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Fundamentals of Analog VLSI Design
Exercise 7 - Problem

Gain and Bandwidth Enhancement

Author

Christian Enz (christian.enz@epfl.ch)

Published

05.11.2025

# 1 Problem 1: Gain Enhancement

Figure 1: Schematic of gain enhancement amplifier.

The amplifier shown in Figure 1 allows to enhance the DC gain by reducing the load conductance seen by the differential pair M1a-M1b [1] [2].

## 1.1 Small-signal analysis

Figure 2: Negative conductance.

The cross-coupled pair M3a-M3b which is redrawn in Figure 2, allow to implement a negative conductance. When combined with transistors M2a-M2b, it allows to reduce the conductance seen by the differential pair M1a-M1b.

- Draw the small-signal schematic of the circuit shown in Figure 2.
- Derive the small-signal differential conductance \(G = \Delta I_{in}/\Delta V_{in}\), where \(\Delta I_{in} \triangleq \Delta I_1-\Delta I_2\) assuming a perfect matching between M3a and M3b.
- Draw the small-signal circuit of the amplifier shown in Figure 1 including the output conductances in differential mode assuming a perfect matching between all the transistors in one current branch and the other. Hint: simplify the circuit by using the fact that the voltage at the common source node of M1a-M1b does not change in differential mode with perfect matching.
- From the above analysis, derive the small-signal differential transfer function \[\begin{equation} A_d(s) \triangleq \frac{V_{od}(s)}{V_{id}(s)}, \end{equation}\] where \(V_{od} \triangleq V_{o1}-V_{o2}\) and \(V_{id} \triangleq V_{i1}-V_{i2}\) are the output and input differential voltages, respectively.
- Deduce the expression of the DC gain and the dominant pole.
- What is the condition on \(G_{m3}\) for the pole to remain in the left half-plane?
- What is the gain-enhancement factor \(K\) compared to the voltage gain without the cross-coupled pair M3a-M3b?
- What is the value of the DC gain for \(G_{m3}=G_{m2}\)? How does it compare to a common-stage stage?
- Find the gain-bandwidth product \(\omega_u\). Is the gain-bandwidth product also enhanced like the DC gain?

## 1.2 Noise analysis

- Draw the small-signal schematic of Figure 1 including all the noise sources but with \(\Delta V_{id}=0\). Hint: simplify the circuit by using the fact that the voltage at the common source node of M1a-M1b does not change in differential mode with perfect matching.
- Derive the output noise resistance \(R_{nout}\).
- Deduce the input-referred noise resistance \(R_{nin}\).
- Calculate the input-referred thermal noise resistance \(R_{nt}\). How does it compare to the noise of the same amplifier without M3a-M3b?
- Calculate the input-referred flicker noise resistance \(R_{nf}\). How does it compare to the noise of the same amplifier without M3a-M3b?

# 2 Problem 2: Bandwidth enhancement

Figure 3: Enhanced \(G_m\) differential OTA

The differential OTA shown in Figure 3 shows a larger gain-bandwidth product compared to the case without the cross-coupled transistors M2a-M2b [1] [3].

## 2.1 Small-signal analysis

Figure 4: Enhanced \(G_m\) differential transconductor

The OTA of Figure 3 uses the differential transconductor shown in Figure 4 which allows to increase the equivalent transconductance compared to the case of a simple differential pair.

- Draw the small-signal schematic of Figure 4. Hint: simplify the circuit by using the fact that the voltage at the common source node of M2a-M2b does not change in differential mode with perfect matching.
- Derive the small-signal equivalent transconductance \[\begin{equation} G_{meq} \triangleq \frac{\Delta I_{od}}{\Delta V_{id}}, \end{equation}\] where \(\Delta I_{od} \triangleq \Delta I_1-\Delta I_2\) is the small-signal differential output current and \(\Delta V_{id} \triangleq \Delta V_{i1}-\Delta V_{i2}\) is the small-signal differential input voltage. How does it compare to the transconductance of a simple differential pair?
- Derive the small-signal equivalent transadmittance \[\begin{equation} Y_{meq}(s) \triangleq \frac{\Delta I_{od}(s)}{\Delta V_{id}(s)}, \end{equation}\] accounting for the the parasitic capacitance \(C_p\) at the source node of M1a-M1b. Hint: add a parasitic capacitance \(C_p\) connected between the source node of M1a and ground to the small-signal half-circuit used above.
- Deduce the expression of the dominant pole \(\omega_p\).
- What is the condition on \(G_{m2}\) for the pole \(\omega_p\) to remain in the left half-plane? Hint: express the condition in terms of the transconductance ratio \(\alpha \triangleq G_{ms1}/G_{m2}\).
- Express the \(G_m\)-enhancement factor \(K\) compared to the transconductance of the differential pair M1a-M1b without the cross-coupled transistors M2a-M2b. Hint: express \(K\) in terms of the transconductance ratio \(\alpha\).
- What is the value of \(K\) when M1a-M1b and M2a-M2b are both biased in weak inversion? Hint: M1a-M1b and M2a-M2b share the same bias current \(I_b\).
- Derive the small-signal differential transfer function \[\begin{equation} A_d \triangleq \frac{V_{od}}{V_{id}}. \end{equation}\]
- What is the value of the DC gain for \(G_{ms1}=G_{m2}\)? How does it compare to a common-stage stage?
- Find the gain-bandwidth product \(\omega_u\). Is the gain-bandwidth product also enhanced like the DC gain?

## 2.2 Noise analysis

- Draw the small-signal schematic of Figure 4 including all the noise sources and with \(V_{id}=0\) and neglecting all output conductances. Hint: simplify the circuit by using the fact that the voltage at the common source node of M2a-M2b does not change in differential mode with perfect matching.
- Derive the output noise conductance \(G_{nout}\).
- Deduce the input-referred noise resistance \(R_{nin}\).
- Calculate the input-referred thermal noise resistance \(R_{nt}\). How does it compare to the noise of a simple differential pair?
- Calculate the input-referred flicker noise resistance \(R_{nf}\). How does it compare to the noise of a simple differential pair?

## 3 References

[1]
W. Sansen, “Opamps, gm-blocks or inverters?” in Efficient sensor interfaces, advanced amplifiers and low power RF systems: Advances in analog circuit design 2015, Springer, 2016, pp. 125–138.
[2]
K. B. Ohri and M. J. Callahan, “Integrated PCM codec,” IEEE Journal of Solid-State Circuits, vol. 14, no. 1, pp. 38–46, 1979, doi: 10.1109/JSSC.1979.1051139.
[3]
R. Castello, A. G. Grassi, and S. Donati, “A 500-nA sixth-order bandpass SC filter,” IEEE Journal of Solid-State Circuits, vol. 25, no. 3, pp. 669–676, 1990, doi: 10.1109/4.102659.