Nodal Analysis of Op Amps
Introduction
There is little doubt that the op amp is the most
fundamental of all analogue components and its
creation marked a change in the whole thinking
behind analogue electronics. Since it is such a
widely used device, pretty much any op amp circuit
that the engineer needs to implement has already
been designed and he is faced with merely tailoring
the component values to suit his needs. This
approach, although quick, does not always mean the
designer has a fundamental understanding of the
theory of the circuit operation. This article will
set out to explain, by the simple process of nodal
analysis, how the transfer function of most op amp
circuits can be derived.
The Basics
It is clear that no electronic component is perfect
and the op amp is no exception. However, to keep
things simple, an ideal op amp has been used, but as
with other things in life the ‘supermodel’ is going
to be admired but with constant awareness of the
flaws that lie under the façade.
For the following text, the perfect op amp is one
with infinite input impedance and zero output
impedance. Thus 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 in response to the
input. In most 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. Now, the designer could
continue to assume many things about his design and
end up with a circuit that he can only assume will
never work. However, with real life components the
above assumptions are perfectly acceptable and very
little circuit degradation occurs in moving away
from the ideal.
Nodal Analysis
It was a long time before the op amp was invented
that Kirchoff devised Kirchoff’s law stating that
the current flowing into any junction (or node)
of an electrical circuit is equal to the current
flowing out of it. Any op amp circuit can be broken
down into a series of nodes, all of which will have
a nodal equation related to them. These can then be
combined to form the transfer function.
Considering the circuit at the input of an op amp,
this means that the current flowing towards 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. However, it is worth
noting that the same cannot be said for the output
since the op amp will change the current it sources
or sinks in order to keep the two inputs of the op
amp the same.
Current to Voltage Converter
Putting this to the test, consider the fate of a
simple current to voltage converter (FIG 1). The
input current flows towards the node at the
inverting pin. The current flowing into this node is
equal to the current flowing away from the node, so
current flows through the feedback resistor, R,
since no current flows into the pin. Thus a voltage
is developed across R (= Iin x R) and the
output adjusts in sympathy to keep that current
flowing. From our assumptions in the previous
paragraph, the voltage at the two inputs must remain
the same. If the non inverting input is at 0V, the
output will start at 0V and decrease for an
increasing current input current, to maintain
current flow.
FIG1
Differential Amplifier
Taking this notion one stage further, FIG 2 shows a
differential amplifier. It transfer function can be
calculated by again considering the currents flowing
into and out of the nodes.
FIG2
Firstly consider the current flowing towards the non
inverting pin. This can be represented by:
Equation 1
Similarly, current flowing away from that node can
be represented by
Equation 2
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
and
so
therefore the voltage V+ is given by
Equation 3
The
nodal equations for the inverting node are just as
straight forward
Equation 4
To find
the transfer function, we know
Equation5
Combining Equation 3 and 5 into
Equation 4 gives
so
In other words the output is dependent on the
differential voltage across the inputs and the gain
setting resistors, as we would expect.
Wien Bridge Oscillator
The above worked example is based on a circuit using
only resistors. The technique of nodal analysis can
be used to analyse circuits with reactive components
too. In the same way we considered the conductances
of resistors, with reactive components the equations
are made easier by considering their admittances.
Thus a capacitor has an admittance of sC. Note that
the Laplace nomenclature is used, since again it
makes the equations look easier and the
psychological effects of this are considerable. We
could equally use jw in place of s if we
wanted to get an idea of the phase effects of a
circuit and this will be done later.
Boldly going forth with the above supposition, a
Wien Bridge Oscillator can now be analysed. FIG3
shows the generic configuration of this circuit.
Again, to keep the equations simple most engineers
keep the resistor values equal and the capacitor
values equal. In this circuit we have both parallel
and series networks, so it makes no difference to
the simplicity of the maths if admittances or
reactances are used. The following analysis will
keep with the preceding text and use admittances.
FIG3
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
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.
Firstly, for oscillation to occur there must be zero
phase shift from the input to the output. This only
happens at one frequency (when ).
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). Secondly, at this frequency the
ratio of V_{out} to V_{+ }(hence V_{})
has to be 3. Anything less than 3 and the
oscillation will decay. Anything greater than 3 and
the output will saturate. This dictates the ratio of
G_{f} to G_{i} to maintain
oscillation. R_{f} must be equal to
precisely twice the value of R_{i}.
Conclusions
Using Kirchoff’s law the currents flowing into and
out of the nodes around the op amp can be translated
into equations and from this the transfer function
can be derived. The above examples use admittances
instead of impedances, but the principles are the
same and it is left to the engineer to decide which
is more suitable. Once the equations have been
derived, the maths (depending on the complexity of
the circuit) is moderately straight forward to
obtain the transfer function. Then the power of
maths processing programs can be unleashed on the
equations to find when for example instability
occurs, or the susceptability of the circuit to
component variations, if this is desired.
