The ideal operational amplifier (A-level only)

What you'll learn:

• What an operational amplifier (op-amp) is and how it connects to a circuit.
• The two defining characteristics of an ideal op-amp.
• How to use the open-loop transfer function to calculate output voltage.
• How an op-amp functions as a simple comparator to make decisions in circuits.

The Op-Amp as a System Building Block

Inside a real operational amplifier, there are dozens of tiny transistors, resistors, and capacitors working together. Fortunately, in A-level Physics, we do not need to understand its internal circuitry.

Instead, we treat the op-amp as a system building block. This means we view it as a "black box" that performs a specific, reliable function. By combining op-amps with standard components like resistors and capacitors, we can easily build complex analogue signal processing circuits (like amplifiers, filters, and oscillators).

Signal and Power Connections

An op-amp is represented in circuit diagrams as a triangle pointing in the direction of signal flow (from left to right).

There are five main connections you need to know:

1. Inverting input ($-$): A signal applied here is amplified and inverted (flipped upside down or multiplied by a negative number).
2. Non-inverting input ($+$): A signal applied here is amplified but keeps its original polarity.
3. Output ($V_{out}$): The single pin where the processed signal leaves the op-amp.
4. Positive power supply ($+V_s$): Op-amps are active components; they need external power to work. This is typically connected to a positive DC voltage, like +15 V.
5. Negative power supply ($-V_s$): This is typically connected to a negative DC voltage, like -15 V, or sometimes to 0 V (ground).

Characteristics of an Ideal Op-Amp

Real op-amps are incredibly complex and have minor imperfections. To make designing circuits easier, we use a theoretical model called the ideal op-amp. You must memorize its two main characteristics:

1. Infinite Input Resistance

The resistance between the two input terminals is assumed to be infinite.

• Practical consequence: Absolutely zero current flows into or out of the inverting or non-inverting inputs. The op-amp "looks" at the voltage of the incoming signals without drawing any electrical current from them.

2. Infinite Open-Loop Gain

Gain is the multiplier by which an amplifier increases a signal. An ideal op-amp has an infinite open-loop gain.

• Practical consequence: Any voltage difference between the two inputs, no matter how microscopically small, will be multiplied by infinity. In a real circuit, this causes the output voltage to instantly shoot up to the maximum possible value the power supply allows.

The Open-Loop Transfer Function

For a real op-amp, the open-loop gain is not quite infinite, but it is typically extremely large (often around $10^5$). We call this open-loop gain $A_{OL}$.

The op-amp works by amplifying the difference between the voltages at its two inputs. This relationship is given by the open-loop transfer function:

$$V_{out}=A_{OL}(V_{+}-V_{-})$$

• $V_{out}$ is the output voltage.
• $A_{OL}$ is the open-loop gain (no units).
• $V_{+}$ is the voltage at the non-inverting input.
• $V_{-}$ is the voltage at the inverting input.

Saturation

An amplifier cannot create energy out of nowhere. The output voltage $V_{out}$ can never exceed the power supply voltages ($+V_s$ and $-V_s$).

If the formula calculates a $V_{out}$ that is greater than $+V_s$, the op-amp simply outputs $+V_s$. We say the op-amp is saturated.

Notice how narrow the central diagonal line is. Because $A_{OL}$ is so massive, even a tiny difference of a few microvolts between $V_{+}$ and $V_{-}$ is enough to drive the output firmly into saturation.

The Op-Amp as a Comparator

Because the open-loop gain is so huge, an op-amp in an open-loop configuration acts essentially as a digital switch. It compares the two input voltages and makes a binary decision:

• If $V_{+}>V_{-}$, the output swings instantly to $+V_s$ (positive saturation).
• If $V_{+}<V_{-}$, the output swings instantly to $-V_s$ (negative saturation).

This specific application is called a comparator. Comparators are the heart of sensor circuits (like automated streetlights or thermostats). Usually, one input is connected to a fixed reference voltage, and the other is connected to a sensor (like a thermistor or LDR) in a potential divider.