#

Fundamentals of Analog VLSI Design
Exercise 5 - Solution

The Cascode Gain Stage

Author

Christian Enz (christian.enz@epfl.ch)

Published

15.10.2025

# 1 Cascode stage

Figure 1: Schematic of the cascode gain stage.

## 1.1 Analysis

### 1.1.1 Small-signal analysis

Figure 2: Small-signal schematic of the cascode stage of Figure 1 for the calculation of the equivalent transconductance.

The small-signal schematic corresponding to Figure 1 for the derivation of the equivalent transconductance is shown in Figure 2. Note that the output is short-circuited to ground for calculating the short-circuit output current and the corresponding transconductance. The equivalent transconductance is then given by \[\begin{equation} G_{meq} \triangleq \left.\frac{\Delta I_{out}}{\Delta V_{in}}\right|_{\Delta V_{out}=0} = G_{m1} \cdot \frac{G_{ms2} + G_{ds2}}{G_{ms2} + G_{ds1} + G_{ds2}} \cong G_{m1}, \end{equation}\] which shows that assuming \(G_{ms2} \gg G_{ds1}, G_{ds2}\), the equivalent transconductance of the cascode stage is equal to the transconductance of the driver transistor M1. This result is expected since the cascode transistor is a common gate stage which has a unity current gain so that the current coming from the driver transistor M1 is directly directed to the output.

Figure 3: Small-signal schematic of the cascode stage of Figure 1 for the calculation of the output conductance.

The small-signal schematic for the calculation of the output conductance is shown in Figure 3. The output conductance is given by \[\begin{equation} G_{out} \triangleq \frac{\Delta I_{out}}{\Delta V_{out}} = \frac{G_{ds1} G_{ds2}}{G_{ms2} + G_{ds1} + G_{ds2}} \cong \frac{G_{ds1}}{G_{ms2}/G_{ds2}}, \end{equation}\] which is equal to the output conductance of M1, \(G_{ds1}\), divided by the voltage gain of the cascode \(G_{ms2}/G_{ds2}\). This means that, at low-frequency, the output conductance of a single transistor can be reduced by adding a cascode stage at the cost of some voltage headroom to maintain M2 in saturation.

Figure 4: Small-signal schematic of the cascode stage of Figure 1 for the calculation of the output admittance including the parasitic capacitance \(C_{p1}\).
Figure 5: Effect of the parasitic capacitance \(C_{p1}\) on the output admittance \(Y_{out}\).

The impact of the parasitic capacitance at node 1 on the output admittance can be investigated by adding the capacitance as shown in Figure 4. The output admittance then becomes \[\begin{equation} Y_{out} = G_{out} \cdot \frac{1+s/\omega_z}{1+s/\omega_p}, \end{equation}\] where \[\begin{align} G_{out} &\cong \frac{G_{ds1} G_{ds2}}{G_{ms2}},\\ \omega_z &\triangleq \frac{G_{ds1}}{C_{p1}},\\ \omega_p &\triangleq \frac{G_{ms2}}{C_{p1}}. \end{align}\]

The magnitude of \(Y_{out}\) versus frequency is sketched in Figure 5. At low-frequency, i.e. for \(\omega \ll \omega_z \ll \omega_p\), the output admittance is equal to the output conductance \(Y_{out} \cong G_{out}\). However for \(\omega_z \ll \omega_p \ll \omega\), the output admittance increases to the output conductance of M2 \(Y_{out} \cong G_{ds2}\). We see that the cascode effect is lost. This can easily be understood since for \(\omega_p \ll \omega\), the cascode node 1 is shortened to the ac ground and the voltage controlling the source of M2 is zero leaving the output conductance of M2 only in the small-signal schematic of Figure 4.

### 1.1.2 Noise analysis

Figure 6: Small-signal schematic of the cascode stage of Figure 1 for the noise calculation.

To calculate the output noise PSD we can use the schematic shown in Figure 6. The output noise current is then given by \[\begin{equation} I_{nout} = \frac{G_{ms2} + G_{ds2}}{G_{ms2} + G_{ds1} + G_{ds2}} \cdot I_{n1} + \frac{G_{ds1}}{G_{ms2} + G_{ds1} + G_{ds2}} \cdot I_{n2} \cong I_{n1} + \frac{I_{n2}}{G_{ms2}/G_{ds1}}, \end{equation}\] for \(G_{ms2} \gg G_{ds1}, G_{ds2}\). We see that the contribution to the output noise current of the cascode stage \(I_{n2}\) is actually divided by \(G_{ms2}/G_{ds1}\). Provided that this gain can be made sufficiently large and that both transistor have the same noise, the noise of the cascode stage can be made negligible compared to the noise due to M1. Ultimately if \(G_{ds1}=0\), the noise current \(I_{n2}\) circulates in the cascode transistor M2 (\(G_{ms2}\) in the small-signal schematic of Figure 6) and hence does not reach the output.

