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The ideal op-amp and the comparator

An operational amplifier acts on the difference between its two inputs, and the ideal one has infinite open-loop gain and infinite bandwidth. Its input resistance is infinite and its output resistance zero. Open loop, a few hundred microvolts pin the output to a rail, which makes the circuit a comparator.

The ideal op-amp and the comparator

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Year 13AQA 3.13.2.1, 3.13.3.2, 3.13.4.4

Operational amplifiers, part 1 of 2. Part 2 is Negative feedback amplifiers.

Builds on Potential dividers and The MOSFET as a switch and Sensing circuits and the potentiometer.

IN THIS TOPIC

  • State the properties of the ideal op-amp and use the open-loop relation.
  • Explain comparator action and saturation, say which input carries the signal, and say what a comparator is for.

COMMON MISCONCEPTION

An op-amp with an open-loop gain of 100 000 multiplies any input by 100 000.

Only for inputs under about 130 μV. An open-loop gain of 100 000 multiplied by more than that asks for an output beyond the supply rails, so the amplifier saturates. Useful gain comes from negative feedback, and then the resistors set it, not the chip.

The ideal op-amp and its supply rails

An operational amplifier amplifies the difference between its two inputs, the non-inverting input V+V_{+} and the inverting input V-V_{-}. At A-level it is treated as ideal, and the ideal has four properties worth learning as a list. Its open-loop gain AOLA_{OL} is infinite. Its input resistance is infinite, so neither input draws current. Its output resistance is zero, so the output does not sag under load. Its bandwidth is infinite. A representative general-purpose device manages about 105, a few megohms and tens of ohms; the numbers vary from chip to chip, and the data sheet is the authority for any particular part.

Each property has a use. Infinite input resistance justifies the assumption that no current enters either input, on which every derivation here depends. Infinite gain is what holds the two inputs at almost equal voltages once negative feedback is applied. Zero output resistance lets one stage drive the next without the output sagging. Infinite bandwidth is an idealisation: a real op-amp's gain falls as the frequency rises, and the gain-bandwidth product quantifies that fall.

Vout=AOL(V+-V-)V_{out} = A_{OL}(V_{+} - V_{-})IN THE AQA DATA BOOKLET

With AOLA_{OL} = 105 and an output that can reach 13 V before the supply rails stop it, a difference of 13/105 = 130 μV already drives the output the whole way. Anything larger asks for a voltage beyond the rails, and the amplifier cannot deliver it. The output sticks a volt or so inside the rail and stays there, saturated. The open-loop gain therefore does not multiply an ordinary input signal by 105: for any input larger than a few hundred microvolts the output is already at a rail.

The op-amp open loop: output against input difference is two saturated rails at +13 V and −13 V joined by a near-vertical ramp only 260 μV of input wide
Describe the drawing

The open-loop characteristic of an op-amp with a gain of ten to the five between thirteen-volt rails, on a millivolt input axis. The output is a flat rail at plus thirteen volts wherever the input difference exceeds 130 microvolts and a flat rail at minus thirteen below minus 130 microvolts, the comparator's two answers; joining them through the origin is a near-vertical ramp of slope ten to the five whose whole width, marked between dashed lines, is 260 microvolts, the thin linear strip negative feedback will later exploit.

FIG. 1The open-loop characteristic: output against the input difference V+ − V− for a gain of 105 between ±13 V rails. The sloping segment spans only 260 μV of input, invisible at any practical scale, so the plot is two flat rails joined by what looks like a vertical step at zero.

Saturation put to work: the comparator

Being saturated is useful if the question you are asking is which input is larger. Wire the signal to the non-inverting input V+V_{+} and the fixed reference to the inverting input V-V_{-}. Then V+-V-V_{+} - V_{-} is positive whenever the signal is above the reference, so the output sits at the positive rail, and it is negative whenever the signal is below the reference, so the output sits at the negative rail. Swap the two connections, reference on V+V_{+} and signal on V-V_{-}, and both statements reverse. The circuit is a comparator: it turns a smoothly varying input voltage into a two-level output. State which input carries the signal whenever a comparator is described.

With no feedback the op-amp is a comparator: as the signal crosses the reference, the output slams to one rail or the other
Describe the drawing

A sine wave on the non-inverting input crossing a dashed reference level held on the inverting input, and beneath it the comparator's output: a square wave pinned at plus thirteen volts whenever the signal is above the reference and at minus thirteen whenever it is below, switching exactly at the crossings.

FIG. 2A sine wave on the non-inverting input crossing a fixed reference held on the inverting input, and the comparator's output beneath it: pinned at +13 V while the signal is above the reference and −13 V while it is below, switching exactly at the crossings.

The switch-over is sharp because the whole swing from one rail to the other takes only 26/105 = 260 μV of input difference. A thermostat is a comparator with a thermistor divider on one input and a preset divider on the other. A light-operated alarm is the same circuit with an LDR, and a frost warning is the same circuit again with the inputs swapped, which is the polarity reversal above.

Open loop, then, the op-amp compares rather than amplifies. Returning part of the output to the inverting input applies negative feedback, which reduces the gain to a value set by two resistors and increases the bandwidth.

Giving the gain away on purpose has moved to the next part of this lesson.

The inverting amplifier and its virtual earth has moved to the next part of this lesson.

The non-inverting amplifier has moved to the next part of this lesson.

What the bargain costs: gain-bandwidth product has moved to the next part of this lesson.

ASSESSMENT FOCUS

  • The four ideal properties are a list, so write them as one. Infinite open-loop gain, infinite input resistance, zero output resistance, infinite bandwidth.
  • Comparator answers need the reference named, the input it sits on stated, and both rails given. With the signal on V+V_{+} and the reference on V-V_{-} the output is at the positive rail above the reference and at the negative rail below it; swapping the inputs reverses both, and either way the change takes only a few hundred microvolts.
  • Saturation is a sentence as well as a number. Say that the output the gain demands would pass the supply rails, so it sits a volt or so inside the nearer rail and stops following the input.
  • The input difference that swings the output from rail to rail is the rail-to-rail voltage over the open-loop gain, a few hundred microvolts. Quoting that number supports the statement that the switching is sharp.

CHECK YOURSELF

An op-amp with an open-loop gain of 1.0 × 105 runs with no feedback from rails that let its output reach ±13 V, and its inverting input is held at a 5.00 V reference. Find the input difference that just drives the output to a rail, state the output when the non-inverting input is at 5.10 V and at 4.90 V, and explain why the output is almost never found between the rails.

Show a hint

Divide the rail voltage by the open-loop gain, then compare the sign of V+-V-V_{+} - V_{-} with zero.

Show the answer

Saturation needs V+-V-V_{+} - V_{-} = 13/(1.0 × 105) = 130 μV, either side of zero.

At 5.10 V the difference is +0.10 V, some 770 times more than needed, so the output sits at the positive rail, about +13 V.

At 4.90 V the difference is −0.10 V and the output sits at the negative rail, about −13 V.

The whole in-between band is only 260 μV of input wide, and no realistic signal pauses inside a window that narrow, so the output is effectively two-valued. That is comparator action.

An op-amp amplifies the difference between its inputs, and the ideal one has infinite gain, infinite input resistance, zero output resistance and infinite bandwidth.

With no feedback the op-amp only compares: a few hundred microvolts of difference send the output to a rail.

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  • State the properties of the ideal op-amp and use the open-loop relation.
  • Explain comparator action and saturation, say which input carries the signal, and say what a comparator is for.

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