#

Fundamentals of Analog VLSI Design
Exercise 3 - Solution

Design and Optimization of the Common-source Gain Stage

Author

Christian Enz (christian.enz@epfl.ch)

Published

01.10.2025

# Introduction

Figure 1: Common-source (CS) gain stage.

In this exercise we want to apply the \(G_m/I_D\) design methodology using the inversion coefficient \(IC\) to the simple common source (CS) amplifier shown in Figure 1 for various specifications. For the design we assume a generic 180 nm bulk CMOS process. The physical parameters are given in Table 1, the global process parameters in Table 2 and finally the MOSFET parameters in Table 3.

Important

For this process, the transistor dimensions are rounded to 10nm.

Table 1: Physical parameters

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

Table 2: Global process parameters

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

Table 3: 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 | \(nm\)
\(\lambda\) | 15 | 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 | \(\,nm\)

# 1 Problem 1: Imposing the gain-bandwidth product (unity gain frequency)

In this first example we will design the CS gain stage shown in Figure 1 for the specifications at room temperature given in Table 4 by imposing the inversions coefficient and the load capacitance.

## 1.1 Analysis

Figure 2: Small-signal schematic of the common-source (CS).

The small-signal schematic of the CS amplifier is shown in Figure 2. The transfer function is given by \[\begin{equation} H(s) \triangleq \frac{\Delta V_{out}}{\Delta V_{in}} = \frac{A_{dc}}{1+\frac{s}{\omega_c}}, \end{equation}\] where \(A_{dc} = -G_m/G_{ds}\) is the dc gain and \(\omega_c = G_{ds}/C_L\) the cut-off frequency.

The magnitude of the frequency response is then given by \[\begin{equation} |H(\omega)| = \frac{|A_{dc}|}{\sqrt{1+\left(\frac{\omega}{\omega_c}\right)^2}} \cong \frac{\omega_u}{\omega} \end{equation}\] for \(\omega_c \ll \omega\) where \[\begin{equation} \omega_u = |A_{dc}| \cdot \omega_c = \frac{G_m}{C_L} \end{equation}\] is the unity-gain frequency or gain-bandwidth product.

## 1.2 Specifications

The specifications for Problem 1 are given in Table 4.

Table 4: Specifications for problem 1.

Specification | Symbol | Value | Unit
--- | --- | --- | ---
Gain-bandwidth product | \(GBW\) | 10 | \(MHz\)
Inversion coefficient | \(IC\) | 1 | -
Load capacitance | \(C_L\) | 1 | \(pF\)

## 1.3 Long-channel case

### 1.3.1 Design

For the long-channel case, we take \(L =\) 1 \(\mu m\) which corresponds to an effective length \(L_{eff} =\) 924 \(nm\).

The gain-badwidth product \(GBW\) and the load capacitance \(C_L\) set the required transconductance as \(G_m =\) 62.832 \(\mu A/V\). Assuming a long-channel transistor we can neglect the effect of velocity saturation and calculate the current efficiency for the chosen inversion coefficient as \(G_m\,n U_T/I_D =\) 0.618, from which we get \(I_b =\) 3.344 \(\mu A\). From the bias current and the inversion coefficient we get the specific current as \(I_{spec} =\) 3.344 \(\mu A\) and the aspect ratio \(W/L =\) 4.677. Knowing the length, we get the effective width \(W_{eff} =\) 4.319 \(\mu m\) and drawn width \(W =\) 4.28 \(\mu m\).

We can check what is the resulting dc gain by estimating the output conductance \(G_{ds} = I_b/(\lambda_n\,L) =\) 0.241 \(\mu A/V\). The resulting DC gain is then \(A_{dc} = G_m/G_{ds} =\) -260.391 or 48.313 dB.

We also calculate the area and perimeter of the drain to estimate the junction capacitance and overlap capacitance in order to correct \(C_L\) to get the correct \(GBW\). This results in \(C_{DBJ} =\) 3.6 \(fF\), \(C_{GD} =\) 1.6 \(fF\), \(C_D =\) 5.2 \(fF\). To account for the parasitic capacitance at the drain \(C_D =\) 5.2 \(fF\), we need to reduce the load capacitance. The effective load capacitance that needs to be added is then \(C_{L0} =\) 0.995 \(pF\) instead of \(C_L =\) 1.000 \(pF\).

### 1.3.2 Simulations

Figure 3: Schematic of the common-source (CS) gain stage used for simulation.

We can now check whether we get the correct GBW and DC gain using ngspice simulations with the schematic shown in Figure 3, where transistor \(M_2\) is used to correctly bias the gate of the CS transistor \(M_1\) so that it draws the desired bias current \(I_b\).

The simulations will be performed with the following parameters:

    .param VDD=1.8 Ib=3.344u
    .param CL=994.80f W=4.28u L=1.00u AD=1.73e-12 PD=9.44e-06

Before running the AC simulation, we first need to check the quiescent voltages and currents and the operating point by running an .OP simulation. The node voltages are extracted from the .ic file and presented in Table 5.

Table 5: Operating point information.

Node | Voltage
--- | ---
vdd | 1.8
in | 0.508564
out | 0.508564
1 | 0.508564
2 | 0.508564

The operating information for each transistor can be extracted and are given in Table 6 and Table 7.

Table 6: Operating point information extracted from ngspice .op file for each transistor.

Transistor | \(I_D\;[\mu A]\) | \(I_{spec}\;[\mu A]\) | \(IC\) | \(n\) | \(V_{Dsat}\;[mV]\)
--- | --- | --- | --- | --- | ---
M1 | 3.344 | 3.277 | 1.023 | 1.27 | 156
M2 | 3.344 | 3.277 | 1.023 | 1.27 | 156

Table 7: Small-signal operating point information extracted from ngspice .op file for each transistor.

