Ideal Op-amp (Differential Amplifier)

Ideal Op-amp (Differential Amplifier)

Dec.-09

The differential amplifier is basically an amplifier which amplifies the difference between the two input signals. The ideal op-amp is basically a differential amplifier which is the input stage of an IC op-amp.

Key Point The differential amplifier amplifies the difference between two input voltage signals. Hence it is also called difference amplifier.

Consider an ideal differential amplifier shown in the Fig. 2.4.1.

V₁ and V₂ are the two input signals while V_o is the single ended output. Each signal is measured with respect to the ground.

In an ideal differential amplifier, the output voltage V_o is proportional to the difference between the two input signals. Hence we can write,

V_o ∞ (V₁-V₂) … (2.4.1)

 

1. Differential Gain A_d

From the equation (2.4.1) we can write,

V_o = A_d (V₁-V₂) ...(2.4.2)

where Ad is the constant of proportionality. The Ad is the gain with which differential amplifier amplifies the difference between two input signals. Hence it is called differential gain of the differential amplifier.

Thus, A_d = differential gain

The difference between the two inputs (V₁ - V₂) is generally called difference voltage and denoted as V_d

V_o = A_d V_d  ...(2.4.3)

Hence the differential gain can be expressed as,

A_d = V_o / V_d ...(2.4.4)

Generally the differential gain is expressed in its decibel (dB) value as,

A_d = 20 Log ₁₀ (A_d) in dB ...(2.4.5)

2. Common Mode Gain A_c

If we apply two input voltages which are equal in all the respects to the differential amplifier i.e. V₁ = V₂ then ideally the output voltage V_o = (V₁ - V₂) Ad, must be zero.

But the output voltage of the practical differential amplifier not only depends on the difference voltage but also depends on the average common level of the two inputs. Such an average level of the two input signals is called common mode signal denoted as V_c.

V_c = V₁ + V₂ / 2 … (2.4.6)

Practically, the differential amplifier produces the output voltage proportional to such common mode signal, also.

Key Point The gain with which it amplifies the common mode signal to produce the output is called common mode gain of the differential amplifier denoted as A_c

V_o = A_c V_c  … (2.4.7)

Thus there exists some finite output for V₁ = V₂ due to such common mode gain A_c, in case of practical differential amplifiers.

So the total output of any differential amplifier can be expressed as,

V_o = A_d V_d + A_c V_c … (2.4.8)

This shows that if one input is + 25 µV and other is - 25 µV then the output of the amplifier will not be same, with the inputs as 600 µV and 650 µV, though the difference between the two sets of the inputs is 50 µV. 

Key Point For an ideal differential amplifier, the differential gain Ad must be infinite while the common mode gain must be zero. This ensures zero output for V₁ = V₂.

But due to mismatch in the internal circuitry, there is some output available for V₁ = V₂ and gain A_c is not practically zero. The value of such common mode gain A_c is very very small while the value of the differential gain A_d is always very large.

At this stage, we can define one important parameter of the differential amplifier known as Common Mode Rejection Ratio (CMRR).

 

3. Common Mode Rejection Ratio (CMRR)

When the same voltage is applied to both the inputs, the differential amplifier is said to be operated in a common mode configuration. Many disturbance signals, noise signals appear as a common input signal to both the input terminals of the differential amplifier. Such a common signal should be rejected by the differential amplifier.

The ability of a differential amplifier to reject a common mode signal is expressed by a ratio called Common Mode Rejection Ratio denoted as CMRR .

It is defined as the ratio of the differential voltage gain A_d to common mode voltage gain A_c.

CMRR = ρ = |A_d/A_c| … (2.4.9)

Key Point Ideally the common mode voltage gain is zero, hence the ideal value of CMRR is infinite.

For a practical differential amplifier Ad is large and Ac is small hence the value of CMRR is also very large.

Many a times, CMRR is also expressed in dB, as

 The output voltage can be expressed in terms of CMRR as below :

This equation explains that as CMRR is practically very large, though both Vc and Vd components are present, the output is mostly proportional to the difference signal only. The common mode component is greatly rejected.

 

4. Features of Differential Amplifier

The various features of a differential amplifier are -

1. High differential voltage gain.

2. Low common mode gain.

3. High CMRR.   

4. Two input terminals.

5. High input impedance.       

6. Large bandwidth.

7. Low offset voltages and currents.

8. Low output impedance.

 

Example 2.4.1 The common mode input to a certain differential amplifier, having differential gain of 125 is 4 sin 200 π t V. Determine the common mode output if CMRR is 60 dB.

Solution :

The CMRR in dB is,

Hence the common mode output is

= A_c V_c = 0.125 (4 sin 200 π t) = 0.5 sin (200 π t) V

 

Example 2.4.2 Determine the output voltage of the differential amplifier having input voltages V₁ =1 mV and V₂ = 2 mV. The amplifier has a differential gain of 5000 and CMRR 1000.

Dec.-09, Marks 4

Solution :

V₁ = 1 mV, V₂ = 2 mV, A_d = 5000, CMRR = 1000

Review Questions

1. What is differential amplifier ? Define differential gain, common mode gain and CMRR.

2. For a differential amplifier, two sets of inputs are applied. The first set is V₁ = 50 µV and V₂ = -50 µV and second set is V = 1050 µV and V₂ = 950 µV. If the CMRR is 100, calculate the percentage difference in the output voltage obtained for the two sets of the input signals. If now CMRR is improved to 10000, calculate the percentage difference in the output voltage obtained for the two sets of the input signals.  

[ Ans.: 10 %, 0.1 % ]

3. Determine the output voltage of an op-amp for the input voltages of

V₁ = 150 µV, V₂ =140 µV The amplifier has a differential gain of 4000 and the value of CMRR is i) 100 and ii) 10⁵.      

[ Ans.: 45.8 mV, 40.006 mV ]

4. Determine the output voltage of a differential amplifier for the input voltages of 300 µV and 240 µV. The differential gain of the amplifier is 5000 and the value of the CMRR is i) 100 and ii) 105.  

[Ans.: 313.5 mV, 300.0135 mV]