A matched NMOS source-coupled pair M_1,M_2 is biased by a tail current source…

A matched NMOS source-coupled pair \(M_1,M_2\) is biased by a tail current source \(I_{SS}=400\,\mu\mathrm{A}\) in parallel with a finite small-signal resistance \(R_{SS}=50.0\,\mathrm{k}\Omega\) from the common source node to ground. Each drain is loaded by \(R_D=6.00\,\mathrm{k}\Omega\) to \(V_{DD}=3.30\,\mathrm{V}\). The saturation law is \(i_D=k_n(v_{GS}-V_{tn})^2\) with \(k_n=\tfrac12\mu_n C_{\mathrm{ox}}(W/L)\), \(\mu_n C_{\mathrm{ox}}=180\,\mu\mathrm{A}/\mathrm{V}^2\), \(W/L=25.0\), \(V_{tn}=0.600\,\mathrm{V}\), and \(\lambda=0\). Substrates of \(M_1\) and \(M_2\) are tied to the common source. Define \[ v_{id}=v_{G1}-v_{G2},\qquad v_{icm}=\tfrac12(v_{G1}+v_{G2}),\qquad v_{od}=v_{D1}-v_{D2}. \] Use the standard MOS small-signal model \(i_d=g_m v_{gs}\) with \(g_m\) evaluated at \(I_D=I_{SS}/2\). Ignore \(r_o\) of \(M_1\) and \(M_2\). (a) Draw and label the differential half-circuit and the common-mode half-circuit. State the value of the source degeneration seen in each half-circuit. (b) Derive and evaluate the differential voltage gain \(A_d=v_{od}/v_{id}\) and the single-ended common-mode gain \(A_{cm,\mathrm{se}}=v_{o1}/v_{icm}\). Then form the single-ended common-mode rejection ratio \(\mathrm{CMRR_{se}}=\lvert A_d/2\rvert/\lvert A_{cm,\mathrm{se}}\rvert\) and the fully differential CMRR \(\lvert A_d\rvert/\lvert A_{cm}\rvert\) where \(A_{cm}=v_{od}/v_{icm}\) for matched loads. Report CMRR as a dimensionless ratio and in decibels. (c) Explain, without numerical guesswork, why a perfectly matched pair with matched drain resistors has \(A_{cm}=v_{od}/v_{icm}=0\) even when \(R_{SS}\) is finite, and identify the single element mismatch that would produce a nonzero \(v_{od}\) under a pure common-mode input. Do not introduce a numerical mismatch in this part.

Odd-mode symmetry kills the tail node; even-mode symmetry doubles the tail resistor in the half-circuit.