Transistor | \(n\) | \(G_{ms}\;[\mu A/V]\) | \(G_m\;[\mu A/V]\) | \(G_{mb}\;[\mu A/V]\) | \(G_{ds}\;[\mu A/V]\)
--- | --- | --- | --- | --- | ---
M1 | 1.27 | 78.616 | 61.163 | 17.292 | 0.160
M2 | 1.27 | 78.616 | 61.163 | 17.292 | 0.160

We see that the simulated transconductance of M1 \(G_m =\) 61.163 \(\mu A/V\) is slightly lower than the theoretical estimation \(G_m =\) 62.832 \(\mu A/V\).

We can now proceed with the simulation of the transfer function which is shown in Figure 4.

Figure 4: Simulated gain response compared to theoretical estimation for the long-channel case.

From Figure 4, we see that the gain-bandwidth product \(GBW =\) 9.7 \(MHz\) is slightly lower than the specifications \(GBW =\) 10 \(MHz\) despite we have accounted for the junction capacitance at the output. This comes from the fact that the simulated transconductance \(G_m =\) 61.163 \(\mu A/V\) is slightly lower than the theoretical estimation \(G_m =\) 62.832 \(\mu A/V\).

We also see that there is some discrepancy of the phase at high frequancy (above the \(GBW\)). This coming from the gate-to-drain capacitance (the overla capacitance \(C_{GDo}\)) which introduces a positive zero. We can add this zero in the theoretical transfer leading to the transfer function magnitude and phase are plotted in Figure 5. We now see that we have a better fit at high frequency (dashed blue line). The gate-to-drain capacitance seems to be larger.

Figure 5: Simulated gain response compared to theoretical estimation for the long-channel case.

If we want to meet the specifications we can slightly increase the bias current to \(I_b =\) 3.5 \(\mu A\). The transfer function is now given in Figure 6. We see that we are now right on spec.

Figure 6: Simulated gain response compared to theoretical estimation for the long-channel case.

## 1.4 Short-channel case (including velocity saturation)

### 1.4.1 Design

The required transconductance hasn’t changed because it is set by the desired gain-bandwidth product and load capacitance \(G_m =\) 62.832 \(\mu A/V\).

For the short-channel case we will use the shortest channel length for this technology, namely \(L =\) 0.18 \(\mu m\) corresponding to an effective length \(L_{eff} =\) 104 \(nm\). The velocity parameter for the chosen length is then given by \(\lambda_c = L_{satn}/L_{eff} =\) 0.250. The procedure is then the same: we first get the bias current from the current efficiency but including the effect of velocity saturation \(G_m\,n U_T/I_D =\) 0.606. We see that the current efficiency \(G_m\,n U_T/I_D\) is only slightly smaller than for the long-channel case due to velocity saturation. The bias current is then given by \(I_b =\) 3.410 \(\mu A\). Despite we use a shorter channel-length, the required bias current is about the same than the one from the long-channel case. This is simply due to the fact the effect of velocity saturation is small in moderate and hence has little effect on the current efficiency.

From the bias current and the inversion coefficient we get the specific current as \(I_{spec} =\) 3.410 \(\mu A\) and the aspect ratio \(W_{eff}/L_{eff} =\) 4.770. For the chosen length we get the effective width \(W_{eff} =\) 0.499 \(\mu m\) and drawn width \(W =\) 0.46 \(\mu m\).

We can check what is the resulting dc gain by estimating the output conductance \(G_{ds} = I_b/(\lambda_n\,L_{eff}) =\) 2.186 \(\mu A/V\). The DC gain is now \(A_{dc} =\) -28.740 or 29.170 \(dB\), which, as expected, is significantly lower than for the long-channel case.

We also calculate the area and perimeter of the drain to estimate the junction capacitance and overlap capacitance in order to correct \(C_L\) to get the correct \(GBW\). This results in \(C_{DBJ} =\) 1 \(fF\), \(C_{GD} =\) 0.183 \(fF\), \(C_D =\) 1 \(fF\). The effective load capacitance that needs to be added is then \(C_{L0} =\) 0.999 \(pF\).

### 1.4.2 Simulations

We can check whether we get the correct GBW and dc gain using simulations with the following values of the parameters:

Before running the AC simulation, we first need to check the quiescent voltages and currents and the operating point by running an .OP simulation. The node voltages are extracted from the .ic file and presented in Table 8.

Table 8: Operating point information.

Node | Voltage
--- | ---
vdd | 1.8
in | 0.511076
out | 0.511076
1 | 0.511076
2 | 0.511076

The operating information for each transistor can be extracted and are given in Table 9 and Table 10.

Table 9: Operating point information extracted from ngspice .op file for each transistor.

Transistor | \(I_D\;[\mu A]\) | \(I_{spec}\;[\mu A]\) | \(IC\) | \(n\) | \(V_{Dsat}\;[mV]\)
--- | --- | --- | --- | --- | ---
M1 | 3.500 | 3.273 | 1.072 | 1.27 | 157
M2 | 3.500 | 3.273 | 1.072 | 1.27 | 157

Table 10: Small-signal operating point information extracted from ngspice .op file for each transistor.

Transistor | \(n\) | \(G_{ms}\;[\mu A/V]\) | \(G_m\;[\mu A/V]\) | \(G_{mb}\;[\mu A/V]\) | \(G_{ds}\;[\mu A/V]\)
--- | --- | --- | --- | --- | ---
M1 | 1.27 | 81.190 | 63.169 | 17.855 | 0.167
M2 | 1.27 | 81.190 | 63.169 | 17.855 | 0.167

We see that the simulated transconductance of M1 \(G_m =\) 63.169 \(\mu A/V\) is slightly larger than the theoretical estimation \(G_m =\) 62.832 \(\mu A/V\).

