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Amplifier AC Design Problem
Robert L Rauck

Here is the analysis of an amplifier that shows the development of the gain expressions for an inverting amplifier and illustrates the origin of the difficulty in getting the amplifier to work as desired. This hypothetical amplifier was intended to be a high pass filter with the gain breaking flat (+60dB) at ~ 160 Hz and the maximum frequency of interest being 5kHz. The amplifier does not work as intended and this analysis will show why. This analysis is somewhat simplified in that it ignores Op Amp bias and offset currents and offset voltage since DC performance was not the object of the analysis. Real Op Amps also exhibit additional break frequencies not modeled here (usually near open loop gain cross-over) but again these were irrelevant to this analysis. Parasitic reactances associated with the layout of PC Boards can also affect the high frequency performance of real amplifiers.
Rf = 50.103   C1 = 20.10-6   G0 = 1.105   Cf = 0.02.10-6


            By superposition:
            Equation11 Eq. 1

            By definition:
            VO = -V2.G            where G is the open loop gain of the amplifier

            Equation21 Eq. 2
If we examine Eq. 1, we can define the coefficient of VO as H. Then the coefficient of VS will be 1-H and we can combine Eq. 1 with Eq. 2 and rewrite the expression as follows:
            EquationB1 Collecting terms involving VO
            Equation31 Solving for circuit gain            Eq. 3
This last expression consists of an ideal gain term -(1-H)/H time a correction factor (G.H/(1+G.H)) that accounts for the non-ideal properties of the Op Amp.

            i = 0,0.1..4            f(i) = 10i              Here we set up a range variable that allows us to sweep frequency.

            EquationC1             EquationD1
            EquationE1             in the sinusoidal steady state

            EquationF1             EquationG1

            EquationH1             EquationI1

            Equation41 Eq. 4 This is the closed loop gain of the amplifier. This expression determines the gain and phase shift applied to signals processed by this amplifier.

Now we will assemble expressions for the magnitudes (dB) of actual of amplifier closed loop gain (GCL_MAG(i)), ideal amplifier closed loop gain (GCL1_MAG(i)) assuming a perfect Op Amp and the open loop gain (GOL_MAG(i)) of the chosen Op Amp. We will plot the results.

Eq. 5
Eq. 6
Eq. 7


Here we can see the interplay of amplifier open loop gain (GOL_MAG(i)) and ideal closed loop gain (GCL1_MAG(i)) resulting in actual closed loop gain (GCL_MAG(i)). The amplifier runs out of gain and fails to follow the ideal gain expression above ~ 100Hz.
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