The output noise conductance \(G_{nout}\) is then given by \[\begin{equation} G_{nout}(f) \cong G_{n1}(f) + \left(\frac{G_{ds1}}{G_{ms2}}\right)^2 G_{n2}(f) \end{equation}\] where the noise conductances are given by \[\begin{equation}\label{eqn:Gni} G_{ni}(f) = \gamma_{ni} \cdot G_{mi} + \frac{\rho_n}{f\,W_i\,L_i} \cdot G_{mi}^2 \quad \text{for $i=1,2$}. \end{equation}\]

The input-referred noise is obtained by dividing \(G_{nout}\) by \(G_{meq}^2\), resulting in \[\begin{equation} R_{nin} = \frac{G_{nout}(f)}{G_{meq}^2} \cong \frac{G_{nout}(f)}{G_{m1}^2} \cong \frac{G_{n1}(f)}{G_{m1}^2} + \left(\frac{G_{ds1}}{G_{m1}\,G_{ms2}}\right)^2\;G_{n2}(f). \end{equation}\] It can be decomposed into the thermal and flicker noise components according to \[\begin{equation} R_{nin} = R_{nt} + R_{nf}(f) \end{equation}\] where \(R_{nt}\) is the total input-referred thermal noise given by \[\begin{equation} R_{nt} \cong \frac{\gamma_{n1}}{G_{m1}} + \left(\frac{G_{ds1}}{G_{m1}}\right)^2 \frac{\gamma_{n2}}{n_2\,G_{ms2}} = \frac{\gamma_{n1}}{G_{m1}} \cdot (1+\eta_{th}), \end{equation}\] where \(\eta_{th}\), given by \[\begin{equation} \eta_{th} \cong \frac{\gamma_{n2}}{n_2\,\gamma_{n1}} \cdot \frac{G_{ds1}^2}{G_{m1}\,G_{ms2}}, \end{equation}\] represents the contribution of the cascode transistor M2 to the input-referred thermal noise relative to that of M1. Since \(\gamma_{n1}\) and \(\gamma_{n2}\) are of the same order of magnitude and \(G_{m1},G_{ms2} \gg G_{ds1}\), this results in \(\eta_{th} \ll 1\), meaning that the contribution of M2 is negligible compared to that of M1 (at low frequency).

The noise excess factor of the cascode gain stage is given by \[\begin{equation} \gamma_{cas} \triangleq G_{meq} \cdot R_{nt} \cong G_{m1} \cdot R_{nt} = \gamma_{n1} \cdot (1+\eta_{th}) \cong \gamma_{n1}. \end{equation}\] The thermal noise excess factor of the cascode gain stage is about equal to the thermal noise excess factor of the driver transistor M1.

The input-referred flicker noise \(R_{nf}(f)\) is given by \[\begin{equation}\label{eqn:p1:Rnf} R_{nf}(f) \cong \frac{\rho_n}{f\,W_1 L_1} + \left(\frac{G_{ds1}}{G_{m1}}\right)^2 \frac{\rho_n}{n_2^2 f\,W_2 L_2} = \frac{\rho_n}{f\,W_1 L_1} \cdot (1+\eta_{fl}). \end{equation}\] where \(\eta_{fl}\), given by \[\begin{equation} \eta_{fl} = \left(\frac{G_{ds1}}{n_2\,G_{m1}}\right)^2 \frac{W_1 L_1}{W_2 L_2} \end{equation}\] Assuming M1 and M2 have the same area and \(G_{m1}/G_{ds1} \gg 1\), the contribution of M2 to the input-referred flicker noise is negligible compared to that of M1 and \(\eqref{eqn:p1:Rnf}\) simplifies to \[\begin{equation} R_{nf}(f) \cong \frac{\rho_n}{f\,W_1 L_1}, \end{equation}\] which corresponds to the contribution of M1 only.

Figure 7: Small-signal schematic of the cascode stage of Figure 1 for the calculation of the noise including the effect of the parasitic capacitance \(C_{p1}\).

