KAHIBARO
Table of Contents
The ideal model
An operational amplifier, or op amp, is a circuit element used to amplify voltage. In the ideal op amp model, we do not focus on the internal electronics. We replace the real device by a simple mathematical object with a few perfect properties. This ideal model is extremely useful because it makes circuit analysis much easier and gives results that are often very close to reality in many practical situations.
An op amp has two input terminals and one output terminal. The two inputs are called the noninverting input, marked with $+$, and the inverting input, marked with $-$. The output voltage depends on the difference between the two input voltages.
If the voltages at the two inputs are $V_+$ and $V_-$, the basic op amp relation is
$$
V_{\text{out}} = A\left(V_+ - V_-\right)
$$
where $A$ is the open loop gain. For an ideal op amp, this gain is taken to be infinitely large.
The ideal assumptions
The ideal op amp is defined by a small set of perfect assumptions. These assumptions are not exactly true for real devices, but they are very powerful for analyzing circuits.
For an ideal op amp:
$$
A \to \infty
$$
$$
R_{\text{in}} \to \infty
$$
$$
R_{\text{out}} = 0
$$
This means infinitely large open loop gain, infinite input resistance, and zero output resistance.
Infinite input resistance means no current enters either input terminal:
$$
I_+ = 0, \qquad I_- = 0
$$
Zero output resistance means the op amp acts like an ideal voltage source at its output, able to maintain the required output voltage without internal voltage drop.
Infinite gain means that even a tiny difference between $V_+$ and $V_-$ can produce a large output. In circuits with feedback, this usually forces the two input voltages to become nearly equal.
The differential nature of the op amp
The op amp responds to the difference between its two input voltages, not to either input alone. This is why it is called a differential amplifier at its core.
If $V_+ > V_-$, then the quantity $V_+ - V_-$ is positive, so the output tends to become positive.
If $V_+ < V_-$, then the quantity $V_+ - V_-$ is negative, so the output tends to become negative.
If $V_+ = V_-$, then ideally
$$
V_{\text{out}} = A(0) = 0
$$
for the open loop formula alone. In practical op amp circuits, feedback changes how this condition appears, and the output may take whatever value is needed to keep the inputs nearly equal.
The two golden rules
When ideal op amps are used in common feedback circuits, two very important rules are used again and again.
Ideal op amp rules for linear operation with negative feedback:
- No current enters the input terminals:
$$
I_+ = I_- = 0
$$ - The input voltages are equal:
$$
V_+ = V_-
$$
The second rule needs care. It is not true for every possible op amp circuit. It applies when the op amp is operating in its normal linear region with negative feedback. If there is no feedback, or if the output has reached its maximum or minimum limit, then you cannot blindly use $V_+ = V_-$.
Because the input voltages become equal without being directly connected, this condition is often called a virtual short. It is not a real wire, because current still does not flow between the inputs.
Virtual short and virtual ground
A very common special case occurs when one input is connected to ground. Then, with negative feedback, the other input is forced to have the same voltage.
If for example
$$
V_+ = 0
$$
then the op amp drives its output so that
$$
V_- = 0
$$
as well, provided linear operation holds.
The point at $V_- = 0$ is then called a virtual ground. It is at zero volts, but it is not physically connected to ground.
A virtual ground has zero voltage, but it is not an actual ground connection. Current does not enter the op amp input even if that input is at virtual ground.
Open loop behavior
Without feedback, the ideal op amp is almost impossible to use for smooth amplification because its gain is infinite. The output would jump immediately to its maximum positive or negative value for any nonzero input difference.
This means that in open loop operation, the op amp behaves more like a comparator than a linear amplifier. A tiny positive difference makes the output go as high as it can, and a tiny negative difference makes it go as low as it can.
A simple summary is shown below.
| Condition | Sign of $V_+ - V_-$ | Output tendency |
|---|---|---|
| $V_+ > V_-$ | Positive | Large positive output |
| $V_+ < V_-$ | Negative | Large negative output |
| $V_+ = V_-$ | Zero | Ideally zero by open loop formula |
In real circuits, the output cannot become infinitely large. It is limited by the power supply voltages. The details of output limits belong more to real op amp behavior, so in the ideal model we usually just say the output adjusts as needed unless saturation occurs.
Why negative feedback matters
The ideal op amp model becomes most useful when negative feedback is present. Negative feedback means part of the output is fed back in a way that opposes the input difference.
Suppose the output changes so that $V_-$ becomes too low compared with $V_+$. Then $V_+ - V_-$ becomes positive, and the op amp drives the output in a direction that tends to reduce the difference. If $V_-$ becomes too high, the output moves the other way.
This self correcting action is what allows a very large gain device to produce stable and predictable circuit behavior.
Interpreting infinite input resistance
Infinite input resistance means the op amp does not draw current from the source connected to its inputs. This is very important in circuit design because it prevents the op amp from loading the previous stage.
If a resistor network is connected to an op amp input, the currents in that network must be determined without sending current into the op amp terminal itself.
For example, if a resistor leads to the inverting input, all current through that resistor must continue through other circuit elements, not into the op amp.
Interpreting zero output resistance
Zero output resistance means the output voltage appears exactly at the output terminal, with no internal voltage loss. This allows the output to drive the connected network as an ideal voltage source.
In ideal analysis, the output can supply whatever current is needed to maintain the required output voltage. This is a mathematical convenience. Real op amps have current limits, but those belong to nonideal behavior.
Summary of the ideal model
The ideal op amp is a simplified amplifier with perfect behavior. Its output is controlled by the difference between two input voltages, and its ideal properties make circuit analysis clean and systematic.
| Ideal property | Meaning |
|---|---|
| Infinite gain | A tiny input difference can control the output |
| Infinite input resistance | No current enters either input |
| Zero output resistance | Output behaves like an ideal voltage source |
For ideal op amp analysis in linear circuits with negative feedback, the most useful working rules are
$$
I_+ = I_- = 0
$$
and
$$
V_+ = V_-
$$
Use the second rule only when negative feedback keeps the op amp in its linear operating region.
These simple rules form the foundation for analyzing many important op amp circuits.