# How system operating conditions affect CMOS op amp open-loop gain and output impedance

Operational amplifiers (op amps) are used in many systems across a broad range of applications. These systems operate over a wide range of conditions, including temperature, power supply voltages, common-mode voltages, output loading, and product lifetimes. Op amp open-loop gain (A_{OL}) and open-loop output impedance (Z_{O}) are the two main small-signal AC parameters that affect the stability and frequency response of an op amp circuit.

In this article we will explore the effects of temperature, power supply voltage, common-mode voltage, output loading, and semiconductor process variation on these two critical parameters. We will provide additional insight beyond what is typically available in manufacturer data sheets regarding how these specifications vary over all of these operating conditions. The variations of these two parameters over the operating conditions can be used to develop a parameter variation circuit analysis model that can be used to create robust designs, using methods as extreme value analysis (EVA), root sum squared (RSS), or Monte Carlo analysis (MCA). The final results are listed in summary tables, which can be used to help identify possible design problems and help to create feasible alternatives.

Simplified small-signal AC model for op amps

A simplified small-signal AC model for an op amp configured for a non-inverting gain is shown in Figure 1. In this simplified model, the differential input voltage (V_{DIFF}) is amplified by the A_{OL} gain block and then passes through Z_{O} to the amplifier’s output pin. Although shown in Figure 1, input impedance (R_{IN}) is not discussed in this article.

While an ideal op amp has infinite open-loop gain and infinite bandwidth capabilities, real op amps have design and manufacturing limitations on both the maximum gain and bandwidth. These limitations are expressed in the magnitude and phase curves of the op amp A_{OL}. Ideal op amps feature zero output impedance, which is not possible in real op amp designs. The actual non-zero output impedance is described in the op amp’s Z_{O} curve.

The magnitude and phase of the op amp A_{OL} and feedback factor (β) will determine the final closed loop gain (A_{CL}) of the circuit as described in the familiar Equation 1.

The product of A_{OL} and β (A_{OL}* β) typically is called loop gain. This critical circuit parameter defines the accuracy and linearity of the closed-loop gain. The loop-gain phase at the intersection of the A_{OL} and β curves is referred to as the circuit phase margin, which is critical in the stability analysis of the circuit. The phase margin determines the circuit’s percent overshoot and settling time in response to an input voltage or output current step.

Interactions between the Z_{O} and the output load affect the final open-loop gain presented at the amplifier’s output pin. These interactions can change the system phase margin, possibly resulting in circuit stability issues.

The modern single-supply, three-stage rail-to-rail output CMOS op amp is significantly evolved from its older high-voltage BJT predecessors, and features complex A_{OL} and Z_{O} behavior, which is detailed in the following sections (when we discuss op amps we’re referring to modern three-stage CMOS amplifiers with complex A_{OL} and Z_{O} behavior).

CMOS op amp A_{OL}

The frequency behavior of an op amp A_{OL} magnitude and phase is displayed in Figure 2.

A_{OL_DC} is the DC change in output voltage (V_{OUT}) versus the change in the differential input voltage (V_{DIFF}). In Figure 2 the value of A_{OL_DC} is 195 dB, as expressed in the following equation:

A_{OL_DC}=195 dB

The frequency where A_{OL} equals 1 V/V, or 0 dB, is defined as the unity-gain crossover frequency and is marked as f_{u} in Figure 2:

f_{U}=1.8 MHz

The frequency behavior of A_{OL} is largely defined by the low-frequency dominant pole located at frequency ω_{1} or f_{1}. At the dominant pole frequency, A_{OL} has decreased 3 dB from A_{OL_DC} and the phase has shifted by -45 degrees:

f_{1}=0.37 mHz

The complete frequency behavior of op amp A_{OL} curves also can be shaped by mid-frequency pole-zero pairs, and higher frequency zeros and poles. In Figure 2, f_{XP1} and f_{XZ1} describes a mid-frequency pole-zero pair. Additionally, there is a zero at f_{XZ2} and a high-frequency triple-pole, f_{XP2}. These poles and zeros in the A_{OL} transfer function determine the f_{U} frequency of 1.8 MHz. The following equations list the frequencies of these poles and zeros:

f_{XP1}=142 kHz

f_{XZ1}=274 kHz

f_{XZ2}=1.24 MHz

f_{XP2}=4.88 MHz

To create a robust design, you need to understand how A_{OL} changes as the system operating conditions change. System operating conditions that affect the performance of the A_{OL} curve include temperature, output load, power supply voltage, and process variation.

Temperature effects on A_{OL}

As an example, we’ll focus on how the operating temperature affects the frequency behavior of the op amp A_{OL} curve. Our reasoning is that, out of all system operating conditions, temperature commonly has the largest impact and varies in many applications (the effects of the other operating conditions are summarized in Table 1).