Similarly, the impact of the parasitic capacitance on the noise can be calculated from Figure 7. The output noise conductance is then given by \[\begin{equation} G_{nout} = |H_{n1}(\omega)|^2 \cdot G_{n1} + |H_{n2}(\omega)|^2 \cdot G_{n2}, \end{equation}\] where \[\begin{align} H_{n1}(s) &= \frac{1}{1+s/\omega_p}\\ H_{n2}(s) &= \frac{G_{ds1}}{G_{ms2}} \cdot \frac{1+s/\omega_z}{1+s/\omega_p}. \end{align}\]

Figure 8: Magnitude of the noise transfer functions \(H_{n1}\) and \(H_{n2}\) versus frequency.

The magnitude of \(H_{n1}\) and \(H_{n2}\) versus frequency are sketched in Figure 8. For \(\omega \ll \omega_z \ll \omega_p\), \(H_{n1} \cong 1\) and \(H_{n2} \cong G_{ds1}/G_{ms2}\) which is the result obtained above. However, for \(\omega_z \ll \omega_p \ll \omega\), \(H_{n1} \cong \omega_p/s\) and \(H_{n2} \cong 1\). We see that the cascode effect is lost since the noise of M2 is no more divided by the cascode gain \(G_{ms2}/G_{ds1}\) but is entirely transferred to the output.

As a conclusion, adding a cascode stage reduces the output conductance without penalty on the noise, but at the cost of a slight voltage overhead for maintaining M2 in saturation. This is only true for \(\omega < \omega_z = G_{ds1}/C_{p1}\). For frequencies \(\omega \gg \omega_p = G_{ms2}/C_{p1}\) the cascode effect is lost. Note that in order to maximize \(G_{ms2}\) at a given current and minimize its saturation voltage, M2 should be biased in weak inversion.

## 1.2 Design

We want to size the circuit of Figure 1 for the specifications given in Table 1. We need to find the minimum current and size the transistor to achieve these specs. We will design the amplifier for a generic 180nm bulk CMOS process. The physical parameters are given in Table 2, the global process parameters in Table 3 and finally the MOSFET parameters in Table 4.

Table 1: Specifications for the differential pair with resistive loads.

Specification | Symbol | Value | Unit
--- | --- | --- | ---
DC gain | \(A_{{dc}}\) | 80 | \(dB\)
DC gain | \(A_{{dc}}\) | 10000 | -
Gain-bandwidth product | \(GBW\) | 10 | \(MHz\)
Load capacitance | \(C_L\) | 1 | \(pF\)

Table 2: Physical parameters

Parameter | Value | Unit
--- | --- | ---
\(T\) | 300 | \(K\)
\(U_T\) | 25.875 | \(mV\)

Table 3: Global process parameters

Parameter | Value | Unit
--- | --- | ---
\(V_{DD}\) | 1.8 | \(V\)
\(C_{ox}\) | 8.443 | \(\frac{fF}{\mu m^2}\)
\(W_{min}\) | 200 | \(nm\)
\(L_{min}\) | 180 | \(nm\)

Table 4: Transistor process parameters

Parameter | NMOS | PMOS | Unit
--- | --- | --- | ---
sEKV parameters |  |  |
\(n\) | 1.27 | 1.31 | -
\(I_{{spec\Box}}\) | 715 | 173 | \(nA\)
\(V_{{T0}}\) | 0.455 | 0.445 | \(V\)
\(L_{{sat}}\) | 26 | 36 | \(nm\)
\(\lambda\) | 13 | 20 | \(\frac{{V}}{{\mu m}}\)
Overlap capacitances parameters |  |  |
\(C_{{GDo}}\) | 0.366 | 0.329 | \(\frac{{fF}}{{\mu m}}\)
\(C_{{GSo}}\) | 0.366 | 0.329 | \(\frac{{fF}}{{\mu m}}\)
\(C_{{GBo}}\) | 0 | 0 | \(\frac{{fF}}{{\mu m}}\)
Junction capacitances parameters |  |  |
\(C_J\) | 1 | 1.121 | \(\frac{{fF}}{{\mu m^2}}\)
\(C_{{JSW}}\) | 0.2 | 0.248 | \(\frac{{fF}}{{\mu m}}\)
Flicker noise parameters |  |  |
\(K_F\) | 8.1e-24 | 6.8e-23 | \(J\)
\(AF\) | 1 | 1 | -
\(\rho\) | 0.05794 | 0.4828 | \(\frac{{V \cdot m^2}}{{A \cdot s}}\)
Matching parameters |  |  |
\(A_{{VT}}\) | 5 | 5 | \(mV \cdot \mu m\)
\(A_{{\beta}}\) | 1 | 1 | \(\% \cdot \mu m\)
Source and drain sheet resistance parameter |  |  |
\(R_{{sh}}\) | 600 | 2386 | \(\frac{{\Omega}}{{\mu m}}\)
Width and length parameters |  |  |
\(\Delta W\) | 39 | 54 | \(\,nm\)
\(\Delta L\) | -76 | -72 | \(\,nm\)

