A differential amplifier is a circuit that amplifies the difference
between two input signals while rejecting any common mode signal. One
common implementation of a differential amplifier is the Bipolar
Junction Transistor (BJT) differential amplifier. Differential amplifier are used primary building block of operational amplifier.

CMMR stands for Common Mode Rejection Ratio, and it is a measure of how well a differential amplifier rejects common-mode signals. Common-mode signals are signals that are present on both inputs of a differential amplifier, such as noise or interference picked up from the environment.

Here, we will discuss how to improve CMRR of basic BJT differential amplifier.

**Basic BJT Differential Amplifier Circuit Diagram**

The following shows the circuit diagram of dual input balanced output BJT differential amplifier.

In
the circuit, Q1 and Q2 are the two BJTs that are connected in a
differential configuration. The input signals, V1 and V2, are applied to
the base terminals of Q1 and Q2, respectively. The current bias is
provided by the emitter resistance RE, which is connected to the
emitters of both BJTs. The output signal, Vout, is taken between the
collector of Q2 and Q1 and the differential output is proportional to
the difference between V1 and V2. The operation of this dual input
balanced output is explained in details in the tutorial Basic BJT Differential Amplifier Construction and Analysis.

### CMRR of BJT differential amplifier

The CMMR of a differential amplifier is defined as the ratio of the differential mode gain to the common mode gain. The differential mode gain is the gain of the amplifier for differential signals applied to the inputs, while the common mode gain is the gain of the amplifier for signals that are present on both inputs.

\( CMRR = \frac{A_d}{A_c} \) ---->(1)

The CMMR is typically expressed in decibels (dB) and is calculated as follows:

\(CMMR (dB) = 20 log \frac{A_d}{A_c} \)

where, \(A_d\) is the differential mode gain and \(A_c\) is the common mode gain.

A high CMMR is desirable in a differential amplifier because it means that the amplifier can amplify the desired differential signal while rejecting any common-mode noise or interference.

For example, if a differential amplifier has a differential mode gain of 100 and a common mode gain of 1, the CMMR would be:

CMMR (dB) = 20 log (100 / 1) = 40 dB

This means that the amplifier can reject common-mode signals up to 40 dB below the differential signal. In practice, a CMMR of at least 60 dB is typically desired for most applications.

#### Framing the Problems to improve CMMR

Here we will discuss the problem to improving CMMR and its solutions.

For the dual input balanced output differential amplifier the differential mode gain and the common mode gain are derived as follows.

\(A_d = \frac{-h_{fe}Rc}{h_{ie}+R_s}\) --------->(2)

where, Rs = R1 = R2, the base input resistance

\(A_c = \frac{-h_{fe}R_c}{R_s+h_{ie}+2R_E(1+h_{fe})} \) ---------->(3)

From equation(1) we can conclude that the CMRR is improved if we increase the differential gain \(A_d\) and/or if we decrease the common mode gain \(A_c\). From equation(2), we can see that \(A_d\) does not depend on the emitter resistance \(R_E\). So there is no negative feedback in case of DIBO differential amplifier. However, the common mode gain depends on the emitter resistor value \(R_E\) and this resistance introduces negative feedback. The negative feedback magnitude is directly proportional to the emitter resistance, higher emitter resistance higher negative feedback and vice versa. Higher negative feedback means less common mode gain. Less common mode means more CMRR.

Similarly, we have to understand the effect of the emitter resistance \(R_E\) on the biasing Q-point. For the above BJT differential amplifier the Q-point collector current and the collector to emitter voltage are:

\(I_{CQ} = \frac{V_{EE}-V_{BE}}{2R_E}\) ------->(4)

and, \(V_{CEQ}=V_{CC}+V_{BE}-I_{CQ}R_C\) ------->(5)

As can be seen from(4) and (5), the biasing Q-point \(I_{CQ}\) and \(V_{CEQ}\) depends on the emitter resistance , \(R_E\). That is to improve common mode gain, the common mode gain has to be increased which can be done by decreasing emitter resistance. However, when emitter resistance is decreased, the biasing emitter current and hence biasing collector current will be increased. To bring the biasing current back to its original value the negative supply has to be changed but the negative supply has limits.

So in order improve the CMMR we need other ways such as constant current source to provide stable biasing current. The following are popular methods to improve the CMRR value of a differential amplifier.

- Constant Current Bias
- Current Mirror Circuit
- Active Load

**Summary**:

In summary, BJT differential amplifier is one way of implementing differential amplifier using bipolar junction transistor. Differential amplifier are used primary building block of operational amplifier. The CMMR of a differential amplifier is an important parameter that measures the ability of the amplifier to reject common-mode signals. A high CMMR is desirable for most applications, and it is typically expressed in decibels (dB). The problem of improving CMRR is to increase emitter resistance but increasing emitter resistance will disturb the biasing point. Solution to increasing emitter biasing resistance and at the same time provide stability is either to use constant current bias, current mirror circuit or using active load.

**References**:

[1] Dual inputs unbalanced output BJT differential amplifier

[2] Single input balanced output BJT differential amplifier

[3] Single input unbalanced output BJT differential amplifier

[4] Differential Amplifier Differential and Common Mode Operation