Operational Amplifier: Ideal Concept
An operational amplifier, or op-amp, is a high-gain differential amplifier. It compares the voltage at the non-inverting input with the voltage at the inverting input and drives its output according to that difference.
The ideal model is deliberately simple. It lets you analyze many practical circuits with only feedback rules, Ohm's law, and Kirchhoff's laws before adding real device limitations.
Learning Objectives
By the end of this lesson, you should be able to explain the ideal op-amp assumptions, use the differential gain equation, identify negative and positive feedback, predict saturation, and apply the virtual short and input-current rules correctly.
Op-Amp Symbol and Equation
An op-amp has two signal inputs and one output:
- non-inverting input, marked
+; - inverting input, marked
-; - output;
- positive and negative supply pins, often omitted from simplified symbols.
The open-loop equation is:
[
V_{OUT}=A_{OL}(V_+ - V_-)
]
where (A_{OL}) is open-loop voltage gain in V/V, (V_+) is the non-inverting input voltage, and (V_-) is the inverting input voltage.
Ideal Op-Amp Assumptions
The ideal op-amp model assumes:
| Property | Ideal value | Practical meaning |
|---|---|---|
| Open-loop gain | (A_{OL}=\infty) | tiny input difference can move the output |
| Input impedance | (Z_{IN}=\infty) | input current is zero |
| Output impedance | (Z_{OUT}=0) | output behaves like a perfect voltage source |
| Bandwidth | infinite | gain does not fall with frequency |
| Slew rate | infinite | output changes instantly |
| Input offset | 0 V | equal inputs produce no output error |
| CMRR | infinite | common input voltage is rejected |
These are analysis assumptions, not datasheet claims.
The Two Golden Rules
For an ideal op-amp with negative feedback and operating in its linear range:
- (V_+ \approx V_-)
- (I_+ = I_- = 0)
The first rule is often called a virtual short. It does not mean the two input pins are physically shorted. It means feedback drives the output until the input voltages are nearly equal.
The second rule means no current enters either input terminal in the ideal model. Currents flow through the external feedback network, not into the op-amp inputs.
Why Feedback Matters
Without feedback, the open-loop gain is so large that microvolts of input difference can drive the output to a rail. With negative feedback, the output is fed back to the inverting input, forcing a stable relationship between input and output.
Negative feedback is used for linear amplifiers, buffers, active filters, integrators, and many signal-conditioning circuits. Positive feedback reinforces the input difference and is used intentionally in comparators with hysteresis, Schmitt triggers, and oscillators.
Saturation and Supply Rails
The ideal equation can predict impossible voltages if supply limits are ignored. A real op-amp powered from (+5\text{ V}) and (0\text{ V}) cannot output (12\text{ V}). Its output is limited by the supply rails and by output swing specifications.
If (V_+ > V_-), the output moves positive. If (V_+ < V_-), the output moves negative. With no stabilizing negative feedback, it usually saturates high or low.
Rail-to-rail op-amps can swing closer to the rails than older devices, but they still have limits that depend on load current, supply voltage, and temperature.
Worked Example: Tiny Difference, Huge Open-Loop Output
Assume an op-amp has:
- (A_{OL}=200000\text{ V/V})
- (V_+ - V_- = 100\ \mu\text{V})
Then:
[
V_{OUT}=200000 \times 100\ \mu\text{V}=20\text{ V}
]
If the op-amp is powered from (+5\text{ V}) and (0\text{ V}), it cannot reach 20 V. It saturates near the positive rail. This is why open-loop op-amps behave more like comparators than linear amplifiers.
Where the Ideal Model Works
The model works well when:
- negative feedback is present;
- the output is not saturated;
- signal frequency is well inside the op-amp bandwidth;
- required output current is within the datasheet limit;
- input common-mode voltage is inside the allowed range.
It is especially useful for first-pass analysis of buffers, inverting amplifiers, non-inverting amplifiers, summing amplifiers, difference amplifiers, and many active filters.
Practical Checks
Before trusting an op-amp design, check:
- supply rails allow the required input common-mode range and output swing;
- feedback is negative, not accidentally positive;
- every input has a DC bias path;
- gain and frequency fit the gain-bandwidth product;
- output load current is within the device rating;
- decoupling capacitors are close to the supply pins.
Common Mistakes
- Applying the virtual short rule when the op-amp has no negative feedback.
- Assuming (V_+=V_-) when the output is saturated.
- Forgetting the supply pins in a single-supply circuit.
- Using an op-amp as a comparator without hysteresis or output logic compatibility checks.
- Assuming rail-to-rail means exactly to both rails under any load.
Summary
An ideal op-amp is a high-gain differential amplifier with infinite input impedance, zero output impedance, infinite bandwidth, and no offset. In negative feedback and linear operation, its inputs sit at nearly the same voltage and draw no current. Those two rules make op-amp circuits predictable, but only after supply rails, saturation, and real-device limits are checked.
Further Reading
- Texas Instruments, "Op Amps for Everyone."
- Analog Devices, "Op Amp Applications Handbook."
- Microchip, "Operational Amplifier Basics."