The gain-bandwidth product direcly sets the transconductance of M1 as \(G_{m1} =\) 62.832 \(\mu A/V\). Imposing an inversion coefficient \(IC_1 =\) 1 allows to derive the required bias current as \(I_b = G_{m1}\,n\,U_T/gmsid(IC1) =\) 3.344 \(\mu A\). We then can find the specific current \(I_{spec1} =\) 3.344 \(\mu A\) and aspect ratio \(W_1/L_1 =\) 4.677.

The DC gain then sets the output conductance as \(G_{out} =\) 6.283 \(nA/V\). To minimize the voltage headroom needed for M2 and maximize its current efficiency, we will bias M2 in weak inversion setting its inversion coefficient to \(IC_2 =\) 0.1. Knowing \(IC_2\) and \(I_b\) we can deduce its source transconductance as \(G_{ms2} =\) 118.405 \(\mu A/V\). If we split the overall voltage gain equally between M1 and M2 we get \(G_{m1}/G_{ds1} = G_{ms2}/G_{ds2} = \sqrt{A_{dc}} =\) 100. We now can deduce \(G_{ds1} =\) 0.628 \(\mu A/V\) and \(G_{ds2} =\) 1.184 \(\mu A/V\). This leads to \(L_1 =\) 0.49 \(\mu m\) and \(L_2 =\) 0.29 \(\mu m\).

It turns out that the length of M1 and M2 are actually independent of their transconductances \(G_{m1}\) and \(G_{ms2}\). Indeed, from the output conductance we get the transistor length as \[\begin{equation}\label{eqn:l1} L_1 = \frac{I_b}{\lambda_n\,G_{ds1}} \end{equation}\] The output conductance \(G_{ds1}\) is set by the required DC gain as \[\begin{equation}\label{eqn:cascode_gds1} G_{ds1} = \frac{G_{m1}}{\sqrt{A_{dc}}} = \frac{2\pi\,GBW\,C_L}{\sqrt{A_{dc}}} \end{equation}\] Now the bias current is given by \[\begin{equation}\label{eqn:ib} I_b = G_{m1}\,\frac{n_1 U_T}{gmsid(IC_1)} \end{equation}\] Replacing \(\eqref{eqn:ib}\) and \(\eqref{eqn:cascode_gds1}\) in \(\eqref{eqn:l1}\) results in \[\begin{equation} L_1 = \frac{n_1 U_T\,\sqrt{A_{dc}}}{\lambda_n\,gmsid(IC_1)}, \end{equation}\] which is independent of the transconductance \(G_{m1}\).

Similarly for M2 we get \[\begin{equation} L_2 = \frac{U_T\,\sqrt{A_{dc}}}{\lambda_n\,gmsid(IC_2)}, \end{equation}\] which is independent of the source transconductance \(G_{ms2}\) of M2.

We now get the transistor width as \(W_1 =\) 1.88 \(\mu m\) and \(W_2 =\) 10.12 \(\mu m\).

We still need to set the bias voltage for the cascode transistor. We want to bias M1 at the onset of saturation \(V_{DSsat1}\). The \(V_{GS2}\) voltage of M2 can be approximated by \(V_{T0n}\). The bias voltage is then given by \[\begin{equation} V_{b2} = V_{GS2} +V_{DSsat1} \end{equation}\] which is 0.571 \(V\). We will set it to \(V_{b2} =\) 0.650 \(V\) so we have some margin.

Table 5: Transistor size and bias information.

Transistor | \(W\;[\mu m]\) | \(L\;[\mu m]\) | \(I_D\;[\mu A]\) | \(I_{{spec}}\;[\mu A]\) | \(IC\) | \(V_G-V_{{T0}}\;[mV]\) | \(V_{{DSsat}}\;[mV]\)
--- | --- | --- | --- | --- | --- | --- | ---
M1 | 1.88 | 0.49 | 3.344 | 2.763 | 1.2 | 22 | 118
M2 | 10.12 | 0.29 | 3.344 | 24.682 | 0.1 | -38 | 105
M3 | 1.88 | 0.49 | 3.344 | 2.763 | 1.2 | 22 | 118
M4 | 10.12 | 0.29 | 3.344 | 24.682 | 0.1 | -38 | 105

Table 6: Transistor small-signal and thermal noise parameters.

