What is transistor matching and why
Let's start with a bit of theory, taking an NPN transistor as example (but feel free to skip this part).
In the active region (VC > VB > VE) the emitter current IE is described by the equation (Ebers-Moll model):
where IS is the saturation current, VBE = VB - VE is the base-emitter voltage, and VT is the thermal voltage defined by the absolute temperature T (in Kelvin)
or about 25 mV at room temperature.
That is a lot of info, but the main point is that the current passing through the transistor (note that the emitter current is almost equal to the collector current for a high-gain transistor) depends on the base-emitter voltage, and the parameter IS. This parameter depends on the details of how the transistor is made, and can be different from one transistor to the next (it also changes with temperature).
When making a differential amplifier, the linearity (and offset voltage) of the amplifier relies on both transistors having the same voltage-to-current characteristic, and so having the same IS. So, the goal is to find a pair transistors with (nearly) equal IS.
where IS is the saturation current, VBE = VB - VE is the base-emitter voltage, and VT is the thermal voltage defined by the absolute temperature T (in Kelvin)
or about 25 mV at room temperature.
That is a lot of info, but the main point is that the current passing through the transistor (note that the emitter current is almost equal to the collector current for a high-gain transistor) depends on the base-emitter voltage, and the parameter IS. This parameter depends on the details of how the transistor is made, and can be different from one transistor to the next (it also changes with temperature).
When making a differential amplifier, the linearity (and offset voltage) of the amplifier relies on both transistors having the same voltage-to-current characteristic, and so having the same IS. So, the goal is to find a pair transistors with (nearly) equal IS.
How to measure IS
We cannot measure IS directly, but instead we can pass a known current IE through the transistor, and measure the voltage VBE. Assuming the temperature does not change, matching VBE is then the same as matching IS directly. The traditional solution is to build an accurate current source, use an accurate multimeter to measure VBE, and try to stabilize the temperature in the test setup.
A much better way was presented by Ian Fritz in this article. The article gives a detailed description, but the basic idea is to directly measure the difference between two transistors directly, operated with the same power supply. This greatly reduces the required accuracy of the current source, as well as the required temperature stability. However, relative temperature between the two transistors is still critical.
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| Transistor matching circuit by Ian Fritz |
I reproduced the circuit here for clarity, but all credit and copyright goes to Ian Fritz. Rref and D1 act like a simple voltage regulator, setting VC to a fixed value (about 0.6 V). R1 and R2 determine the currents of the transistors. It is a good idea to match R1 and R2 (measure a bunch of resistors with a multimeter and pick the best pair) to ensure equal emitter currents. We measure the differential voltage between the emitter terminals with a multimeter (set to the most sensitive setting, preferably microvolts). Because the transistor base terminals are connected together, the measurement is the difference in VBE of Q1 and Q2. If the multimeter reads 0 V, we know the transistors are matched!
If you are familiar with differential amplifiers, there is actually a more intuitive way to explain this circuit. This circuit is simply a differential amplifier with 0 V differential input (both inputs are grounded). If the (differential) output is also 0 V, it means you have a good match.
Ian Fritz uses this technique to find a match to within 50 microvolts, but for basic differential amplifiers we don't care that much; some sources give 2 mV as a requirement. In stead of trying all possible pairs until you find a match, it is much easier to define one transistor to be your reference transistor, and leave it always in place as Q1. Then put all other transistors in as Q2 one by one, and record the voltage. You will typically find values of about 1 or 2 mV. Then, from the recorded values you can pick good matched pairs.
My setup and findings
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| My transistor matching setup |
I built the circuit on a breadboard, using two 9V batteries as power supply, and a batch of 25 BC547C transistors. It is immediately clear that temperature stability is key, and just touching a transistor will easily give 10mV difference or more. So you have to put in a transistor, and wait until Q1 and Q2 are again at equal temperature (wait until the reading stabilizes). To speed up the waiting, I put a small fan next to the setup. Ideally you would avoid touching the transistors, but getting them in place with tweezers is annoying. I tried wearing gloves to reduces thermal contact, but it hardly helped and was cumbersome. So, a few minutes of waiting for each transistor is needed. After a while, you get something like this, where the numbers are measured in mV:
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| Measured transistors with their measured VBE (in mV) relative to the 'reference transistor' |
The good news is that this technique is extremely simple, and the batch was still tested within an hour. Most transistors were actually very closely matched (owing to modern fabrication facilities, it was probably a lot worse back in the 60's), and I could sort almost all of them into pairs that are matched to within 0.1 or 0.2 mV.