Operational Amplifier
Theoretical Model
So far, we have generated a high level model of the op-amp. Let’s now expand to a bit more detail. We can model the basic operation of an op-amp with the following circuit.
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On the left hand side of the circuit we have the non-inverting input given by IN+(t) and the inverting input given by IN-(t). The difference between those two inputs is given by vin(t) = (IN+(t) - IN-(t)) which is shown as the voltage across the resistor Ri where the subscript i indicates 'input'. Now let’s look at the right hand side of the circuit. Let’s start with the red diamond. That’s a symbol you might not have seen before. It indicates a voltage-controlled voltage source. By ‘voltage-controlled’ it is meant that the output of the voltage source is not fixed but rather is specified by another voltage in the circuit. A formula is used to specify the voltage; in this case the formula is: AOLvin.
The last circuit element is the series output resistor Ro. Because the output of the op-amp is open-circuited as is shown in Figure 3-1, the output voltage v0(t), will be equal to the formula we saw earlier.
What’s missing from this model?
- Show Answer
- This model does not saturate and therefore does not clip the output voltage at the rail voltages. In fact the model makes no mention of the rail voltages at all.
Let's carry on with the model we have started developing. We'll separate it into two parts: the input and the output.
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Let's look at the input (left-hand) part first. Assume some arbitrary circuitry connected to the input of the op-amp. We will model that circuitry as a voltage source Vs in series with some resistance RTh1. We're modeling that source circuitry as a Thévenin equivalent circuit.
Now let’s look at the right hand side of the circuit. The voltage-controlled voltage source is in series with the output resistor Ro and that is in turn in series with a load resistor RL. The output voltage vo(t) is measured across the load resistor.
We will now extend this model to what is called an 'ideal' op-amp. In the ideal op-amp the input resistance Ri approaches infinity. As the input resistance Ri approaches infinity, the current flowing through the input resistor approaches zero. This means that the currents entering the inverting and non-inverting inputs also approach zero.
It was previously stated that the open-loop gain AOL was very large; we will now let it approach infinity. The output vo(t), however, must remain a bounded number. Given that AOL is heading off to infinity, the only way that can happen is if the difference in the input voltages vin(t) = v+(t) - v-(t) approaches zero. This in turn means that v+(t) = v-(t). Finally, in order for vo(t) = AOL vin(t), the output resistance Ro must also approach zero.
So in summary, the ideal op-amp has the following characteristics:
- the input resistance Ri → ∞,
- the open-loop gain AOL → ∞,
- the output resistance Ro → 0,
- the input currents i+(t) = i-(t)=0, and
- the input voltages v+(t) = v-(t).