Transistor | \(G_{{spec}}\;[\mu A/V]\) | \(G_{{ms}}\;[\mu A/V]\) | \(G_m\;[\mu A/V]\) | \(G_{{ds}}\;[nA/V]\) | \(\gamma_n\)
--- | --- | --- | --- | --- | ---
M1 | 106.796 | 75.656 | 59.508 | 529.950 | 0.724
M2 | 953.893 | 115.312 | 90.700 | 877.208 | 0.659
M3 | 106.796 | 75.656 | 59.508 | 529.950 | 0.724
M4 | 953.893 | 115.312 | 90.700 | 877.208 | 0.659

## 1.3 Simulations

Figure 9: Schematic used for the simulation of the transfer function.

We can now can check the design by simulation using the circuit shown in Figure 9. The purpose of M3 and M4 is to set the DC gate voltage of M1 such that its current is equal to the bias current. Of course M3 is identical to M1 and M4 is identical to M2.

We first will check the operating point. The node voltages are given in Table 7. We can check that all transistors are biased in saturation. The transistor operating point information are given in Table 8 and Table 9. We see that the simulated transconductances are very close to the theoretical values given in Table 6.

Table 7: OTA node voltages with the OTA in open-loop without offset correction.

Node | Voltage
--- | ---
vdd | 1.8
in | 0.527778
out | 0.527778
vb2 | 0.65
1 | 0.176378
2 | 0.176378
3 | 0.527778

Table 8: Large-signal operating point information extracted from ngspice .op file for each transistor.

Transistor | \(I_D\;[nA]\) | \(I_{spec}\;[nA]\) | \(IC\) [-] | \(n\) [-] | \(V_{DSsat}\;[mV]\)
--- | --- | --- | --- | --- | ---
M1 | 3.344 | 3.182 | 1.060 | 1.27 | 157
M2 | 3.344 | 33.928 | 0.099 | 1.27 | 120
M3 | 3.344 | 3.182 | 1.060 | 1.27 | 157
M4 | 3.344 | 33.928 | 0.099 | 1.27 | 120

Table 9: Small-signal operating point information extracted from ngspice .op file for each transistor.

Transistor | \(G_m\;[\mu A/V]\) | \(G_{ms}\;[\mu A/V]\) | \(G_{ds}\;[nA/V]\)
--- | --- | --- | ---
M1 | 60.093 | 77.541 | 878.705
M2 | 93.586 | 117.423 | 992.781
M3 | 60.093 | 77.541 | 878.705
M4 | 93.586 | 117.423 | 992.781

### 1.3.1 Transfer function

We now can simulate the transfer function which is plotted in Figure 10. We see that the simulation results are close to the theoretical vaues. The \(GBW\) is slightly lower than specs despite the simulated transconductance \(G_{m1}\) is larger. This is coming from the self-loading. Indeed, since M2 is biased in weak inversion, it is a fairly large transistor and therefore its paraisitc capacitance at the drain adds to the load capaciance. We could estimate this parasitic capacitance and add it to the load capacitance.

Figure 10: Simulated gain response compared to theoretical estimation.

### 1.3.2 Input-referred noise

We can compare the theoretical input-referred noise to that obtained from simulations. The simulation results are presented in Figure 11.

Figure 11: Simulated input-referred noise PSD compared to theoretical estimation.

The simulation confirms that, below the gain-bandwidth product \(GBW =\) 10 \(MHz\), the square root of the input-referred flicker noise of M2 is about 40 dB lower than that of M1.

# 2 Regulated cascode

Figure 12: Schematic of the regulated cascode [1].

The schematic of Figure 12 corresponds to the regulated cascode, where an additional CS gain stage M3 is added [1].

## 2.1 Small-signal analysis

Figure 13: Small-signal schematic of the regulated cascode.

The small-signal equivalent schematic of the regulated cascode of Figure 12 is presented in Figure 13.

Solving the KCL equations accounting for all the output conductances leads to the full expression of the equivalent transconductance given by \[\begin{equation} G_{meq} = G_{m1}\cdot\frac{G_{m2}\,G_{m3}+G_{ds3}\,(G_{ms2}+G_{ds2})}{G_{m2}\,G_{m3}+G_{ds3}\,(G_{ms2}+G_{ds2}+G_{ds1})} \end{equation}\] which for \(G_{ms2} \gg G_{ds2}, G_{ds1}\) reduces to \[\begin{equation} G_{meq} \cong G_{m1}. \end{equation}\] This is the same result as the cascode stage.