    .param VDD=1.8 Ib=3.410u
    .param CL=999f W=0.46u L=0.18u AD=2.00e-13 PD=1.80e-06

The transfer function magnitude and phase are plotted in Figure 7, which shows a reasonable fit between the simulation and the theoretical estimation. The simulated gain-bandwidth product \(GBW\) is slightly lower than target and the simulated DC gain is slightly larger than the theoretical estimation.

Figure 7: Simulated gain response compared to theoretical estimation for the short-channel case.

From Figure 7, we see that the simulated DC gain is slightly larger than the theoretical estimation. We also see that the gain-bandwidth product specs is not met. If we want to meet the specifications we can slightly increase the bias current to \(I_b =\) 4.0 \(\mu A\). The transfer function is now given in Figure 8. We see that we are now right on spec.

Note

We also see that for the short-channel case, we don’t see the zero which has moved to much higher frequency. This is due to the much smaller transistor width which significantly reduces the gate-to-drain capacitance.

Figure 8: Simulated gain response compared to theoretical estimation for the short-channel case.

# 2 Problem 2: Imposing the power consumption

## 2.1 Specifications

We now impose the power consumption, the inversion coefficient and the load capacitance according to the specifications given in Table 11.

Table 11: Specifications for problem 2.

Specification | Symbol | Value | Unit
--- | --- | --- | ---
Power consumption | \(P\) | 20 | \(\mu W\)
Inversion coefficient | \(IC\) | 1 | -
Load capacitance | \(C_L\) | 1 | \(pF\)

## 2.2 Long-channel case

### 2.2.1 Design

For the long-channel case, we choose \(L =\) 1 \(\mu m\) which corresponds to an effective width \(L_{eff} =\) 0.924 \(\mu m\). The maximum available current is directly given by \(I_b = P/V_{DD} =\) 11.111 \(\mu A\). Since the inversion coefficient is chosen as \(IC =\) 1, we get the specific current as \(I_{spec} =\) 11.111 \(\mu A\). The normalized transconductance is the given by the chosen inversion coefficient as \(g_{ms} = G_{ms}/G_{spec} =\) 0.618, from which we derive the transconductance \(G_m =\) 208.747 \(\mu A/V\) and the gain-bandwidth product for the given load capacitance \(GBW = G_m/C_L =\) 33.223 \(MHz\). The aspect ratio is then given by \(W_{eff}/L_{eff} =\) 15.540. The effective width \(W_{eff} =\) 14.359 \(\mu m\) and drawn width \(W =\) 14.32 \(\mu m\).

We can calculate the dc gain by first estimating the output conductance \(G_{ds} = I_b/(\lambda_n\,L_{eff}) =\) 0.802 \(\mu A/V\). The dc gain is then given by \(A_{dc} =G_m/G_{ds} =\) -260.391 or 48.313 \(dB\).

We also calculate the area and perimeter of the drain to estimate the junction capacitance and overlap capacitance in order to correct \(C_L\) to get the correct \(GBW\). This results in \(C_{DBJ} =\) 12 \(fF\), \(C_{GD} =\) 5.248 \(fF\), \(C_D =\) 17 \(fF\). The effective load capacitance that needs to be added is then \(C_{L0} =\) 0.983 \(pF\).

### 2.2.2 Simulations

We can now check whether we get the correct GBW and DC gain using ngspice simulations with the schematic shown in Figure 3. The simulations will be performed with the following parameters:

    .param VDD=1.8 Ib=11.111u
    .param CL=983.10f W=14.32u L=1.00u AD=5.74e-12 PD=2.95e-05

Before running the AC simulation, we first need to check the quiescent voltages and currents and the operating point by running an .OP simulation. The node voltages are extracted from the .ic file and presented in Table 5.

Table 12: Operating point information.

Node | Voltage
--- | ---
vdd | 1.8
in | 0.508531
out | 0.508531
1 | 0.508531
2 | 0.508531

The operating information for each transistor can be extracted and are given in Table 6 and Table 7.

Table 13: Operating point information extracted from ngspice .op file for each transistor.

Transistor | \(I_D\;[\mu A]\) | \(I_{spec}\;[\mu A]\) | \(IC\) | \(n\) | \(V_{Dsat}\;[mV]\)
--- | --- | --- | --- | --- | ---
M1 | 11.111 | 10.895 | 1.022 | 1.27 | 156
M2 | 11.111 | 10.895 | 1.022 | 1.27 | 156

Table 14: Small-signal operating point information extracted from ngspice .op file for each transistor.

Transistor | \(n\) | \(G_{ms}\;[\mu A/V]\) | \(G_m\;[\mu A/V]\) | \(G_{mb}\;[\mu A/V]\) | \(G_{ds}\;[\mu A/V]\)
--- | --- | --- | --- | --- | ---
M1 | 1.27 | 261.256 | 203.257 | 57.466 | 0.533
M2 | 1.27 | 261.256 | 203.257 | 57.466 | 0.533

We see that the simulated transconductance of M1 \(G_m =\) 203.257 \(\mu A/V\) is slightly lower than the theoretical estimation \(G_m =\) 208.747 \(\mu A/V\).

The transfer function magnitude and phase are plotted in Figure 5, which shows a good fit between the simulation and the theoretical estimation.

Figure 9: Simulated gain response compared to theoretical estimation for the long-channel case.

