Differential Pair

Analog (Integrated) Circuit Design

7 Differential Pair

Differential Pair

Figure 29: A differential pair.

Like the current mirror in Section 6, the differential pair is a ubiquitous building block often used in integrated circuit design (Blumlein 1937). The fundamental structure is given in Figure 29.

Differential Pair

1. a purely differential voltage, and
2. a common-mode voltage,

In order to understand its operation it is instructive to separate the input condition into

and see what the impact on the output currents is.

7.1 Differential Operation of the Diffpair

Differential Operation of the Diffpair

The differential input voltage \(V_\mathrm{in} = V_\mathrm{in,p} - V_\mathrm{in,n}\), so that \[ V_\mathrm{in,p} = V_\mathrm{CM} + \frac{V_\mathrm{in}}{2} \] and \[ V_\mathrm{in,n} = V_\mathrm{CM} - \frac{V_\mathrm{in}}{2}. \]

The output current on each side is then given by (neglecting \(g_\mathrm{ds}\) and \(g_\mathrm{mb}\) of \(M_1\) and \(M_2\)) \[ I_\mathrm{out,p} = g_\mathrm{m1} \left( \frac{V_\mathrm{in}}{2} \right) \] and \[ I_\mathrm{out,n} = g_\mathrm{m2} \left( -\frac{V_\mathrm{in}}{2} \right). \]

For a differential mode of operation we assume that the input common-mode voltage is constant, i.e., \((V_\mathrm{in,p} + V_\mathrm{in,n})/2 = V_\mathrm{CM}\).

For a small-signal differential drive the potential at the tail point stays constant and we can treat it as a virtual ground.

Differential Operation of the Diffpair

The differential output current \(I_\mathrm{out}\) is then given by \[ I_\mathrm{out} = I_\mathrm{out,p} - I_\mathrm{out,n} = g_\mathrm{m}V_\mathrm{in} \tag{12}\]

Usually we assume symmetry in the differential pair, so \(g_\mathrm{m1} = g_\mathrm{m2} = g_\mathrm{m}\).

We see in Equation 12 that the differential output current is simply the differential input voltage multiplied by the \(g_\mathrm{m}\) of the individual transistor. We also note that the bottom conductance \(g_\mathrm{tail}\) plays no role for the small-signal differential operation.

7.2 Common-Mode Operation of the Diffpair

Common-Mode Operation of the Diffpair

If this is the case, then the output currents are not a function of the common-mode input voltage (\(I_\mathrm{tail}\) is set by the tail current source), and \[ I_\mathrm{out,p} = I_\mathrm{out,n} = I_\mathrm{out} = \frac{I_\mathrm{tail}}{2}. \]

Usually, the source conductance \(g_\mathrm{tail}\) is realized by a current source and ideally should be \(g_\mathrm{tail} = 0\).

However, if we assume a realistic tail current source then \(g_\mathrm{tail} > 0\). For analysis we can simply look at a half circuit since the circuit operation is symmetric. In order to simplify the analysis a bit we remove all capacitors from the MOSFET small-signal model and set \(g_\mathrm{ds}= g_\mathrm{mb}= 0\). We then arrive at the small-signal equivalent circuit shown in Figure 30 (note that we set \(V_\mathrm{in,p} = V_\mathrm{in,n} = V_\mathrm{in}\) and \(I_\mathrm{out,p} = I_\mathrm{out,n} = I_\mathrm{out}\) under symmetry considerations). As both transistors inject their drain current into the shared tail node, the half circuit sees an effective tail conductance of \(g_\mathrm{tail}/2\).

Common-Mode Operation of the Diffpair

Figure 30: Small-signal model of the differential pair half-circuit in common-mode operation.

Common-Mode Operation of the Diffpair

Formulating KVL for the input-side loop we get \[ V_\mathrm{in} = V_\mathrm{gs}+ \frac{2 I_\mathrm{d}}{g_\mathrm{tail}}. \]

With \(I_\mathrm{out} = I_\mathrm{d}= g_\mathrm{m}V_\mathrm{gs}\) we arrive at \[ I_\mathrm{out} = \frac{g_\mathrm{m}\cdot g_\mathrm{tail}}{2 g_\mathrm{m}+ g_\mathrm{tail}} V_\mathrm{in} \tag{13}\]

Interpreting Equation 13 we can distinguish the following extreme cases:

Common-Mode Operation of the Diffpair

1. If \(g_\mathrm{tail} = 0\) (ideal tail current source) then \(I_\mathrm{out} = 0\), the common-mode voltage variation from the input is suppressed and does not show up at the common-mode output current (which is constant due to the ideal tail current source). This is usually the case that we want to achieve.

Common-Mode Operation of the Diffpair

2. If \(g_\mathrm{tail} \rightarrow \infty\) then \(I_\mathrm{out} = g_\mathrm{m}V_\mathrm{in}\), which means the output current is a function of the MOSFET \(g_\mathrm{m}\). If everything is perfectly matched, then the differential output current is zero, but the common-mode output current changes according to the common-mode input voltage. In special cases this can be a wanted behavior, this configuration is called a “pseudo-differential pair.”
3. For the typical case that \(g_\mathrm{m}\gg g_\mathrm{tail}\) then \(I_\mathrm{out} \approx (g_\mathrm{tail}/2) V_\mathrm{in}\).

Note that Equation 13 is closely related to the general case of a degenerated common-source transistor stage (see Figure 31). The effective transconductance \(g_\mathrm{m}'\) of a degenerated CS stage is given by

Common-Mode Operation of the Diffpair

\[ g_\mathrm{m}' = \frac{I_\mathrm{out}}{V_\mathrm{in}} = \frac{1}{g_\mathrm{m}^{-1} + R_\mathrm{degen}}. \]

For the differential-pair half circuit in common-mode operation, the effective source degeneration is \(R_\mathrm{degen} = 2/g_\mathrm{tail}\), yielding Equation 13.

When no degeneration resistor is used, then \(g_\mathrm{m}' = g_\mathrm{m}\). If a degeneration of \(R_\mathrm{degen} \gg g_\mathrm{m}^{-1}\) is used then \(g_\mathrm{m}' = 1 / R_\mathrm{degen}\). This result is worth memorizing.

Common-Mode Operation of the Diffpair

Figure 31: A MOSFET common-source amplifier with resistive degeneration.

References

Blumlein, A. D. 1937. Thermionic Valve Amplifying Circuit. U.S. Patent 2,185,367.

References