Determine the range of differential mode gam when R2 is adju

Determine the range of differential mode gam when R2 is adjusted between 0Ohm - 100Ohm

Solution

The Differential Mode Range is the maximum signal that can be applied between the + and – inputs without overloading the probe amplifier, resulting in “clipping” or distortion of the waveform measured by the oscilloscope.

Gain of an amplifier is defined as V_OUT/V_IN. For the special case of a differential amplifier, the input V_IN is the difference between its two input terminals, which is equal to (V₁-V₂) as shown in the following diagram.

So the gain of this differential amplifier is

Gain = VOUT/(V1-V2). -------------- (1)

We can find the expression of V_OUT in term of V₁ and V₂ by using superposition theorem:

VOUT = [R3/(R1+R3)] [(R4 + R2)/R2] V1 - [R4/R2] V2 -------------- (2)

However, we will not be able to re-arrange this expression in the form of eqn (1) to find the gain of the amplifier (except in the special case of R₁ = R₂ and R₃ = R₄).
  

Instead of applying superposition theorem with V₁ and V₂ separately, a better way is to first combined V₁ and V₂ in a different format, viz. (V₁-V₂). This is known as the differential mode input - V_d. Associated with this differential mode component will be the common mode input - V_cm., which is equal to the average value of V₁ and V₂.

Differential mode component : Vd = (V1-V2)

Common mode component : V_cm = (V₁+V₂)/2

By using these alternate representation of the input components (V_d and V_cm) instead of the original components (V₁ and V₂), we can re-express eqn (2) in terms of V_d and V_cm as follows.

Vcm = (V1+V2)/2     Þ      2Vcm = V1 + V2 ---- (3)

Since                                     V_d = V₁ - V₂         ---- (4)

Therefore

(3) + (4)                    Þ        V1 = Vcm + Vd/2 ---- (5)

and               (3) - (4)                    Þ         V₂ = V_cm - V_d/2 ---- (6)
  

Substitute eqns (5) & (6) into eqn (2) :

                      V_OUT   =   1/2[R₃/(R₁+R₃)] [(R₄ + R₂)/R₂ + R₄/R₂] V_d +

                    [R3/(R1+R3)] [(R4 + R2)/R2 - R4/R2] Vcm -------------- (7)
  

From this expression, we can find the gain of the differential amplifier

Gain = VOUT/(V1-V2)

        = V_OUT/V_d

        = 1/2[R₃/(R₁+R₃)] [(R₄ + R₂)/R₂ + R₄/R₂]

This gain is known as the Differential Gain (A_d) as it is based on the differential input alone, i.e.

                     A_d = 1/2[R₃/(R₁+R₃)] [(R₄ + R₂)/R₂ + R₄/R₂]