## 2.3 Short-channel case (including velocity saturation)

### 2.3.1 Design

For the short-channel case we will use the shortest channel length for this technology, namely \(L =\) 180 \(nm\). The velocity parameter for the chosen length is then given by \(\lambda_c = L_{satn}/L =\) 0.250. The procedure is basically the same except that we now need to account for velocity saturation in the estimation of the transconductance. Since the inversion coefficient is chosen as \(IC =\) 1, we get the specific current as \(I_{spec} =\) 11.111 \(\mu A\). The normalized transconductance is the given by the chosen inversion coefficient as \(g_{ms} = G_{ms}/G_{spec} =\) 0.606, from which we derive the transconductance \(G_m =\) 0.205 \(mA/V\) and the gain-bandwidth product for the given load capacitance \(GBW = G_m/C_L =\) 32.580 \(MHz\). The aspect ratio is then given by \(W_{eff}/L_{eff} =\) 15.540. The effective width \(W_{eff} =\) 1.619 \(\mu m\) and drawn width \(W =\) 1.58 \(\mu m\).

We can calculate the dc gain by first estimating the output conductance \(G_{ds} = I_b/(\lambda_n\,L_{eff}) =\) 7.123 \(\mu A/V\). The dc gain is then given by \(A_{dc} =G_m/G_{ds} =\) -28.740 or 29.170 \(dB\).

We also calculate the area and perimeter of the drain to estimate the junction capacitance and overlap capacitance in order to correct \(C_L\) to get the correct \(GBW\). This results in \(C_{DBJ} =\) 1.5 \(fF\), \(C_{GD} =\) 0.579 \(fF\), \(C_D =\) 2.0 \(fF\). The effective load capacitance that needs to be added is then \(C_{L0} =\) 0.998 \(pF\).

### 2.3.2 Simulations

We can check whether we get the correct GBW and dc gain using simulations with the following values of the parameters:

    .param VDD=1.8 Ib=11.111u
    .param CL=997.97f W=1.58u L=0.18u AD=6.48e-13 PD=4.04e-06

The transfer function magnitude and phase are plotted in Figure 10, which shows a reasonable fit between the simulation and the theoretical estimation.

Figure 10: Simulated gain response compared to theoretical estimation for the short-channel case.

# 3 Problem 3: Imposing the gain bandwidth product and the current budget

## 3.1 Specifications

We now don’t impose the inversion coefficient but the gain bandwidth product and the current budget. We keep the same load capacitance and consider a long channel. The specifications are now given in Table 15.

Table 15: Specifications for problem 3.

Specification | Symbol | Value | Unit
--- | --- | --- | ---
Gain-bandwidth product | \(GBW\) | 1 | \(MHz\)
Bias current | \(I_b\) | 400 | \(nA\)
Load capacitance | \(C_L\) | 1 | \(pF\)
Transistor length | \(L\) | 1 | \(\mu m\)

## 3.2 Design

We first get the transconductance required to achieve the target gain-bandwidth product as \(G_m =\) 6.283 \(\mu A/V\). Knowing the bias current and the transconductance we can calculate the current efficiency as \(G_m\,n U_T/I_D =\) 0.517, from which we can derive the corresponding inversion coefficient \(IC =\) 1.810. We see that we are in the moderate inversion region. The specific current is then given by \(I_{spec} =\) 221 \(nA\), the aspect ratio by \(W_{eff}/L_{eff} =\) 0.309, the effective width by \(W_{eff} =\) 289 \(nm\) and finally the drawn width \(W =\) 250 \(nm\).

We can calculate the dc gain by first estimating the output conductance \(G_{ds} = I_b/(\lambda_n\,L_{eff}) =\) 28.860 \(nA/V\). The dc gain is then given by \(A_{dc} =G_m/G_{ds} =\) -217.712 or 46.758 \(dB\).

We also calculate the area and perimeter of the drain to estimate the junction capacitance and overlap capacitance in order to correct \(C_L\) to get the correct \(GBW\). This results in \(C_{DBJ} =\) 391.2 \(aF\), \(C_{GD} =\) 105.918 \(aF\), \(C_D =\) 497.1 \(aF\). The effective load capacitance that needs to be added is then \(C_{L0} =\) 999.503 \(fF\). We observe that the parasitic capacitance at the drain are negligible compared to the load capacitance.

## 3.3 Simulations

We can now check whether we get the correct GBW and DC gain using ngspice simulations with the schematic shown in Figure 3. The simulations will be performed with the following parameters:

    .param VDD=1.8 Ib=0.400u
    .param CL=999.50f W=0.25u L=1.00u AD=1.16e-13 PD=1.38e-06

The transfer function magnitude and phase are plotted in Figure 5, which shows a good fit between the simulation and the theoretical estimation.

Figure 11: Simulated gain response compared to theoretical estimation for the long-channel case.

# 4 Problem 4: Imposing the input-referred noise

## 4.1 Specifications

In this example, we impose the input-referred noise by setting the white noise and the corner frequency. The specifications are given in Table 16.

Table 16: Specifications for problem 4.

Specification | Symbol | Value | Unit
--- | --- | --- | ---
Corner frequency | \(f_k\) | 100 | \(kHz\)
Input referred white PSD | \(\sqrt{{S_{{n,th}}}}\) | 30 | \(\frac{{nV}}{{\sqrt{{Hz}}}}\)
| \(10\,\log(S_{{n,th}})\) | -150.46 | \(\frac{{dBv}}{{\sqrt{{Hz}}}}\)
Inversion coefficient | \(IC\) | 1 | -
Load capacitance | \(C_L\) | 1 | \(pF\)

## 4.2 Design

We need the flicker noise parameter \(K_f =\) 8.100e-24 \(J\) and the oxide capacitance per unit area \(C_{ox} =\) 8.443 \(fF/\mu m^2\).

Before the transconductance is derived from the thermal noise specification, we need to know the thermal noise excess factor \(\gamma_n\) for the given \(IC\) (we assume long-channel case) which is \(\gamma_n =\) 0.717. The transconductance is then given by \(G_m = 4 kT \gamma_n/S_{nin,th} =\) 13.186 \(\mu A/V\), where \(S_{nin,th} = V_{nin,th}^2\). Once we have the transconductance, the procedure is similar to the previous one.

