MOSFET Small Signal Modeling

This problem assumes a MOSFET circuit to the right, with the following parameters: Ith = 125nA, K = 80 uA/V2, VT = 0.6V, VA = 25V.  The drain voltage is biased at 2.5V; assume the MOSFET is in saturation throughout this problem.  Also assume k = 1.

(a) What is the minimum drain voltage bias required to keep this MOSFET in saturation as a function of gate voltage? This should be for both subthreshold and above-threshold operation.

 In subthreshold, minimum drain voltage is 100mV. 

In above threshold, minimum drain voltage is V_g - V_T

(b)  Compute and draw the small signal model of this MOSFET circuit (not including the capacitors yet).   Evaluate the transconductance at a bias current of 100nA and 10mA. 

I_th = 125nA, so the 100nA bias is for subthreshold biases.

g_m = I / U_T in subthreshold, and sqrt( 2 K I) in above threshold.
g_m (100nA) = 4mA/V, g_m (10 mA) = 40 mA/V,
No current (therefore an infinite resistance) going into the gate terminal).

r_o = V_A / I
r_o (100nA) = 250MHm, r_o (10 mA) = 2.5MOhm

(c)  Qualitatively, how would this model change for a BJT device instead of a MOSFET transistor (assume the collector replaces the drain and the base replaces the gate).

The model is identical to the subthreshold MOSFET model, except with the inclusion of the input resistance looking into the base (we have no current going into the gate).

(d) Solve for the expression for the maximum gain from this device?  Evaluate again at bias currents of 100nA and 10mA.

A_V = I / U_T in subthreshold.
A_v (100nA) = 1000, A_v (10 mA) = g_m (10 mA) r_o (10 mA) = 100

(e) Solve for the expression of unity gain frequency when the output is driving a 2fF load.   Plot the log of unity-gain frequency (f_T) versus log of channel current. Extra credit: Add an additional curve to the graph illustrating the effects of velocity saturation.

f_T = g_m / (2 p 2fF)
f_T (100nA) = 4mA/V / (6.28 * 2fF) = 318.47MHz
f_T (10 mA) = 40 mA/V / (6.28 * 2fF) = 3.1847GHz

Velocity saturation will degrade the g_m in above threshold operation, but not in subthreshold operation. After a certain gate overdrive, g_m becomes independent of current, and then for even larger increases, g_m decreases with increasing current.

(f) Calculate the power consumed at a bias current of 100nA and 10mA.

Power at 100nA bias current: 100nA * 2.5V = 0.25 mW
Power at 10mA bias current: 100mA * 2.5V = 25 mW