4.3. Transconductance Parameter

The transconductance parameter, g_m, was introduced in Unit 2 in the treatment on the rudimentary electronic amplifier; it is the proportionality constant of the linear relationship between the output (responding) current and the input (control) voltage [(2.3)]. For the MOSFET, NMOS, or PMOS, I_d = g_mV_gs, where I_d is into the drain for both transistor types. An ideal transistor can be modeled with this alone. A simple model, which includes no other components, would often be adequate for making estimates of circuit performance.

To obtain an expression for g_m as a function of the general form i_D = f(v_GS, v_DS, v_SB) [e.g., (3.8)], we use the definition [from (4.1)]

Equation 4.2

Using (3.8) to express i_D, the resulting relation for g_m is

Equation 4.3

where [(3.7)] and V_effn = V_GS – V_tno. Note that the use of V_DS is consistent with the partial derivative taken with respect to v_GS, that is, V_ds = 0. Also, the use of V_tno implies that v_SB = 0. In general, V_SB could be nonzero, although in the definition of g_m, V_sb must be zero. For the case of nonzero V_SB (bias), one substitutes for V_tno a constant V_tn(V_SB) in the g_m expression.

Alternative forms for the g_m expression can be obtained from (3.8), which is, solving for V_effn,

Equation 4.4

Using (3.8), (4.3), and (4.4), g_m takes on altogether three forms:

Equation 4.5

Usually, in initial design, k_n replaces to eliminate the V_DS dependence without a serious penalty in accuracy.

Using the simple linear transistor model, an expression for the circuit transconductance, G_m, for the circuit of Fig. 4.1 will now be obtained. The input loop equation for an applied gate signal voltage, V_g, is

Equation 4.6

which is, with V_gs = I_d/g_m,

Equation 4.7

and

Equation 4.8

The far right-hand side uses (4.5). For example, for a 1-V drop across R_S and V_effn = 0.5 V, G_m = g_m/5. Note that the ratio of the signal voltage drop across R_S and signal voltage V_gs is g_mR_S:1. The G_m concept is utilized routinely in MOSFET circuits (and BJT circuits), which gives the effective reduced transconductance, referred to V_g, in the presence of the source resistor.