Open-loop gain of an operational amplifier

Most voltage feedback (VFB) op amps have a very high open-loop voltage gain (usually called AVOL, sometimes abbreviated AV). Common values ​​range from 100,000 to 1,000,000, with high-precision parts ranging from 10 to 100 times that. Some fast op amps have much lower open-loop gains, but gains below a few thousand are not suitable for high-precision applications. Also note that open-loop gain is not highly stable over temperature and can vary greatly between different parts of the same type, so the gain value must be very high.

Voltage feedback op amps work in voltage-in/voltage-out mode, and their open-loop gain is a dimensionless ratio, so no units are required. However, when the value is small, for convenience, the data sheet will express the gain in V/mV or V/μV instead of V/V. Voltage gain can also be expressed in dB, and the conversion relationship is dB = 20×logAVOL. Therefore, an open-loop gain of 1V/μV is equivalent to 120 dB, and so on.

Current feedback (CFB) operational amplifiers use current input and voltage output, so their open-loop transconductance gain is expressed in V/A or Ω (or kΩ, MΩ). The gain value is usually between several hundred kΩ and several tens of MΩ.

Basic feedback principles dictate that the DC open-loop gain AVOL of a precision amplifier must be high to maintain accuracy. This can be seen by examining the closed-loop gain formula, which includes the error caused by finite gain. The closed-loop gain formula including the finite gain error is as follows:

Where β is the feedback loop attenuation, also known as the feedback factor (the voltage attenuation of the feedback network). The noise gain is equal to 1/β, so the formula can also be expressed in other forms. Combining the two terms on the right side of the formula and substituting NG (noise gain) into it, we get

The following formula:

Equations 1 and 2 are equivalent and either can be used. As mentioned previously, the noise gain (NG) is simply the gain obtained from a small voltage source in series with the op amp input and is the ideal amplifier signal gain in non-inverting mode. If

If AVOL is infinite, the closed-loop gain is exactly equal to the noise gain 1/β.

However, since NG << AVOL and AVOL is a finite value, there is a closed-loop gain error, which can be estimated as follows:

Note that the percentage gain error in Equation 3 is directly proportional to the noise gain, so the finite AVOL has less impact on low gains. Some examples can illustrate the key points of the above gain relationship.

Open-Loop Gain Uncertainty

In Figure 1 below, the first example with a noise gain of 1000 shows that the closed-loop gain error is about 0.05% for an open-loop gain of 2 million. Note that if the open-loop gain remains constant over temperature, output load, and voltage, the 0.05% gain error can be easily calibrated out of the measurement, resulting in no overall system gain error. However, if the open-loop gain changes, the resulting closed-loop gain will also change. This results in gain uncertainty. In the second example, AVOL is reduced to 300,000, resulting in a gain error of 0.33%. This situation results in a gain uncertainty of 0.28% in the closed-loop gain. In most applications, the gain resistors of the circuit are the largest source of absolute gain error when using a good amplifier, but it should be noted that gain uncertainty cannot be calibrated out.

Figure 1: Open-loop gain variation causes closed-loop gain uncertainty

Changes in output level and output load are the most common causes of changes in the open-loop gain of an op amp. Changes in signal level in the open-loop gain cause nonlinearities in the closed-loop gain transfer function and cannot be removed during system calibration. Most op amps have fixed loads, so changes in AVOL with load are generally not important. However, the sensitivity of AVOL to the output signal level can increase at higher load currents.

The severity of nonlinearity varies greatly between different device types and is not usually specified in the data sheet. However, a minimum AVOL is usually specified, and choosing an op amp with a high AVOL can minimize the occurrence of gain nonlinearity errors. Gain nonlinearity can come from many sources, depending on the design of the op amp. One common source is thermal feedback (e.g., from a hot output stage to the input stage). If temperature changes are the only cause of nonlinearity errors, reducing the output load may help. To verify this, measure the nonlinearity under no-load conditions and compare it to loaded conditions.

Measuring Open-Loop Gain Nonlinearity

Figure 2 below shows an oscilloscope XY display test circuit for measuring DC open-loop gain nonlinearity. The precautions discussed above with respect to the offset voltage test circuit should also be observed in this circuit. The amplifier’s signal gain is set to –1. The open-loop gain is defined as the change in output voltage divided by the change in input offset voltage. However, for large values ​​of AVOL, the actual offset voltage may only change by a few microvolts over the entire output voltage swing. Therefore, with a divider consisting of a 10Ω resistor and RG (1 MΩ), the node voltage VY is calculated as follows:

The value of RG is chosen so that VY achieves a measurable voltage, depending on the expected value of VOS.

Figure 2: Circuit for measuring open-loop gain nonlinearity

The ±10 V ramp generator output, multiplied by a signal gain of –1, causes the op amp output voltage, VX, to swing between +10 V and –10 V. Because the offset voltage adds the gain factor, an offset adjustment potentiometer is added to set the initial output offset to zero. The resistor values ​​are chosen to offset input offset voltages up to ±10 mV. A stable 10 V reference (such as the AD688) should be used at each end of the potentiometer to prevent output drift. It should also be noted that due to the low corner frequency of the open-loop gain, the ramp generator frequency must be low, probably no more than a fraction of 1 Hz (for example, 0.1 Hz for the OP177).

The graph on the right side of Figure 2 shows the relationship between VY and VX. If there is no gain nonlinearity, the graph should show a straight line with a constant slope, and AVOL is calculated as follows:

If nonlinearity exists, AVOL will change dynamically as the output signal changes. The approximate value of the open-loop gain nonlinearity is calculated based on the maximum and minimum values ​​of AVOL within the output voltage range, as follows:

The closed-loop gain nonlinearity is calculated by multiplying the open-loop gain nonlinearity by the noise gain NG, as shown below:

Ideally, the relationship between VOS and VX is a straight line with a constant slope, and the inverse of the slope is the open-loop gain AVOL. A horizontal line with a slope of zero indicates infinite open-loop gain. In actual op amps, the slope will change over the output range due to factors such as nonlinearity and thermal feedback. In fact, the slope can even change sign.

Figure 3 shows the relationship between VY (and VOS) and VX in the OP177 precision op amp. The relationship is shown for 2 kΩ and 10 kΩ loads. The inverse of the slope is calculated based on the endpoints, and the average value of AVOL is about 8 million. The maximum and minimum values ​​of AVOL over the output voltage range are measured to be approximately 9.1 million and 5.7 million, respectively. The corresponding open-loop gain nonlinearity is about 0.07 ppm. Therefore, when the noise gain is 100, the corresponding closed-loop gain nonlinearity is about 7 ppm.

Figure 3: OP177 gain nonlinearity

Of course, these nonlinear measurements are most useful in high-precision DC circuits. However, they are also useful in wide bandwidth applications such as audio. For example, the XY display technique in Figure 2 can easily reveal the crossover distortion of a poorly designed op amp output stage.