8.3. Signal Voltage Gain of the Ideal Differential Amplifier Stage

This initial assessment of the gain of the differential amplifier neglects the effects of the bias resistor, R_bias, and of the output resistance of the transistors as well as the body effect. These effects are considered below. To generalize the discussion to include the possibility of inverting and noninverting modes of operation, we base the discussion on the circuit (Fig. 8.2) with a drain resistor for both transistors. The inverting and noninverting modes take, respectively, the outputs at V_d1 = V_o1 (a_v1) and V_d2 = V_o2 (a_v2). Also, this is the configuration that is studied in the project on the differential amplifier. Note that the input could arbitrarily be moved to the gate of M₂, in which case the output taken at V_d1 becomes the noninverting mode, and so forth.

Assume for the present that the amplifier is perfectly symmetrical such that, for example, I_D1 = I_D2 and g_m1 = g_m2. A signal voltage V_i is applied to the gate of M₁ as shown in the diagram. The voltage divides between the two transistors (from gate 1 to gate 2) to give

Equation 8.11

Since V_gs1 = |V_sg2| (due to the symmetry of this case), then

Equation 8.12

The negative V_gs2 results from the fact that the total v_GS2 is reduced from bias V_GS2 upon the application of a positive signal voltage, V_i, at the gate of M₁ (i.e., v_GS2 = V_GS2 – V_i/2). It follows that total i_D2 decreases, which leads to a positive signal voltage at the drain, V_d2. Thus, the amplification is noninverting.

As always, the relation between the signal drain current and the signal drain-source voltage is (4.1), which is (with g_ds = g_mb = 0)

I_d = g_mV_gs

Eliminating V_gs in (4.1), applied to M₂, with (8.12), the drain current for M₂ is related to the input voltage, V_i, by

Equation 8.13

Combining (8.13) with V_o2 = –I_d2R_D2, the output voltage is related to the input voltage by

Equation 8.14

such that the gain is

Equation 8.15

Note that there is no minus in the relation; that is, this is the noninverting amplifier mode. Also note that the gain for this amplifier is one-half of that for the common-source amplifier. This is a reasonable price to pay for not having a bypass capacitor. The amplification can be viewed as a source-follower stage in cascade with a common-base stage, neither of which is an inverting stage.

The inverting mode of the amplifier is for the output taken at the drain of M₁. For this case, V_o1 = –I_d1R_D1, and from (8.12) and (4.1) applied to M₁, I_d1 = g_m1V_g/2 and the voltage relationship is

Equation 8.16

This leads to the inverting gain, which is

Equation 8.17