Impedance of a Degenerated Circuit

I had to review the series and parallel RLC circuits for my RFIC class. In typical analog IC regime, I rarely deal with L so that I didn't have a chance to study it, but in RFIC I have one more component, L. Is this bless or curse? Well, of course, it depends on how to use it. Since I have one more degree of freedom to tweek a knob so that there is more freedom to design circuits. However, I have one more degree of freedom that make me consider many other consequences to consider.

Today, I would like to discuss about a source load degenerated circuit. This is a particularly important in a LNA design because inductively degenerated LNA gives possibly good noise figure while this provides good input matching as well. The side effect of this, a good one, is that it also boost Gm (transconductance) so that it helps the LNA gain. Let's see why it give me that!

If you derive the input impedance equation, it will look like above. One interesting term is (gm*Z)/(jwCgs). Depending on Z, it can be resistance, capacitance, and negative resistance.

The reason is that if you consider the term's magnitude, (gm*Z)/(wCgs). However, it has 1/j multiplied. What does 1/j term do? This term rotates (gm*Z)/(wCgs) by-90 degree shift in phase diagram (rotate by 90 degree in counterclockwise) - Re and Im for x-axis and y-axis, respectively.

Keep in mind what 1/j term does.

Let Z = jwL (inductor),

The last term will be (gm*L)/Cgs - resistance!, but it is NOT a real physical resistor so that it does not generate any thermal noise! This is a great advantage! If you make L such that it gives you the right resistor for good input matching at resonant frequency, it is good for noise and matching - you threw a stone, and caught two birds!! Is that all? Don't be so greedy~ Well, there is one more advantage....since this is series RLC circuit, voltage across L and C will be boosted by Q times - will be explained in detail later.

In the same manner, you can substitute Z = 1/(jwC) and Z = R.

The last term, then, will give you capactance and negative resistance, respectivley.

Why? Since 1/jw term give you -90 degree rotation ( couterclockwise 90 degree rotation),

Z=jwL (postive part of the y-axis) --> resistance

Z=R (postive part of the x-axis) --> capacitance

Z=1/(jwC) (negative part of the y-axis) --> negative resistance

Hope this is not very confusing any more...

Now, let's review how this also boost gm by Q times --> Gm = Q*gm! (gm: transistor transconductance and Gm: system transconductance)

From series RLC circuit analysis, we know that at resonant frequency the voltage across the capacitor and the inductor is Q times larger than what is applied. Thus, the voltage across Cgs is boosted by Q time at resonant frequency.

Remember Q boosting and input matching work at resonant frequency (or its neighbor frequencies)

That's all for today!