Having imposed the inversion coefficient \(IC =\) 1, we can deduce the current efficiency as \(G_m\,n U_T/I_D =\) 0.618. By definition of the current efficiency we can get the corresponding bias current \(I_b =\) 702 \(nA\). The specific current and aspect ratio are then given by \(I_{spec} =\) 702 \(nA\) and \(W_{eff}/L_{eff} =\) 0.982.

The corner frequency will actually set the flicker noise which is inversely proportional to the gate are. We can therefore calculate the gate area from the flicker noise remembering that the corner frequency is the frequency at which the flicker noise becomes equal to the white noise. The gate area is obtained as \(W_{eff}\,L_{eff} =\) 10.660 \(\mu m^2\). Knowing the \(W/L\) ratio we can calculate the effective width and length as \(W_{eff} =\) 3.235 \(\mu m\) and \(L_{eff} =\) 3.295 \(\mu m\). The drawn width and length follow as \(W =\) 3.20 \(\mu m\) and \(L =\) 3.37 \(\mu m\).

The gain-bandwidth product is given by \(GBW =\) 2.099 \(MHz\).

We can calculate the dc gain by first estimating the output conductance \(G_{ds} = I_b/(\lambda_n\,L_{eff}) =\) 0.014 \(\mu A/V\). The dc gain is then given by \(A_{dc} =G_m/G_{ds} =\) -928.687 or 59.357 \(dB\).

We also calculate the area and perimeter of the drain to estimate the junction capacitance and overlap capacitance in order to correct \(C_L\) to get the correct \(GBW\). This results in \(C_{DBJ} =\) 2.7 \(fF\), \(C_{GD} =\) 1.186 \(fF\), \(C_D =\) 3.9 \(fF\). The effective load capacitance that needs to be added is then \(C_{L0} =\) 0.996 \(pF\).

## 4.3 Simulations

We can now check whether we get the correct GBW and DC gain using ngspice simulations with the schematic shown in Figure 3. The simulations will be performed with the following parameters:

    .param VDD=1.8 Ib=702n
    .param CL=996.07f W=3.20u L=3.37u AD=1.29e-12 PD=7.27e-06

The transfer function magnitude and phase are plotted in Figure 13, which shows a good fit between the simulation and the theoretical estimation.

Figure 12: Simulated gain response compared to theoretical estimation for the long-channel case.

We can now check the input-referred noise PSD which is plotted in Figure 13. We can see that simulation perfectly matches the theoretical estimation.

Figure 13: Simulated input-referred noise PSD compared to theoretical estimation.

# 5 Problem 5: Minimum current for a given gain-bandwidth product with self-loading

Figure 14: Schematic of the common-source (CS) illustrating the self-loading effect.

In this problem we want to find the minimum current required to achieve a given gain-bandwidth product accounting for the effect of self-loading as illustrated in Figure 14.

The total transistor parasitic capacitance at the drain can be written as \[\begin{equation} C_D = C_{D0} + C_{DW} \cdot W. \end{equation}\] with \[\begin{align} C_{D0} &= 4 H_{dif} \cdot C_{JSW},\\ C_{DW} &= 2(H_{dif} \cdot C_J + C_{JSW}) + C_{GDe}, \end{align}\] where \(C_{GDe}\) is the extrinsic capacitance per unit width which is made of the overlap capacitance \(C_{GDo}\). In more advanced technologies we also need to add the fringing-field capacitance \(C_{GDf}\). In this 180nm technology the fringing-field capacitance can be neglected so that \(C_{GDe} \cong C_{GDo}\).

The part \(C_{D0}\) of the total parasitic capacitance at the drain \(C_D\) that doesn’t scale with \(W\) needs to be added to \(C_{L0}\) so that the total load capacitance that does not scale with \(W\) is given by \[\begin{equation} C_L = C_{L0} + C_{D0}. \end{equation}\]

It can be shown [1] that when accounting for the self-loading effect, there is a minimum current for achieving a given gain-bandwidth product. The otptimum value of the inversion coefficient \(IC_{opt}\) corresponding to this minimum bias current is given by \[\begin{equation}\label{eqn:icopt} IC_{opt} = \left(\sqrt{\Theta \cdot (1+\Theta)} + \Theta + \frac{1}{2}\right)^2 - \frac{1}{4} = 2 \Theta \cdot (1+\Theta) + (1+2\Theta) \cdot \sqrt{\Theta \cdot (1+\Theta)}, \end{equation}\] where \[\begin{equation} \Theta \triangleq \frac{\omega_u}{\omega_W}, \end{equation}\] with \[\begin{align} \omega_u &\triangleq 2 \pi\,GBW,\\ \omega_W &\triangleq \frac{I_{spec\Box}}{n U_T \cdot C_{DW} \cdot L}. \end{align}\]

The parameter \(\Theta\) is usually \(\Theta \ll 1\) and therefore \(\eqref{eqn:icopt}\) reduces to \[\begin{equation} IC_{opt} \cong 2 \Theta + \sqrt{\Theta}. \end{equation}\]

We can define a normalized current and aspect ratio as \[\begin{align} i_b &\triangleq \frac{I_b}{I_{pec\Box} \cdot \Omega},\\ AR &\triangleq \frac{W/L}{\Omega}, \end{align}\] where \[\begin{align} \Omega &\triangleq \frac{\omega_u}{\omega_L},\\ \omega_L &\triangleq \frac{I_{spec\Box}}{n U_T \cdot C_L}. \end{align}\]

The minimum normalized current \(i_{bopt}\) and aspect ratio \(AR_{opt}\) only depend on \(\Theta\) according to \[\begin{align} i_{bopt} &\triangleq i_b(IC_{opt}) = \sqrt{4 IC_{opt} + 1} = 1 + 2\Theta +2\sqrt{\Theta \cdot (1+\Theta)},\\ AR_{opt} &\triangleq AR(IC_{opt}) = \frac{\sqrt{4 IC_{opt} + 1}}{IC_{opt}} = \frac{1}{\sqrt{\Theta \cdot (1+\Theta)}}. \end{align}\]

## 5.1 Specifications

The specifications for Problem 5 are given in Table 17.

