4.5. Output Conductance Parameter

The output conductance accounts for the finite slope of the output characteristic, an example of which is shown in Fig. 4.5. The plot for the SPICE formulation [(3.8)] of the active region is also shown (applicable to the device for V_DS > V_effn). Both plots are for V_GS = constant (i.e., V_gs = 0 and g_mV_gs = 0).

In the ideal case, for any v_DS in the active region, the current is the same for a given V_GS, and the drain current is simply I_d = g_mV_gs. The real case, though, obviously possesses a current dependence on v_DS. The linear model treats g_m as a constant (calculated at the bias values) and includes the effect of the nonzero slope with the output conductance, g_ds, such that I_d = g_mV_gs + g_dsV_ds (without or neglecting the body effect). The circuit shown in Fig. 4.6 includes the g_ds component.

An expression for the output conductance is obtained from the definition

Equation 4.12

Again using (3.8), the result is

Equation 4.13

where V_effn = V_GS – V_tno is constant and where I_D is the dc (bias) current. The last term (on the right-hand side) is the form that is generally used in practice for an initial design, as it does not require a value for V_DS. Note that g_ds is not a conductance in the physical sense but has the correct dimensions and behaves in the circuit like a conductance.

We will now obtain the circuit transconductance for the case where the effect of g_ds is included. Upon application of an input signal, V_g, a signal drain current, I_d, will flow in the output circuit. This causes a signal voltage to appear across R_S and R_D, which is equal to the voltage to V_ds. That is,

Equation 4.14

The associated current through the output resistance is thus

Equation 4.15

The effect is to reduce the current through R_S and R_D and thus reduce the circuit transconductance of the common-source amplifier stage. The output current with the current of (4.15) subtracted from the basic g_mV_gs is

Equation 4.16

which is, when solved for V_gs,

Equation 4.17

Again using (4.6), which is V_g = V_gs + I_dR_S, the new circuit transconductance is

Equation 4.18

With a 1 - V drop across R_S and a 5 - V drop across R_D, and with λ_n = 1/50 V, the new G_m is g_m/5.12, compared with g_m/5 when neglecting g_ds [(4.8)].

In general, the complete circuit includes, in addition, the body-effect transconductance current source of Figs. 4.3 and 4.4. The omission of this current source in Fig. 4.6 implies that V_sb = 0 because the source and body are connected. This connection is possible to implement in special cases such as in some of our MOSFET lab projects where only one transistor on the chip is used or for the case of a differential stage where the source of two transistors is at the same node. It is also possible to eliminate the body-effect current source by bypassing the source resistor with a bypass capacitor. The capacitor places the source at signal ground. However, in this case, the dc threshold voltage is still affected by V_SB = I_DR_S.

In Unit 8, the circuit transconductance equivalent to (4.8) and (4.18), but which includes the body effect [(8.49)], is given as

where η is defined in (4.11). For g_ds = 0 and η = 0, (4.19) reduces to (4.8). Note that including the body effect will in general have more effect on G_m than including g_ds, as η can be on the order of 0.2. In all cases where we can calculate the circuit transconductance, G_m, the magnitude of the voltage gain is obtainable from –G_mR_D (common-source stage) and G_mR_S(source-follower stage).