Appendix: Useful Circuit Theorems

Analog (Integrated) Circuit Design

17 Appendix: Useful Circuit Theorems

17.1 Miller’s Theorem

Miller’s Theorem

Figure 82: An impedance connected between two nodes A and B.

Using Miller’s theorem we can find the equivalent circuit of an impedance connected between two nodes, if we know the transfer function between these nodes. The original situation is shown in Figure 82, and the equivalent circuit is shown in Figure 83.

Miller’s Theorem

Figure 83: An equivalent circuit using Miller’s theorem.

Miller’s Theorem

Using Miller’s theorem (Sheikholeslami 2015) we can calculate \[ Z_1 = \frac{Z}{1 - A} = \frac{Z}{1 - V_\mathrm{B} / V_\mathrm{A}} \] and \[ Z_2 = \frac{Z}{1 - A^{-1}} = \frac{Z}{1 - V_\mathrm{A} / V_\mathrm{B}} \] to arrive at an equivalent circuit, given that \(A = V_\mathrm{B} / V_\mathrm{A}\) is the voltage gain between nodes A and B.

A derivation of this theorem is relatively straightforward considering the current through \(Z\) when looking into the impedance from either node A or node B and calculating an equivalent impedance causing the same current.

Note that if \(V_\mathrm{A} = V_\mathrm{B}\) then there is no current flow through \(Z\), and accordingly the impedances \(Z_1 = Z_2 = \infty\).

Miller’s theorem can be quite handy when an impedance is strapped between two nodes, and we want to break this connection in a calculation, e.g., considering the effect of \(C_\mathrm{GD}\) in a MOSFET.

Miller’s Theorem

Miller’s Secret

Note that Miller’s compensation is so much more than just making a big capacitor out of a small one. There are layers upon layers of subtlety, and huge hidden benefits which can be read in (Mangelsdorf 2025a) and (Mangelsdorf 2025b).

17.2 Bode’s Noise Theorem

Bode’s Noise Theorem

\[ \overline{V_\mathrm{n}^2} = kT \left( \frac{1}{C_\infty} - \frac{1}{C_0} \right), \]

The total integrated noise of any (no matter how complicated) \(RLC\) network (interpreted as a one-port) is given by

where \(C_\infty\) is the capacitance looking into the network with all resistors and inductors open-circuited, and \(C_0\) is the capacitance looking into the circuit when all inductors and resistors are shorted (Pavan 2019).

Reference (Pavan 2019) is an excellent read deriving Bode’s noise theorem from different angles.

References

Mangelsdorf, Chris. 2025a. “Miller’s Secret [Shop Talk: What You Didn’t Learn in School].” IEEE Solid-State Circuits Magazine 17 (1): 21–27. https://doi.org/10.1109/MSSC.2024.3503792.

Mangelsdorf, Chris. 2025b. “Miller’s Wrong Half-Plane Zero [Shop Talk: What You Didn’t Learn in School].” IEEE Solid-State Circuits Magazine 17 (2): 19–29. https://doi.org/10.1109/MSSC.2025.3562109.

Pavan, Shanthi. 2019. “An Alternative Approach to Bode’s Noise Theorem.” IEEE Transactions on Circuits and Systems II: Express Briefs 66 (5): 738–42. https://doi.org/10.1109/TCSII.2019.2907860.

Sheikholeslami, Ali. 2015. “Miller’s Theorem [Circuit Intuitions].” IEEE Solid-State Circuits Magazine 7 (3): 9–10. https://doi.org/10.1109/mssc.2015.2446457.

References