$2\mathrm{.}8$ Nonideal $($i$.$e$.,$ real$)$ operational amplifiers respond to both the differential and common$-$mode components of their input signals $($refer to Fig. $2\mathrm{.}4$ for signal representation$).$ Thus the output voltage of the op amp can be expressed as

$v_o=A_d{v}_{ld}+A_{cm}{v}_{\text{l}\text{c}\text{m}}$

where $A_d$ is the differential gain $($referred to simply as $A$ in the text$)$ and $A_{cm}$ is the common$-$mode gain $($assumed to be zero in the text$).$ The op amp's effectiveness in rejecting common$-$mode signals is measured by its CMRR$,$ defined as

$CMRR=20log|\frac{A_d}{A_{cm}}|$

Consider an op amp whose internal structure is of the type shown in Fig. E$2\mathrm{.}3$ except for a mismatch $G_m$ between the transconductances of the two channels; that is$,$

$G_{m1}=G_m-\frac{1}{2}{G}_m$

$G_{m2}=G_m+\frac{1}{2}{G}_m$

Find expressions for $A_d,A_{cm},$ and CMRR$.$ What is the maximum permitted percentage mismatch between the two $G_m$ values if a minimum CMRR of $60$ dB is required?