Appendix: Middlebrook’s Method

Analog (Integrated) Circuit Design

16 Appendix: Middlebrook’s Method

Appendix: Middlebrook’s Method

Figure 80: Middlebrook voltage loop gain simulation.

If we want to do a closed-loop gain analysis (for stability or other investigations), we have the need to break the loop at one point, apply a stimulus, and monitor the response on the other end. By doing this we want to keep the loading (i.e. the impedance) on both ends similar to the original case. To achieve this, we break the loop at one point by inserting (1) an ac voltage source, and (2) attaching an ac current source, as shown in Figure 80 and Figure 81. The derivation of this approach is presented in (Middlebrook 1975), and has the big advantage that loading is not changed compared to the closed-loop situation, and the bias points are also unchanged.

Appendix: Middlebrook’s Method

Figure 81: Middlebrook current loop gain simulation.

Appendix: Middlebrook’s Method

For both cases we do an ac analysis, and find the corresponding transfer functions \(T_\mathrm{v}\) and \(T_\mathrm{i}\) as \[ T_\mathrm{v} = -\frac{V_\mathrm{r}}{V_\mathrm{f}} \] and \[ T_\mathrm{i} = -\frac{I_\mathrm{r}}{I_\mathrm{f}}. \]

Then, we can calculate the open-loop transfer function \(T(s) = H_\mathrm{ol}(s)\) as \[ T(s) = \frac{T_\mathrm{v} T_\mathrm{i} - 1}{T_\mathrm{v} + T_\mathrm{i} + 2} = \frac{V_\mathrm{r} I_\mathrm{r} - V_\mathrm{f} I_\mathrm{f}}{2 V_\mathrm{f} I_\mathrm{f} - V_\mathrm{r} I_\mathrm{f} - V_\mathrm{f} I_\mathrm{r}}. \]

The four ac quantities \(V_\mathrm{f}\), \(V_\mathrm{r}\), \(I_\mathrm{f}\), and \(I_\mathrm{r}\) we can readily find by circuit simulation or calculation.

Please note that Middlebrook’s method works well for \(|T| \gg 1\), so it will show inaccuracies at the crossover frequency of the open-loop gain at \(|T(s)| \approx 1\). An improved method (slightly more complicated) can be found in (Tian et al. 2001).

References

Middlebrook, R. D. 1975. “Measurement of loop gain in feedback systems.” International Journal of Electronics 38 (4): 485–512. https://doi.org/10.1080/00207217508920421.

Tian, M., V. Visvanathan, J. Hantgan, and K. Kundert. 2001. “Striving for small-signal stability.” IEEE Circuits and Devices Magazine 17 (1): 31–41.

References