Q.If common mode gain and differential mode gain for a differential amplifier are 0.1 & 500 respectively, find CMRR in dB.

An op-amp is meant to amplify only the difference between its two inputs. Any signal appearing equally on both inputs — noise, supply hum, temperature drift — is a common-mode signal and should ideally produce no output. The Common Mode Rejection Ratio (CMRR) quantifies how well a real device achieves this.

CMRR is defined as the ratio of the differential-mode voltage gain $A_d$ to the common-mode voltage gain $A_c$:

Because this ratio is often very large, it is normally quoted in decibels:

A high CMRR means the wanted difference signal is amplified far more strongly than any common-mode interference, so the amplifier stays accurate even in a noisy environment. For an ideal op-amp $A_c=0$, giving infinite CMRR; a practical device such as the IC 741 has a CMRR of roughly 90 dB.

For example, if a differential amplifier has $A_d=500$ and $A_c=0.1$, then $\text{CMRR}=500/0.1=5000$, which in decibels is $20\log_{10}(5000)\approx74\ \text{dB}$. The larger this figure, the better the amplifier discriminates a real difference signal from unwanted common-mode disturbances — a key quality metric when selecting an op-amp for instrumentation.