The output conductance accounting for all transistor’s output conductances is given by \[\begin{equation} G_{out} = G_{ds1} \cdot \frac{G_{ds2}\,G_{ds3}}{G_{m2}\,G_{m3}+G_{ds3}\,(G_{ms2}+G_{ds2}+G_{ds1})}, \end{equation}\] which for \(G_{ms2} \gg G_{ds2}, G_{ds1}\) reduces to \[\begin{equation}\label{eqn:sol2:gout} G_{out} \cong G_{ds1}\cdot\frac{G_{ds2}}{G_{m2}}\cdot\frac{G_{ds3}}{G_{m3}}. \end{equation}\] Equation \(\eqref{eqn:sol2:gout}\) shows that the output conductance of the regulated cascode is equal to the output conductance of M1 divided by the square of a transistor self-gain, i.e. \((G_m/G_{ds})^2\), this is \(G_m/G_{ds}\) smaller than that of the simple cascode.

The voltage gain is given by \[\begin{equation} A_v \triangleq \frac{\Delta V_{out}}{\Delta V_{in}} = -\frac{G_{meq}}{G_{out}} = -G_{m1}\cdot\frac{G_{m2}\,G_{m3}+G_{ds3}\,(G_{ms2}+G_{ds2})}{G_{ds1}\,G_{ds2}\,G_{ds3}}, \end{equation}\] which for \(G_{ms2} \gg G_{ds2}, G_{ds1}\) and reduces to \[\begin{equation}\label{eqn:sol2:av} A_v \cong -\frac{G_{m1}}{G_{ds1}}\cdot\frac{G_{m2}}{G_{ds2}}\cdot\frac{G_{m3}}{G_{ds3}}. \end{equation}\] Equation \(\eqref{eqn:sol2:av}\) shows that the voltage gain of the regulated cascode is equivalent of that of a cascade of three stages, i.e. \((G_m/G_{ds})^3\). However, this gain is achieved at lower current since the driver transistor M1 and the cascode transistor M2 share the same current (current reuse).

## 2.2 Noise analysis

Figure 14: Small-signal schematic of the regulated cascode including the noise sources.

The equivalent small-signal circuit of the regulated cascode including all the noise current sources for the calculation of the noise is shown in Figure 14. The output noise current is given by \[\begin{equation} I_{nout} = H_{n1}\cdot I_{n1} + H_{n2} \cdot I_{n2} + H_{n3} \cdot I_{n3}. \end{equation}\] where \[\begin{align} H_{n1} &= \frac{G_{m2}\,G_{m3}+G_{ds3}\,(G_{ms2}+G_{ds2})}{G_{m2}\,G_{m3}+G_{ds3}\,(G_{ms2}+G_{ds1}+G_{ds2})} \cong 1,\label{eqn:sol2:hn1}\\ H_{n2} &= \frac{G_{ds1}\,G_{ds3}}{G_{m2}\,G_{m3}+G_{ds3}\,(G_{ms2}+G_{ds1}+G_{ds2})} \cong \frac{G_{ds1}\,G_{ds3}}{G_{m2}\,G_{m3}},\label{eqn:sol2:hn2}\\ H_{n3} &= -\frac{G_{m2}\,G_{ds1}}{G_{m2}\,G_{m3}+G_{ds3}\,(G_{ms2}+G_{ds1}+G_{ds2})} \cong -\frac{G_{ds1}}{G_{m3}}.\label{eqn:sol2:hn3} \end{align}\] Equation \(\eqref{eqn:sol2:hn1}\) shows that the noise of M1 is transferred directly to the output. Equation \(\eqref{eqn:sol2:hn2}\) shows that the noise of M2 is divided by the square of a transistor self-gain, i.e. divided by \((G_m/G_{ds})^2\). Finally, the noise coming from M3 is divided by \(G_m/G_{ds}\). This means that the noise coming from M2 and M3 are much smaller than the noise coming from M1 and can usually be neglected. We still will derive the output and input-referred noise accounting for all contributions.

The output noise conductance is given by \[\begin{equation} G_{nout} = |H_{n1}|^2 \cdot G_{n1} + |H_{n2}|^2 \cdot G_{n2} + |H_{n3}|^2 \cdot G_{n3} \end{equation}\] where \(G_{ni}\) is given by \(\eqref{eqn:Gni}\).

