Noise analysis in differential mode circuits

The use of differential mode including a differential pair is probably the most important circuit topology in IC. However, people, particulary me, often are confused differential mode circuit noise analysis with its signal analysis. For example, I typically use Half-circuit concept to analyze a differential pair signal analysis (i.e., signal transfer function) for convenience. This half-circuit concept greatly reduce amount of work that I have to undergo to analyze the differential circuit otherwise.

The prime reason that this half-circuit concept is possible is due to the circuit's symmetricity. In order for the circuit to be perfect symmetric, left- and right- side of the circuit components should have the same components and values - ideally no random variation. Another requirement is the input should be 180 out of phase each other so that the drain voltage of the tail current source should be "virtual ground"! (Vin1 and Vin2 are 180 out of phase - same magnitude, but opposite sign!) The input relationship shows that both Vin1 and Vin2 are 100% correlated! This fact should be paid special attention because this fact is important discrimination in noise analysis.

Anyway, in signal analysis, the half-circuit concept is very useful and it is mere double the differential output voltage with differential input voltage. (or, Vout,dm/2 = (......)Vin,dm/2 ---> Thus, Vout,dm = 2(......)Vin,dm).

The noise analysis of the differential pair should be treated differently from the signal analysis case. Since noise sources are typically uncorrelated in the differential pair - all the noise sources in the differential pair (i.e., drain current noise, load resistance noise...etc), the virtual gound concept cannot be used in noise analysis. This fact, the drain voltage of the tail current source is no longer virtual ground, makes it impossible to use half-circuit concept in noise analysis in the differential pair. Thus, we simply derive the noise effect of each source individually! If you still want to use half-circuit concept, which is inappropriate in exact sense, then you should use it with special attentions!!!!


The following is a fine example that I had for midterm exam - yr 2006, fall.

Let's fine the output noise power!
Can we use half circuit concept with special care? Um...yes....
To be exact, I should not use half-circuit concept, which Vin is now shorted and circuit gets divided a half exactly. Then, the noise sources in the half-circuit concept are Vn, (VR1)/2 from R1 thermal noise and VR2.
Notice Vn and R2 are noise sources, which do not have any correlation with the other half-circuit. Thus, the noise power will be added! Due to symmetricity, the noise power will be doubled! (Assumption: noise is additive) However, (VR1)/2 is used for half-circuit concept. This is the part we should give special attention to. Since one noise voltage source has been divided into two for half-circuit concept, the noise voltage, (VR1)/2, is correlated to the other half-circuit. Thus, now in (VR1)/2 perspective, the half-circuit can be treated just like signal analysis case - virtual ground!. Thus, at the output, the noise voltage is doubled in differential mode consideration! This mean in noise power calculation, the noise power due to this term will be quadrapled!!!!
Therefore, the answer is ....

E[Vn,out^2] = A^2{2E[VR2^2] + 2(1+2(R2/R1)^2)E[Vn^2] + 4((R2/R1)^2)E[VR1^2]}
the orange colored number "2" is due to uncorrelated noise sources, in psd, they are added.
the red colored numer "4" indicates that there was correlation in noise source in half-circuit, thus it was treated as signal half-circuit case. Thus, its noise power are quadrapled!

That's all, folks!!!