Table 17: Specifications for problem 5.

Specification | Symbol | Value | Unit
--- | --- | --- | ---
Gain-bandwidth product | \(GBW\) | 25 | \(MHz\)
Load capacitance | \(C_{L0}\) | 20 | \(fF\)
Transistor length | \(L\) | 1 | \(\mu m\)

## 5.2 Design

For accounting for the self-loading effect we need the capacitance per width \(C_{DW} =\) 1.167 \(fF/\mu m\). All the parameters are given in Table 18.

Table 18: Parameters for problem 5.

Parameter | Value | Unit
--- | --- | ---
\(C_{{DW}}\) | 1.167 | \(fF/\mu m\)
\(C_{{D0}}\) | 0.16 | \(fF\)
\(C_{{L0}}\) | 20 | \(fF\)
\(C_L\) | 20.16 | \(fF\)
\(f_L\) | 171.589 | \(MHz\)
\(f_W\) | 2.965 | \(GHz\)
\(\Omega\) | 0.146 | -
\(\Theta\) | 0.00843 | -
\(IC_{{opt}}\) | 0.111 | -
\(i_{{b,opt}}\) | 1.201 | -
\(AR_{{opt}}\) | 10.846 | -
\(\left(W/L\right)_{{opt}}\) | 1.58 | -
\(I_{{b,opt}}\) | 125 | \(nA\)
\(W_{{opt}}\) | 1.46 | \(\mu m\)
\(G_{{m,opt}}\) | 3.456 | \(\mu A/V\)
\(C_{{D,opt}}\) | 1.703 | \(fF\)
\(AD\) | 0.58 | \(\mu m^2\)
\(PD\) | 3.72 | \(\mu m\)
\(C_{{DBJ}}\) | 1.328 | \(fF\)
\(C_F\) | 0.535 | \(fF\)
\(C_{{out}}\) | 21.863 | \(fF\)
\(GBW\) (check) | 25 | \(MHz\)
\(f_z\) | 1.028 | \(GHz\)
\(A_{{dc}}\) | 52.346 | \(dB\)

## 5.3 Simulations

We can now check whether we get the correct GBW and DC gain using ngspice simulations with the schematic shown in Figure 3. The simulations will be performed with the following parameters:

    .param VDD=1.8 Ib=125n
    .param CL=20.00f W=1.42u L=1.00u AD=5.84e-13 PD=3.72e-06

Before running the AC simulation, we first need to check the quiescent voltages and currents and the operating point by running an .OP simulation. The node voltages are extracted from the .ic file and presented in Table 19.

Table 19: Operating point information.

Node | Voltage
--- | ---
vdd | 1.8
in | 0.412978
out | 0.412978
1 | 0.412978
2 | 0.412978

The operating information for each transistor can be extracted and are given in Table 20 and Table 21.

Table 20: Operating point information extracted from ngspice .op file for each transistor.

Transistor | \(I_D\;[\mu A]\) | \(I_{spec}\;[\mu A]\) | \(IC\) | \(n\) | \(V_{Dsat}\;[mV]\)
--- | --- | --- | --- | --- | ---
M1 | 0.125 | 1.147 | 0.109 | 1.27 | 121
M2 | 0.125 | 1.147 | 0.109 | 1.27 | 121

Table 21: Small-signal operating point information extracted from ngspice .op file for each transistor.

Transistor | \(n\) | \(G_{ms}\;[\mu A/V]\) | \(G_m\;[\mu A/V]\) | \(G_{mb}\;[\mu A/V]\) | \(G_{ds}\;[\mu A/V]\)
--- | --- | --- | --- | --- | ---
M1 | 1.27 | 4.385 | 3.395 | 0.981 | 0.008
M2 | 1.27 | 4.385 | 3.395 | 0.981 | 0.008

We see that the simulated transconductance of M1 \(G_m =\) 3.395 \(\mu A/V\) is close to the theoretical estimation \(G_m =\) 3.456 \(\mu A/V\).

The transfer function maginitude and phase are plotted in Figure 15, which shows a good fit between the simulation and the theoretical estimation. The gain-bandwidth product \(GBW\) is right on target. However, the simulated DC gain is slightly larger than the theoretical estimation. We also see some discrepancy at higher frequency. This is due to the zero introduced by the gate-to-drain capacitance \(C_{GD}\).

Figure 15: Simulated gain response compared to theoretical estimation for the long-channel case.

