Bootstrapping is great

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Bootstrapping is great

Sep 23, 2022 4 min read

Bootstrapping is a technique that has always fascinated me. In few words, by feeding an in-phase signal to the other end of the resistor, you can increase its value many times (up to infinity but limited by how much the in-phase signal is attenuated). It's an application of the Miller theorem. While this is also basically positive feedback and has to be done carefully, it seems great, because it allows a resistor to be large for AC signals but its actual value for DC.

As an example, it is in both theory and practice a good way to counter one problem of the good old bipolar transistor when used in high impedance circuits: the base current results in a significant voltage drop in the bias resistor(s), throwing off that bias in a beta-dependent fashion. To give some context, a collector current of 450μA (think of the usual center-biased emitter follower with 10k resistor and 9V supply) means a voltage drop of 2.25V across a 1M bias resistor for a transistor with a current gain of 200.

BootBuffer

Bootstrapping to the rescue! By feeding the other end of the bias resistor with the emitter signal, we can get the same ~1M input impedance with a 220k bias resistor, resulting in a fifth of the voltage drop across it. Here's what that looks like in practice:

The reason for R1 and R2 to not be equal is mostly to compensate for the Vbe drop and put the emitter at 4.5V. The voltage dropped across R3 is even smaller at this current than assumed by that back-of-the-paper calculation. Even two 10k resistors will be fine there.

The input impedance was about 1M at all frequencies of interest with a 3904 transistor, both in simulation and in reality. Using an high hfe transistor like the 5088 results in close to 2M, and still way above 1M even with 100k R3 (which makes the voltage drop tiny indeed). The frequency response is flat between 20 and 20kHz and the distortion is very low, or at least as low as you can expect from any single-ended buffer (<0.1% with instrument levels).

Finally there's a match for JFET followers at these impedances, all with just one extra capacitor (or maybe no extra capacitors, you would want to decouple your bias divider anyway). Of course op-amp buffers sit above both, but if you're considering discrete buffers you probably are concerned about the footprint, the current draw, the output swing or something else.

BootBoost

Can we apply this same idea to a common-emitter boost? Of course! This arrangement is similar to one Roland used in their first pedals, to give an example.

The principle is similar to the previous one, except that there's now a degenerated common emitter, with the bootstrap signal still taken from the emitter. No degeneration doesn't just mean too much gain, it also means no emitter signal to take advantage of. Also the input impedance would drop to the floor again, since the upper limit of this circuit is the R5 resistor multiplied by the current gain in parallel