14.2. Current-Source Output Resistance and Common-Mode Gain

As discussed in Unit 8, the common-mode gain is the gain for the same signal being applied to both inputs. In Unit 8.6 the common-mode voltage gain expression, (8.26), was obtained for the bias resistor, R_bias. This is

To apply the common-mode gain equation to the circuit of Fig. 14.1, resistor R_bias is replaced by r_ds11. The result for the common-mode gain is

Equation 14.2

The approximate form is consistent with neglecting the change of the gate – source voltage compared to the change of voltage at the drain of the current-source transistor, M₁₁. Intuitively, (14.2) indicates that for r_ds11 → ∞, transistor M₁₁, is a pure current source, and i_D1 = i_D2 = I_D1 = I_D2 regardless of the magnitude of common voltage applied to both gates. In this case, signals I_d1 = I_d2 = 0 for any (realistic) common-mode input signal.

Common-mode inputs can be a form of noise, and therefore the ideal opamp would reject these signals entirely. An important consideration is the extent of rejection based on the common-mode gain relative to the differential amplifier gain. This is quantitatively assessed with the common-mode rejection ratio, which is the ratio of the differential amplifier gain with differential output divided by the common-mode gain, (14.2). The gain for this case was obtained in Unit 8.5 as (8.24), which is a_vd12 = –g_mR_D.

Using (8.24) and (14.2), the common-mode rejection ratio is

Equation 14.3

With g_m1 = 2I_D1/V_effn1 [(4.5)], and for the output resistance of M₁₁, r_ds11 ≈ 1/λ_n2I_D1 [(4.15)], the common-mode rejection ratio is

Equation 14.4

By comparison, for resistor bias, the equivalent result is

Equation 14.5

where R_bias is the bias resistor of, for example, the circuits of Figs. 13.1 and 13.2.