From Figure 15 we see that accounting for the gate-to-drain capacitance which corresponds actually to the overlap capacitance \(C_{GDo}\), still underestimates the loction of the zero. This is due to a larger feedback capacitance. This is actually coming from the fact that we have ignored the transcapacitance \(C_m\) in the small-signal analysis. In the quasi-static (QS) circuit of the MOS transistor, the transconductance needs to be replaced by the a transadmittance \(Y_m\) given by \[\begin{equation} Y_m = G_m + s\,C_m \end{equation}\] where \(C_m\) is the gate transcapacitance given by \[\begin{equation} C_m = C_{OX} \cdot c_m \end{equation}\] where \(C_{OX} \triangleq W\,L\,C_{ox}\) and \(c_m\) is the normalized gate transcapacitance given by [2] \[\begin{equation} c_m = \frac{q_s-q_d}{15}\;\frac{4 q_s^2+4 q_d^2+12 q_s q_d+10 q_s+10 q_d+5}{(q_s+q_d+q)^3}, \end{equation}\] which in saturation (\(q_d=0\)) reduces to \[\begin{equation} c_{m_{sat}} \cong \frac{4 q_s^2+10 q_s+5}{(q_s+1)^3}. \end{equation}\] For a long-channel transistor, \(q_s\) is simply given by \[\begin{equation} q_s = \frac{\sqrt{4 IC+1}-1}{2}. \end{equation}\]

The corresponding transfer function including the effect of the gate transcapacitance \(C_m\) is then given by \[\begin{equation} A(s) \triangleq \frac{\Delta V_{out}}{\Delta V_{in}} = A_{dc} \cdot \frac{1-s/\omega_z}{1+s/\omega_p} \end{equation}\] where \[\begin{align} A_{dc} &= -G_m \cdot R_{ds},\\ \omega_z &= \frac{G_m}{C_F+C_m},\\ \omega_p &= \frac{1}{R_{ds}\,C_{out}}, \end{align}\] with \(C_{out} = C_L+C_F\) the total load capacitance. We see that the transcapacitance \(C_m\) directly adds to the feedback capacitance \(C_F\), but does not add to the load capacitance \(C_{out}\). Therefore, the expression of the pole remains unchanged. The DC gain and the gain-bandwidth hence also remain unaffected by the transcapacitance.

The theoretical transfer function including the effect of the gate transcapacitance is shown in Figure 16. We now see that the theoretical estimation perfectly matches the simulation including at high frequency.

Figure 16: Simulated gain response compared to theoretical estimation for the long-channel case.

# 6 Problem 6: Minimum current for a given DC gain and gain-bandwidth product with self-loading

In this problem we want to find the minimum current required to achieve a given gain-bandwidth product and a given dc gain accounting for the effect of self-loading. We can actually use the additional degree of freedom, namely the transistor length \(L\) (which has been arbitrarily set in the previous example), to set the DC gain. To this purpose we can use the simple output conductance model given by \[\begin{equation} G_{ds} \cong \frac{I_D}{\lambda \cdot L}. \end{equation}\]

It can be shown that there is also an optimum value of the inversion coefficient for which the bias current becomes minimum [1] \[\begin{equation} IC_{opt} = \frac{\sqrt{\xi\;}}{\left(1-\sqrt{\xi\;}\right)^2}, \end{equation}\] where parameter \(\xi\) brings together the technology parameters and the specs according to \[\begin{equation} \xi \triangleq \underbrace{A_{dc} \cdot \omega_u}_{\textsf{specifications}} \cdot \underbrace{\frac{C_{DW} \cdot (n U_T)^2}{I_{spec\Box} \cdot \lambda}}_{\textsf{technology parameters}}. \end{equation}\]

It can be shown that parameter \(\xi\) needs to satisfy the following inequality \[\begin{equation*} \xi \triangleq A_{dc} \cdot \omega_u \cdot \frac{C_{DW} \cdot (n U_T)^2}{I_{spec\Box} \cdot \lambda} < 1. \end{equation*}\] This means that we can check whether the specifications on \(A_{dc}\) and \(\omega_u\) can be achieved for a given technology (i.e. for a set of \(I_{spec\Box}\), \(\lambda\) and \(C_{DW}\)). The closer \(\xi\) gets to 1, the larger the required current and size.

We can define the normalized bias current \(i_b\), transistor width \(w\) and length \(\ell\) according to \[\begin{align} i_b &\triangleq \frac{I_b}{I_{norm}} = \frac{g_{ms} \cdot IC}{g_{ms}^2 - \xi \cdot IC} = \frac{g_{ms}/IC}{(g_{ms}/IC)^2 - \xi/IC},\\ w &\triangleq \frac{W}{W_{norm}} = \frac{IC}{g_{ms}^2 - \xi \cdot IC},\\ \ell &\triangleq \frac{L}{L_{norm}} = \frac{IC}{g_{ms}}, \end{align}\] where \[\begin{align} I_{norm} & \triangleq n\,U_T \cdot C_{L0} \cdot \omega_u,\\ W_{norm} &\triangleq \frac{C_{L0} \cdot (n U_T)^2}{I_{spec\Box} \cdot \lambda} \cdot A_{dc} \cdot \omega_u,\\ L_{norm} & \triangleq \frac{n U_T}{\lambda} \cdot A_{dc}. \end{align}\]

The minimum normalized current is then given by \[\begin{equation} i_{b,opt} \triangleq i_b(IC_{opt}) = \frac{1}{\left(1-\sqrt{\xi\;}\right)^2} = \frac{1+2 IC_{opt}+\sqrt{4 IC_{opt}+1}}{2}. \end{equation}\]

The optimum normalized width, length and transconductance are given by \[\begin{align} w_{opt} &\triangleq w(IC_{opt}) = \frac{1}{\sqrt{\xi\;}\;\left(1-\sqrt{\xi\;}\right)} = \frac{1+3IC_{opt}+(1+IC_{opt})\,\sqrt{4IC_{opt}+1}}{2IC_{opt}},\\ \ell_{opt} &\triangleq \ell(IC_{opt}) = \frac{1}{1-\sqrt{\xi\;}} = \frac{1+\sqrt{4IC_{opt}+1}}{2},\\ g_{m,opt} &\triangleq g_m(IC_{opt}) = \frac{1}{1-\sqrt{\xi\;}} = \frac{1+\sqrt{4IC_{opt}+1}}{2}. \end{align}\]

Note that the optimum current, width, length and transconductance only depend on \(\xi\).

