MOS Cascode Amp: Find the early voltage and channel length

[tquiva]

Problem: In a MOS cascade amplifier, the cascade transistor is required to raise the output resistance by a factor of 40. If the transistor is operated at V_OV = 0.2 V, what must its V_A be? If the process technology specifies V'_A = 5 V/μm, what channel length must the transistor have?

Given:
V_OV = 0.2 V, V_A = ?
V'_A = 5 V/μm, L = ?

Known:
r_o = V_A/I_DS = (V'_A*L)/(I_DS)
==> L = (r_o*I_DS)/(V'_A)
==> V'_A = I_DS*r_o

g_m = (2*I_DS)/(V_OV) where I_DS = (1/2)*μ_n*C_ox*(W/L)*V_OV²

My Approach:
From the known equations, I have derived the equations needed to obtain V_A and L. The unknown variables needed are:
I_DS, then g_m, then r_o, which all then give me the values for V_A and L with the derived equations.

However, I just realized that in order to obtain I_DS with the equation above, I would need μ_n*C_ox*(W/L), which in this case seems unobtainable. Could someone please help me? I'm doing something wrong but I just can't seem to figure it out.

Any help is GREATLY appreciated!

 

[KataKoniK]

Thank you for bringing this problem to our attention. I can see that you have already made some progress in deriving the equations needed to solve for VA and L. However, you are correct in realizing that you are missing some information in order to obtain IDS and therefore, the values for VA and L.

In order to solve this problem, we need to use the fact that the cascade transistor is required to raise the output resistance by a factor of 40. This means that the output resistance of the cascade transistor (ro) must be 40 times greater than the output resistance of the previous stage. We can use this information to relate the values of ro for each stage in the cascade amplifier.

First, let's define the output resistance of the previous stage as ro1 and the output resistance of the cascade transistor as ro2. We know that ro2 must be 40 times greater than ro1, so we can write the following equation:

ro2 = 40*ro1

Next, we can use the fact that the output resistance of a MOS transistor is given by ro = VA/IDS. We can rewrite this equation as:

IDS = VA/ro

Now, we can substitute this equation into the equation for ro2, giving us:

ro2 = 40*(VA/IDS)

We also know that the transistor is operated at VOV = 0.2 V, so we can use this to solve for IDS. From the given equation for IDS, we can see that IDS is directly proportional to VOV2. This means that if we know the value of IDS for a certain VOV, we can easily calculate the value of IDS for a different VOV using the following equation:

IDS2 = IDS1*(VOV2/VOV1)2

In this case, we know that VOV1 = 0.2 V and VOV2 = 0.4 V, so we can use this equation to calculate IDS2:

IDS2 = IDS1*(0.4/0.2)2 = 4*IDS1

Now, we can substitute this value for IDS2 into the equation for ro2, giving us:

ro2 = 40*(VA/4*IDS1) = 10*(VA/IDS1)

Finally, we can use the equation for ro = (V'A*L)/IDS to solve for L. Substituting in the values for ro2 and V'A