The input-referred noise is obtained by dividing \(G_{nout}\) by \(G_{meq}^2\), resulting in \[\begin{equation} R_{nin} = \frac{G_{nout}(f)}{G_{meq}^2}. \end{equation}\] It can be decomposed into a thermal and flicker noise component according to \[\begin{equation} R_{nin} = R_{nt} + R_{nf}(f) \end{equation}\] \(R_{nt}\) is the total input-referred thermal noise resistance which can be written as \[\begin{equation} R_{nt} = \frac{\gamma_{n1}}{G_{m1}} \cdot (1+\eta_{th}), \end{equation}\] where \(\eta_{th}\) represents the contributions of M2 and M3 to \(R_{nt}\) relative to that of M1 \[\begin{equation}\label{eqn:sol2:etath} \eta_{th} \cong \frac{\gamma_{n2}}{\gamma_{n1}} \cdot \frac{G_{ds1}^2\,G_{ds3}^2}{G_{m1}\,G_{m2}\,G_{m3}^2} + \frac{\gamma_{n3}}{\gamma_{n1}} \cdot \frac{G_{ds1}^2}{G_{m1}\,G_{m3}}. \end{equation}\] \(\eta_{th}\) allows to compare the contributions of M2 and M3 to that of M1. It confirms what we already have mentioned above, namely that the contribution of M2 (first term in \(\eqref{eqn:sol2:etath}\)) is completely negligible since it is inversely proportional to \((G_m/G_{ds})^4\). The contribution of M3 (second term in \(\eqref{eqn:sol2:etath}\)) is also very small since it is inversely proportional to \((G_m/G_{ds})^2\).

The input-referred flicker noise resistance is given by \[\begin{equation} R_{nf} = \frac{\rho_n}{f\,W_1\,L_1} \cdot (1 + \eta_{fl}), \end{equation}\] where \(\eta_{fl}\) represents the contributions of M2 and M3 to \(R_{nf}\) relative to the that of M1 \[\begin{equation}\label{eqn:sol2:etafl} \eta_{fl} \cong \left(\frac{G_{ds1}}{G_{m1}} \cdot \frac{G_{ds3}}{G_{m3}}\right)^2 \cdot \frac{W_1\,L_1}{W_2\,L_2} + \left(\frac{G_{ds1}}{G_{m1}}\right)^2 \cdot \frac{W_1\,L_1}{W_3\,L_3}. \end{equation}\] Similarly to the thermal noise, assuming that M1, M2 and M3 have all about the same gate area, \(\eqref{eqn:sol2:etafl}\) shows that the contribution of M2 (first term in \(\eqref{eqn:sol2:etafl}\)) is completely negligible since it is inversely proportional to \((G_m/G_{ds})^4\). The contribution of M3 (second term in \(\eqref{eqn:sol2:etafl}\)) is also very small since it is inversely proportional to \((G_m/G_{ds})^2\).

The equivalent noise excess factor of the regulated cascode can then be written as \[\begin{equation}\label{eqn:rc:gammaneq} \gamma_{neq} \triangleq G_{meq} \cdot R_{nt} \cong \gamma_{n1} \cdot (1 + \eta_{th}) \cong \gamma_{n1}. \end{equation}\]

This shows that, similarly to the cascode gain stage, the noise of the regulated cascode is dominated by the contribution of the driver transistor M1.

# 3 Gain boosting

Note

This section is not part of the exercises, however it is an extension to Problem 2 and is therefore good to know.

Figure 15: The principle of gain-boosting [2] [3].

The regulated cascode is a special case of the more general concept of gain boosting illutrated in Figure 15 [2] [3], where the amplifier is implemented by the simple CS source stage M3 of Figure 12. It has basically the same properties than the regulated cascode except that the gain \(A\) can be made much larger than that of a simple CS stage.

## 3.1 Small-signal analysis

Figure 16: Small-signal equivalent circuit used for the calculation of the equivalent transconductance.

Using the small-signal equivalent circuit shown in Figure 16, it is easy to show that the equivalent transconductance of the gain-boosting circuit of Figure 15 is given by \[\begin{equation}\label{eqn:gb:gm} G_{meq} = G_{m1} \cdot \frac{G_{ds2}+G_{m2}\,(A+n_2)}{G_{ds1}+G_{ds2}+G_{m2}\,(A+n_2)}. \end{equation}\] For \(A \gg n_2\), the output conductances can be neglected \(G_{ds1},G_{ds2} \ll A\,G_{m2}\) and \(\eqref{eqn:gb:gm}\) reduces to \[\begin{equation} G_{meq} \cong G_{m1}. \end{equation}\] Similarly to the cascode and regulated cascode, the equivalent transconductance of the gain boosting circuit is simply equal to the transconductance of the driver transistor M1.

