Keywords: nodal analysis, op amp circuits, circuit analysis, op amps, transfer function of an op amp, opamps, opamp
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TUTORIALS 1939

Abstract: The creation of the op amp introduced a new fundamental component and marked a change in thinking for analog designers. Since it is so widely used, pretty much any op amp circuit that an engineer needs to implement has already been designed and the engineer can merely tailor the component values. This approach, although quick, does not always mean the designer has a fundamental understanding of the theory of the circuit operation. This application note explains how the transfer function of most op amp circuits can be derived by a simple process of nodal analysis.

In particular, we assume infinite input impedance and zero output impedance. The front end of the circuit is not loaded in any way by the op amp and its output can source or sink as much current as needed to faithfully respond to the input. With these assumptions and op amp configurations with negative feedback, the voltage at the two inputs is identical and the output adjusts itself to a voltage to maintain this state.

It is also assumed that the bandwidth of the op amp is sufficient to respond to the needs of the circuit and the open loop gain of the amplifier is infinite.

The performance of modern components is such that in most cases, the above assumptions are perfectly acceptable and very little performance degradation occurs as we move away from the ideal.

Consider the circuit at the input of an op amp. The current flowing toward the input pin is equal to the current flowing away from the pin (since no current flows into the pin due to its infinite input impedance). The same cannot be said for the output, since the op amp can source or sink current.

V = I

As explained earlier, the voltage at inverting input is equal to that applied at the noninverting input because the circuit has negative feedback; that is, the inverting input is fixed at V

V

Consider the current flowing towards the non inverting pin. This can be represented by:

Similarly, current flowing away from that node can be represented by

Combining Equation 1 into Equation 2 gives

Now, life is made easier if we use conductances instead of resistances (it keeps the fractions to a minimum). Thus,

where

so

therefore the voltage V+ is given by

The nodal equations for the inverting node are just as straight forward

To find a transfer function, we know

Combining Equation 3 and Equation 5 into Equation 4 gives

so

In other words, the output depends on the differential voltage across the inputs and the gain-setting resistors, as we would expect.

Boldly going forth with the above supposition, a Wien bridge oscillator can now be analyzed.

Firstly, from Equation 3 the voltage at the inverting pin is

It is worth noting that if two admittances are placed in series, the total admittance is the inverse of the sum of their reciprocals (using the same formula as for two resistors in parallel). Similarly if two admittances are placed in parallel, the total admittance is sum of the admittances. Therefore the admittance from the output of the op amp to the non inverting input is

Likewise the admittance from the non inverting terminal to ground is

Using the methodology from before, it can be shown that (eventually)

Putting s = jw and R = 1/G gives

Therefore, using the principles of nodal analysis, the transfer function for the Wien bridge oscillator has been derived. From this equation two conclusions can be drawn, both of which are well known conditions for oscillation of the Wien bridge oscillator.

First, for oscillation to occur, there must be zero phase shift from the input to the output. This only happens at one frequency (when w = 1/CR). At this frequency the real terms of the numerator cancel and the phase shift represented by the imaginary terms in both numerator and denominator cancel (essentially, if you have no j terms in either numerator or denominator, there is no phase shift). Second, at this frequency the ratio of V

A similar version of this article appeared in the December 2002 issue of