5.2. Amplifier Voltage Gain

This dc (bias) circuit becomes an amplifier now simply by adding a signal source at the gate as in Fig. 5.2. This requires a coupling capacitor, as shown here in the complete circuit, to prevent disturbing the bias upon connecting the input signal to the circuit.

In the amplifier of Project 5, the signal will be superimposed on the bias voltage at the node of V_GG. This can be facilitated with LabVIEW and the DAQ. A capacitor, as in an actual amplifier, is therefore not required. The requirement for having LabVIEW control over both V_GG and V_SS, and the limitation of two output channels, dictates this configuration.

In Project 5 we measure the gain as a function of bias current, I_D. For a SPICE comparison, we need an expression for the gain. For the ideal case, which neglects the output conductance, g_ds, the output current is related to the input voltage by (4.1), which is

I_d = g_mV_gs = g_mV_i

The output signal voltage is, in general,

Equation 5.1

The convention used here for subscript order for signal (linear) variables is common to the NMOS and PMOS. This is consistent with the fact that the linear model does not distinguish between the two types. Thus, for example, the dc terminal voltage for a PMOS is V_SG, but the signal equivalent is V_gs (Fig. 5.3) and the signal input voltage is positive at the input terminal (common-source, gate input). For the PMOS, i_D is defined as positive out of the drain, but the signal output current is into the drain (as in the NMOS). We note that a positive V_gs (V_gs = –V_sg) corresponds to a decrease in the total gate – source voltage, v_SG, which is consistent with a decrease of i_D and positive I_d.

Thus, the negative sign in (5.1) is consistent with the flow of current I_d up through the resistor (Fig. 5.3) for positive V_i = V_gs. The common-source stage is an inverting amplifier and has an inherent 180° phase shift. From (4.1) and (5.1), the gain is

Equation 5.2

where both V_i = V_gs and V_o = V_ds are with respect to ground or the source terminal for the common-source stage.

If the output resistance, 1/g_ds, cannot be neglected (which is the case for the project on PMOS amplifiers), the transistor current, g_mV_i, is shared between the output resistance and R_D. The portion that flows through R_D is (Fig. 5.4)

Equation 5.3

Note again that the signal schematic transistor represents a current source with value g_mV_i, as established in connection with Fig. 4.1. The additional feature of the transistor model is included with the addition of 1/g_ds. This resistance is actually part of the transistor and is between the drain and source of the transistor, but the circuit as given is equivalent, as the source is at ground. Since the output voltage is V_o = –I_RDR_D, the new gain result is

Equation 5.4

Note that this form evolves from ideal transistor current, g_mV_gs, flowing through the parallel combination of the output resistance and R_D.

To facilitate an intuitive grasp of the magnitude of the effect of g_ds, we use the expression for g_ds (4.13) in (5.4), to obtain

Equation 5.5

Note that I_DR_D is the voltage drop across R_D. For example, for a –10-V power supply, we choose I_DR_D ≈ 5 V. A measurement of λ_p for our devices will show that λ_p ≈ 1/20 V, which results in λ_pI_DR_D ≈ 1/4. Thus, the effect of g_ds (= λ_pI_D) for this case is significant.

Finally, we can get an overall current dependence for a_v with the elimination of g_m, using(4.5) with ≈ k_p, which results in

Equation 5.6

Using an alternative form for g_m (= 2I_D/V_effp), also (4.5), the gain expression is

Equation 5.7

where

For simplicity, approximate forms of (4.5) and (4.13) of g_m and g_ds are used here, which are independent of V_SD. For reference, the “exact” and approximate forms of (4.5) and (4.13), respectively, are repeated here:

and

The “exact” equations of g_m and g_ds are used in conjunction with the amplifier projects to compare the computed gain with the measured gain plotted against I_D. This is done in both LabVIEW and Mathcad. Parameters k_p and V_tpo (to get V_effp) will be extracted from the measured dc data, and λ_p will be used as an adjustable parameter to fit the SPICE and measured gain data.