Figure 17: Small-signal equivalent circuit used for the calculation of the output conductance.

The output conductance can be calculated using the small-signal circuit shown in Figure 17, which leads to \[\begin{equation} G_{out} = G_{ds1} \cdot \frac{G_{ds2}}{G_{m2}\,(A+n_2)} \cong G_{ds1} \cdot \frac{G_{ds2}}{G_{m2}} \cdot \frac{1}{A}, \end{equation}\] which corresponds to the output conductance of the regulated cascode in case \(A=G_{m3}/G_{ds3}\), but can be made smaller with a larger amplifier gain \(A\). It can therefore be made very small.

Figure 18: Small-signal equivalent circuit used for the calculation of the voltage gain.

The voltage gain can be calculated with the help of the schematic shown in Figure 18 resulting in \[\begin{equation} A_v = -\frac{G_{m1}\,(G_{ds2}+G_{m2}\,(A+n_2))}{G_{ds1}\,G_{ds2}} = -\frac{G_{m1}}{G_{ds1}} \cdot \left(\frac{G_{m2}}{G_{ds2}} \cdot (A+n_2)+1\right), \end{equation}\] which for \(A \gg n_2\) simplifies to \[\begin{equation}\label{eqn:Av:gb} A_v \cong -\frac{G_{m1}}{G_{ds1}} \cdot \frac{G_{m2}}{G_{ds2}} \cdot A. \end{equation}\] Equation \(\eqref{eqn:Av:gb}\) corresponds to the voltage gain of the regulated cascode in case \(A=G_{m3}/G_{ds3}\), but can be made larger with a larger amplifier gain \(A\).

## 3.2 Noise analysis

Figure 19: Small-signal equivalent circuit used for the calculation of the noise.

The noise can be analyzed by means of the small-signal circuit shown in Figure 19. It is easy to show that the input-referred noise resistance is given by \[\begin{equation}\label{eqn:gb:rnin} R_{nin} \cong \frac{G_{n1}}{G_{m1}^2} + \left(\frac{G_{ds1}}{A\,G_{m1}\,G_{m2}}\right)^2 \cdot G_{n2} + \left(\frac{G_{ds1}}{G_{m1}}\right)^2 \cdot R_{noa}, \end{equation}\] where \(R_{noa}\) is the amplifier equivalent input-referred noise resistance. Similarly to the regulated cascode, the noise of the cascode transistor M2 is divided by the square of the gain \(A\,G_{m1}/G_{ds1}\) and can therefore be completely neglected. The impact of the noise coming from the amplifier depends very much on how this amplifier is implemented. If we want to make its contribution corresponding to the last term in \(\eqref{eqn:gb:rnin}\) smaller than the contribution of M1, we need to have \[\begin{equation}\label{eqn:gb:rnoa} \frac{G_{n1}}{G_{ds1}^2} > R_{noa}. \end{equation}\] If we only consider the thermal noise, then \(G_{n1} = \gamma_{n1}\,G_{m1}\) and \(R_{oa} = \gamma_{noa}/G_{moa}\) and \(\eqref{eqn:gb:rnoa}\) becomes \[\begin{equation} \gamma_{noa} < \gamma_{n1}\,\frac{G_{m1}\,G_{moa}}{G_{ds1}^2} \end{equation}\] which should not be too difficult to achieve assuming the gain of the driver \(G_{m1}/G_{ds1}\) is large enough.

# 4 Conclusion

Cascoding is an eeficient way to increase the output resistance and the voltage gain without spending more current and without degrading the noise at the cost of a slightly reduced output voltage swing. A simple cascode stage achieves the same gain as the cascade of two CS stages, but with less current and less noise.

## 5 References

[1]
E. Sackinger and W. Guggenbuhl, “A high-swing, high-impedance MOS cascode circuit,” IEEE Journal of Solid-State Circuits, vol. 25, no. 1, pp. 289–298, 1990, doi: 10.1109/4.50316.
[2]
K. Bult and G. J. G. M. Geelen, “A fast-settling CMOS op amp for SC circuits with 90-dB DC gain,” IEEE Journal of Solid-State Circuits, vol. 25, no. 6, pp. 1379–1384, 1990, doi: 10.1109/4.62165.
[3]
K. Bult and G. J. G. M. Geelen, “The CMOS gain-boosting technique,” Analog Integrated Circuits and Signal Processing, vol. 1, no. 2, pp. 119–135, 1991.