Optimizing the Tracking Pre-Regulator Design

In my last post I investigated a bit into a novel tracking pre-regulator design by John Barnes (I originally heard of this design in Dave Jones' EEVBlog #329 video). I had modeled mathematically the system rather than using SPICE simulation software as Dave did to play with the design and came up with a fairly nice model for figuring out what parameters to use. This is just a slight update as I investigated the design a bit further. In this post I'm going to present a novel way to make the transistor gain error nearly negligible.

So how do we accomplish this? It's actually quite easy, just set \(R_3 = R_1\). Why does this work? Recall from my last post the equation for calculating pre-regulation offset:

\[ V_{reg} - V_{set} = \frac{\beta R_1 V_{FB}}{(1 + \beta) R_2} + \frac{R_3 V_{FB}}{R_2 \beta} + V_{EB} \]

Let's see what happens if \(R_3 = R_1\).

\[ V_{reg} - V_{set} = \frac{\beta R_1 V_{FB}}{(1 + \beta) R_2} + \frac{R_1 V_{FB}}{R_2 \beta} + V_{EB}\\ V_{reg} - V_{set} = \frac{R_1 V_{FB}}{R_2} \left ( \frac{\beta}{(1 + \beta)} + \frac{1}{\beta} \right ) + V_{EB}\\ V_{reg} - V_{set} = \frac{R_1 V_{FB}}{R_2} \left ( \frac{\beta^2 + \beta + 1}{\beta^2 + \beta} \right ) + V_{EB} \]

Hello, look at the terms with \(\beta\)'s. As the transistor gain gets larger the change in pre-regulation offset drops off extremely rapidly. Let's plot the error:

Figure 1. Absolute error of pre-regulation offset vs. \(\beta\), compared with target \(\beta = 100\)

Even as the gain approaches 10 (typically accepted as fully saturated, in which case this model fails to be accurate), the error is under 0.5%. For more practical gain values, (say \(\beta > 50\)) the error drops below 0.01%. Not bad for just matching two resistors.