## 6.1 Specifications

The specifications for Problem 5 are given in Table 22.

Table 22: Specifications for problem 6.

Specification | Symbol | Value | Unit
--- | --- | --- | ---
DC gain | \(A_{dc}\) | 50 | \(dB\)
Gain-bandwidth product | \(GBW\) | 25 | \(MHz\)
Load capacitance | \(C_{L0}\) | 20 | \(fF\)

## 6.2 Design

We can start by checking that the specifications in Table 22 are feasible for the given 180nm technology. For this we can check that \[\begin{equation*} \xi \triangleq A_{dc} \cdot \omega_u \cdot \frac{C_{DW} \cdot (n U_T)^2}{I_{spec\Box} \cdot \lambda} < 1. \end{equation*}\] For the specifications in Table 22 and the chosen 180nm technology we have \(\xi =\) 0.006 which is smaller than 1.

Table 23: Parameters or problem 6.

Parameter | Value | Unit
--- | --- | ---
\(C_{{DW}}\) | 1.167 | \(fF/\mu m\)
\(C_{{D0}}\) | 0.16 | \(fF\)
\(C_{{L0}}\) | 20 | \(fF\)
\(C_L\) | 20.16 | \(fF\)
\(\xi\) | 0.006 | -
\(IC_{{opt}}\) | 0.09 | -
\(i_{{b,opt}}\) | 1.172 | -
\(w_{{opt}}\) | 14.161 | -
\(\ell_{{opt}}\) | 1.083 | -
\(g_{{m,opt}}\) | 1.083 | -
\(I_{{b,opt}}\) | 122 | \(nA\)
\(\left(W/L\right)_{{opt}}\) | 1.58 | -
\(I_{{norm}}\) | 104 | \(nA\)
\(W_{{norm}}\) | 101 | \(nm\)
\(L_{{norm}}\) | 694 | \(nm\)
\(G_{{m,norm}}\) | 3.167 | \(\mu A/V\)
\(I_{{b,opt}}\) | 122.138 | \(nA\)
\(W_{{opt}}\) | 1.43 | \(\mu m\)
\(L_{{opt}}\) | 750.94 | \(nm\)
\(G_{{m,opt}}\) | 3.429 | \(\mu A/V\)
\(C_{{D,opt}}\) | 1.669 | \(fF\)
\(AD\) | 0.57 | \(\mu m^2\)
\(PD\) | 3.662 | \(\mu m\)
\(C_{{DBJ}}\) | 1.305 | \(fF\)
\(C_F\) | 0.524 | \(fF\)
\(C_{{out}}\) | 21.829 | \(fF\)
\(GBW\) (check) | 25 | \(MHz\)
\(f_z\) | 1.028 | \(GHz\)
\(A_{{dc}}\) | 50 | \(dB\)

## 6.3 Simulations

We can now check whether we get the correct GBW and DC gain using ngspice simulations with the schematic shown in Figure 3. The simulations will be performed with the following parameters:

    .param VDD=1.8 Ib=122n
    .param CL=20.00f W=1.43u L=0.75u AD=5.72e-13 PD=3.66e-06

Before running the AC simulation, we first need to check the quiescent voltages and currents and the operating point by running an .OP simulation. The node voltages are extracted from the .ic file and presented in Table 24.

Table 24: Operating point information.

Node | Voltage
--- | ---
vdd | 1.8
in | 0.4076
out | 0.4076
1 | 0.4076
2 | 0.4076

The operating information for each transistor can be extracted and are given in Table 25 and Table 26.

Table 25: Operating point information extracted from ngspice .op file for each transistor.

Transistor | \(I_D\;[\mu A]\) | \(I_{spec}\;[\mu A]\) | \(IC\) | \(n\) | \(V_{Dsat}\;[mV]\)
--- | --- | --- | --- | --- | ---
M1 | 0.122 | 1.597 | 0.076 | 1.27 | 118
M2 | 0.122 | 1.597 | 0.076 | 1.27 | 118

Table 26: Small-signal operating point information extracted from ngspice .op file for each transistor.

Transistor | \(n\) | \(G_{ms}\;[\mu A/V]\) | \(G_m\;[\mu A/V]\) | \(G_{mb}\;[\mu A/V]\) | \(G_{ds}\;[\mu A/V]\)
--- | --- | --- | --- | --- | ---
M1 | 1.27 | 4.390 | 3.400 | 0.980 | 0.010
M2 | 1.27 | 4.390 | 3.400 | 0.980 | 0.010

We see that the simulated transconductance of M1 \(G_m =\) 3.400 \(\mu A/V\) is close to the theoretical estimation \(G_m =\) 3.429 \(\mu A/V\).

The transfer function magnitude and phase are plotted in Figure 17, which shows an almost perfect fit between the simulation and the theoretical estimation. The DC gain and gain-bandwidth product are right on target. Even the high frequency behavior is well captured because of the good estimation of the total gate-to-drain capacitance including the contribution of the gate transcapacitance.

Figure 17: Simulated gain response compared to theoretical estimation for the long-channel case.

# 7 Conclusion

This exercise has shown various ways to design a single transistor setting its bias current and choosing its width and length for reaching different specifications. It has also illustrated the role of parasitic capacitances and how to estimate them. All the designs have been succesfully checked with simulations using ngspice and the EKV 2.6 compact model.

## 8 References

[1]
C. Enz, “Common-source Open-loop Optimization.” https://github.com/chrisenz/Analog-Circuit-Design/blob/main/Amplifiers/Basic/CS%20OL%20Optimization/CS_OL_optimization.pdf, 2025.
[2]
C. C. Enz and E. A. Vittoz, Charge-Based MOS Transistor Modeling - The EKV Model for Low-Power and RF IC Design, 1st ed. John Wiley, 2006.