Many op amps are specified over an extended operating temperature range of -40 ºC to 150 ºC. The operating temperature affects both the DC and frequency behavior of the A_{OL} curve (Figure 3).

Figure 4 zooms in on the temperature effects on A_{OL_DC}. Over the operating temperature range, AOL_DC can vary from 214 dB to 149 dB. The 65 dB change in A_{OL_DC} changes the op amp loop gain at low frequencies, impacting the accuracy of the circuit’s closed-loop gain.

The variation of the unity-gain frequency, fu, over the operating temperature range is shown in Figure 5. Over the operating temperature, f_{u}, can vary from 1.26 MHz to 2.75 MHz. This variation affects circuit closed-loop bandwidth and loop gain phase margin, which impacts circuit response and settling times.

CMOS op amp ZO

Figure 6 shows the typical frequency behavior of the op amp Z_{O} magnitude.

As mentioned, this op amp features a three-stage architecture, which results in three distinct Z_{O} regions as seen in the Z_{O} magnitude. At low frequencies the Z_{O} curve is defined by a low-frequency resistance value, R_{LOW_F}. As frequency increases, Z_{O} becomes capacitive. In that region, Z_{O} is defined by a low-frequency capacitance value, C_{LOW_F}. At mid-frequencies, Z_{O} becomes resistive again and is defined by a mid-frequency resistance value, R_{MID_F}. Z_{O} then becomes inductive and is defined by an open-loop inductance value, L_{O}. This inductive region is the most important for stability analysis because capacitive loading on the output can interact with the inductance. This results in resonance and stability issues that are difficult to compensate. The inductive region turns resistive again at higher frequencies, and can be defined by a high-frequency resistance value, R_{HIGH_F}. Finally, at high frequencies near the end of the region of interest, Z_{O} turns capacitive again and can be defined by a capacitance, C_{HIGH_F}.

Component values for the op amp featured in Figure 6 are listed in the equations below:

R_{LOW_F}=4.87 MΩ

C_{LOW_F}=1.57 mF

R_{MID_F}=4.09 MΩ

L_{O}=1.23 mH

R_{HIGH_F}=1.03 kΩ

C_{HIGH_F}=20.14 pF

To create a robust design, you need to understand how Z_{O} changes as the system operating conditions change. System operating conditions that affect the performance of the Z_{O} curve include temperature, power supply voltage, common-mode voltage, output loading, and process variation.

Temperature effects Zo

Similar to the section on A_{OL}, here we examine how the operating temperature affects the frequency behavior of the op amp Z_{O} curve because commonly it also has the largest impact of all other operating conditions (effects of the other operating conditions are summarized in Table 2).

The effects of the operating temperature on frequency behavior of the op amp Z_{O} curve over the full -40 ºC to 150 ºC temperature range are shown in Figure 7.

Worst-case analysis

Table 1 lists the variations of A_{OL} over the system operating conditions that have the greatest effect on the curve. The worst-case results are generated by operating the op amp in the worst conditions simultaneously, which is why the shifts are beyond the levels for any of the individual conditions. Figure 8 shows the results generated from the worst-case analysis.

Table 2 lists the variations of Z_{O} over the system operating conditions that have the greatest impact on the curve. Similar to the worst-case A_{OL} results, these results are generated by operating the op amp in the worst scenarios simultaneously. Figure 9 shows the envelope of the results generated from the worst-case analysis.

Conclusion

The A_{OL} and Z_{O} of op amps are two key specifications in the understanding of small-signal behavior of op amps, including circuit closed-loop gain, bandwidth, settling time, and stability. The typical magnitude and phase response of the A_{OL} and Z_{O} curves change with variations in the system operating conditions. Some system operating conditions that affect these parameters are temperature, output load, power supply voltage, common-mode voltage, and semiconductor processing variations.

The changes in A_{OL} and Z_{O} over these system operating conditions were presented in this article over the full operating range of an example op amp. The worst-case changes over the same operating range that may occur were also shown. This provides additional insight beyond what is typically available in manufacturer data sheets.

Variations of these two parameters over the operating conditions can be used to develop a parameter variation circuit analysis model that can be used to create robust designs, using methods as EVA, RSS, or MCA. Engineers can use information in this article to create a robust design over the expected application operating conditions.

Texas Instruments

References:

1. Jerald G. Graeme, Optimizing op amp performance, ISBN: 978-0071590280.

2. Sergio Franco, Design with operational amplifiers and analog integrated circuits, ISBN: 978-0078028168.

3. Miro Oljaca, Collin Wells, Tim Green, Understanding open loop gain of the PGA900 DAC gain amplifier, TI Application Report (SLDA031), April 2015.

4. Miro Oljaca, Collin Wells, Tim Green, Understanding open loop output impedance of the PGA900 DAC gain amplifier, TI Application Note (SLDA033A), May 2015.

Solving op amp stability issues, TI E2E™ Community Forum, October 14, 2015.