some transistors there, not just inductors.

1 kiloHenry at 1Hz has an impedance of 6.3kΩ - high enough to be of small concern in this circuit. These bias inductors could be practically made with lots of turns of very thin wire on a high permeability core, and would have a high ohmic resistance as well, and thus would not be a DC short. That's for real inductors. On the other hand, the SPICE models may have poor numerical behavior, since the DC current will be very high. Some SPICE systems will detect DC shorts via inductors and will raise the inductor's series impedance from the ideal 0Ω, to keep the current finite and make it possible to get a transient solution.

The frequency response of this filter, with response of the original filter shown for reference, looks like this:

It's a decently looking filter if you feel like winding all those inductors, and tweaking the capacitors.

Now, let's replace the inductor L2 with a gyrator, referenced to CL2. The gyrator transforms the capacitance into inductance:

$$L_2 = {C_{L2} \over g^2} = 4.9\mu\,{\rm F} / 10^{-6} = 4.9\,{\rm H}.$$

The scaling constant $$\g\$$ is arbitrary, and is chosen so that reasonable component values can be used in a practical implementation.

simulate this circuit

The same could be done to the remaining inductors.

Now let's implement an "ideal inductor" using a transistor. It's not as good a circuit as a gyrator - we have to "tune" the values a bit.

simulate this circuit

With a few tweaks, ignoring the slight voltage gain change, the response is not too bad for getting rid of a 4.9H inductor:

Now we can use gyrators for L1, L2 and L3. Since L2 needs a tee bias, there's an ideal inductor built around Q7 that provides a DC path for that, while blocking AC.

simulate this circuit

Q1-Q6 are not critical and can be 2N2222 or most any other transistor that will survive 20mA collector current and 30Vceo or more. Instead of using Darlington pairs, single high-gain types like BC547C could be used instead without much degradation.

The DC operating points of the above circuit could use some improvement, e.g. using current sources for bias, but it's a reasonable starting point at least.

There was no tuning of gyrated capacitors - their values are derived directly from the coil inductances and the scaling resistors.

The manually tuned settings were the bias voltages, and the wiper position of R33.

The response looks as follows:

A fixed-gain section can be added after the filter to bring passband gain back up to 0dB.

The gyrators could also be op-amp based, and would perform a bit better, but for a simple speaker filter it doesn't matter that much.

I want to stress that a gyrator-based solution is not the only one nor the best one. It's just a solution that I chose to show. That's all. I find gyrators nifty - as well as negative impedance converters, impedance scalers/multipliers, etc.

• Thanks for this well detailed answer. Before posting this question I did a LTSpice simulation of the first stage including the capacitor C3 and got more or less the same frequency response as yours. So there is no equation to find this frequency band? Could this filter be considered an active band-pass filter? Jan 12, 2023 at 16:05
• @raffs Of course there is an formula. Solve the circuit with some simple assumptions about what the transistor is doing, and use complex impedances. You'll get output voltage as a function of input voltage and frequency. That's the simplest you can do. In this application, the transistor is a simple self-biased voltage buffer. You could replace it with a voltage-controlled voltage source (VCVS) set to the effective gain of &approx;5, with a series output resistor of 100&ohm; - that's about the source impedance as seen from the output side of Q1 (first stage). Jan 13, 2023 at 2:42
• @raffs I’ll also add the design equations for the filter I used here for completeness. Jan 13, 2023 at 12:06
• I'm still learning electronics (especially analog). Currently I'm playing around with AM radio signals, oscillators (tank circuit) and filters. Part of this answer is beyond my knowledge yet. For example: I understand that if the calculations result are the same, inductors could be replaced by transistors and some passive elements but I cannot visualize how to switch between them yet. You mentioned "gyrators" which are totally new to me. After reading this answer I got some direction to go on this topic! Thank you! Jan 16, 2023 at 15:50
• @raffs Maybe it would help by looking at the gyrators at DC and AC separately. They are transistor circuits, and thus must be biased, and to have sufficient gain, they need some collector current. The transistors in this circuit operate at a couple mA of DC collector current. The behavior at AC is a separate aspect - once the DC point is established, their behavior - as observed between collector and emitter - is of a reciprocal of the reference impedance. Recall $X_C={1\over j\omega C};\,\,X_L=j\omega L$, and a gyrator does $C^{-1}\longleftrightarrow L$, up to a constant factor $g$. Jan 17, 2023 at 14:12