How to Derive the Non-Inverting Amplifier Transfer Function

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One of the most common amplifiers in Analog Design is the non-inverting amplifier.

non_inverting_amplifier_1

Figure 1

Its transfer function is

image002 (1)

How do you derive this function?

Let’s first note that we can consider this Op Amp as ideal.  As such, the current in the inverting input is zero (I = 0A, see Figure 2) and the currents through R1 and R2 are equal.

image003 (2)

non_inverting_amplifier_2

Figure 2

Next, we can write an equation for the loop made by Vout, R2, V and Vin.

image005 (3)

From equation (3) the I2 expression is

image006 (4)

In a similar way we can determine the expression for I1.  Equation (5) is the loop equation for R1, V1 and Vin.

image007 (5)

and

image008 (6)

Being an ideal Op Amp, we can consider that the non-inverting input is at the same potential as the inverting input, so V = 0V.  This is due to the high gain of the ideal Op Amp.  When the output is at a level of a few volts, the differential input can be at a level of some tens of microvolts.  Hence, V is very close to zero.

Replacing I1 and I2 in equation (2) and eliminating V, we can write this equation:

image009 (7)

Therefore, the transfer function of the non-inverting amplifier is

image002 (8)

Q. E. D.

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