Cascode Stage

Analog (Integrated) Circuit Design

9 Cascode Stage

Cascode Stage

Figure 42: A MOSFET cascode circuit.

As we have seen in Section 8, the performance of the OTA is generally quite acceptable (see Table 2), but we might want to aim for better output voltage accuracy. As our analysis has shown the output voltage tolerance is limited by the open-loop dc gain \(A_0\) of the OTA (see Equation 14), which in turn is limited by the output conductance of \(M_2\) and \(M_4\) in Figure 32, which is also confirmed by the analytical result in Equation 18.

During the sizing procedure, we have seen that the achievable \(g_\mathrm{m}/ g_\mathrm{ds}\) ratio of a single MOSFET is limited, even if we increase \(L\). We are thus searching for a better option, and here (local) feedback in form of a cascode comes to help.

For analysis of a cascode, we use the following single-transistor stage shown in Figure 42.

Cascode Stage

Figure 43: The MOSFET cascode small-signal model.

In order to derive the operation of the cascode analytically, we draw the small-signal equivalent circuit in Figure 43. We assume that \(V_\mathrm{B}\) is a low-ohmic bias voltage, thus we replace it with ac ground. We further set \(g_\mathrm{mb}= 0\) (and ignore a couple of capacitances for now).

Cascode Stage

Figure 44: The simplified MOSFET cascode small-signal model.

Since the gate is assumed at a fixed potential, we can put \(C_\mathrm{gs}\) in parallel with \(G_\mathrm{S}\) as \(G_\mathrm{S}^{*} = G_\mathrm{S} + s C_\mathrm{gs}\), and we can put \(C_\mathrm{gd}\) in parallel with \(G_\mathrm{D}\) as \(G_\mathrm{D}^{*} = G_\mathrm{D} + s C_\mathrm{gd}\). As a result, we will disregard these capacitors for now, and just consider \(G_\mathrm{S}\) and \(G_\mathrm{D}\), which leads to the simplified small-signal circuit shown in Figure 44.

9.1 Cascode Output Impedance

Cascode Output Impedance

Figure 45: The simplified MOSFET cascode small-signal model for calculation of the output impedance.

As a first step, we want to calculate the output impedance at the drain of the MOSFET (i.e., looking into the drain). For this, we replace \(G_\mathrm{D}\) with a current source. The resulting small-signal equivalent circuit is shown in Figure 45.

Cascode Output Impedance

We realize that \(I_\mathrm{out}\) flows through \(G_\mathrm{S}\) (as there is no other way to go) and drops \(V_\mathrm{gs}\) (note the sign): \[ V_\mathrm{gs}= -\frac{I_\mathrm{out}}{G_\mathrm{S}} \]

Calculating KCL at the output node results in \[ I_\mathrm{out} - g_\mathrm{m}V_\mathrm{gs}- g_\mathrm{ds}V_\mathrm{ds}= 0. \]

Further, \(V_\mathrm{out} = -V_\mathrm{gs}+ V_\mathrm{ds}\).

Cascode Output Impedance

Using the previously found identities, and after a bit of algebraic manipulations we arrive at \[ g_\mathrm{out} = \frac{I_\mathrm{out}}{V_\mathrm{out}} = \frac{g_\mathrm{ds}}{1 + \frac{g_\mathrm{m}+ g_\mathrm{ds}}{G_\mathrm{S}}} = \frac{g_\mathrm{ds}\cdot G_\mathrm{S}}{G_\mathrm{S} + g_\mathrm{m}+ g_\mathrm{ds}} \tag{34}\]

We can further calculate the benefits of a cascode if we assume we put a cascode on top of a common-source transistor stage (thus \(G_\mathrm{S} = g_\mathrm{ds}'\)) and get \[ g_\mathrm{out} = \frac{g_\mathrm{ds}\cdot g_\mathrm{ds}'}{g_\mathrm{ds}' + g_\mathrm{m}+ g_\mathrm{ds}} \approx g_\mathrm{ds}' \frac{g_\mathrm{ds}}{g_\mathrm{m}} \tag{35}\]

We find that if \(G_\mathrm{S} = 0\) (\(R_\mathrm{S} \to \infty\)) then \(g_\mathrm{out} = 0\) (\(r_\mathrm{out} \to \infty\)), and if \(G_\mathrm{S} \to \infty\) (\(R_\mathrm{S} = 0\)) then \(g_\mathrm{out} = g_\mathrm{ds}\).

Cascode Output Impedance

Benefit of Cascode (Output)

The output impedance of the lower MOSFET (\(r_\mathrm{out} = 1 / g_\mathrm{ds}'\)) is increased by the self-gain (\(g_\mathrm{m}/ g_\mathrm{ds}\)) of the cascode transistor! This is a powerful technique to increase the output impedance of a transistor stage by cascoding, much better than increasing \(L\)!

9.2 Cascode Input Impedance

Cascode Input Impedance

Figure 46: The simplified MOSFET cascode small-signal model for calculation of the input impedance.

To calculate the input impedance of a cascode (i.e., looking into the source) we replace \(G_\mathrm{S}\) with a current source. The resulting small-signal equivalent circuit is shown in Figure 46.

Cascode Input Impedance

Formulating KCL at the input node results in \[ I_\mathrm{in} + g_\mathrm{ds}V_\mathrm{ds}+ g_\mathrm{m}V_\mathrm{gs}= 0. \]

After some manipulation we find that \[ g_\mathrm{in} = \frac{I_\mathrm{in}}{V_\mathrm{in}} = \frac{(g_\mathrm{m}+ g_\mathrm{ds}) \cdot G_\mathrm{D}}{g_\mathrm{ds}+ G_\mathrm{D}}. \tag{36}\]

We note that \(V_\mathrm{gs}= -V_\mathrm{in}\) and that \(I_\mathrm{in}\) flows through \(G_\mathrm{D}\) (again, there is no other way to go), resulting in \(V_\mathrm{D} = I_\mathrm{in} / G_\mathrm{D}\).

Note that \(V_\mathrm{ds}= V_\mathrm{D} - V_\mathrm{in}\).

Setting \(G_\mathrm{D} = 0\) (\(R_\mathrm{D} \to \infty\)) results in \(g_\mathrm{in} = 0\) (\(r_\mathrm{in} \to \infty\)) as well, so the input impedance of the cascode becomes very large when the drain impedance is large.

However, setting \(G_\mathrm{D} = \infty\) (\(R_\mathrm{D} = 0\)) results in the well-known result of

Cascode Input Impedance

\[ g_\mathrm{in} = g_\mathrm{m}+ g_\mathrm{ds}\approx g_\mathrm{m}, \]

which means that the input impedance looking into a cascode is \(r_\mathrm{in}\approx g_\mathrm{m}^{-1}\).

Cascode Input Impedance

Benefit of Cascode (Input)

This has the practical benefit that a capacitance connected at this node results in a high-frequency pole, which is often not critical in terms of stability. Further, the voltage swing at a cascode input node is small due to the often small impedance, and this minimizes the Miller effect at connected inter-node capacitors (see Section 17.1).

Cascode Input Impedance