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The first installment of this article series discussed the need to verify SPICE model accuracy and how to measure the open- and closed-loop small-signal AC output impedance of operational amplifier (op amp) models. Here in part 2, I’ll explain how to verify the parameters of an op amp that define its small-signal bandwidth or frequency response while the op amp is in its linear operating region: open-loop gain/phase (Aol), closed-loop gain (Acl) and small-signal step response.

**Open-loop gain/phase – Aol**

Aol is the gain stage that acts on the differential signal applied to the op amp input pins and is undoubtedly the most fundamental parameter of any op amp. It affects nearly all aspects of small-signal or linear operation, including gain-bandwidth product, stability response, rise and fall time, settling time, and even input offset voltage. In essence, Aol is what makes an amplifier an amplifier. To reiterate a point from the first installment in this series, Aol interacts with the open-loop small-signal output impedance (Zo) across frequency to create the op amp’s overall AC response.

Let’s return to our simplified small-signal open-loop model of an op amp from part 1, shown in **Figure 1**.

Figure 1

In this model, a differential input signal (Ve) develops across the op amp’s input resistance (Rdiff). Aol amplifies Ve to generate the ideal output voltage (Vo), which flows through Zo before appearing at the output pin (Vout).

As implied earlier, Aol is not simply an ideal gain block that behaves the same at all frequencies. In practice, real-world limitations and choices of modern op amp design cause most products to exhibit an Aol response with predictable gain and phase components, as shown in **Figure 2**. Both gain and phase curves are represented on the same set of axes, with the gain curve labeled G and the phase curve labeled φ.

Figure 2

To better analyze the characteristics of open-loop gain, let’s divide **Figure 2** into three distinct regions, as shown in **Figure 3**.

Figure 3

The first region is the low-frequency region, highlighted in red. In this region, the gain and phase components are both fairly constant. An op amp operating on signals at these low frequencies is as close as possible to ideal, with a very large maximum gain (usually >100dB or 100,000V/V) and no serious concerns in terms of stability.

The low-frequency region ends at a point in frequency known as the dominant pole (fp1). At this frequency, a pole placed in Aol by design causes the gain and phase curves to change. Right at fp1, the gain reduces by −3dB and the phase shifts by −45 degrees. After fp1, the gain continues to roll off at −20dB per decade and the phase shifts by a total of -90 degrees before remaining constant.

Let’s call this second region the roll-off region, highlighted in green; it is in this region where op amps most commonly operate. In the roll-off region, it’s possible to configure op amps with negative feedback for stable operation with various amounts of closed-loop gain (Acl), as long as the desired Acl is less than Aol at that frequency.

The roll-off region ends at a frequency called the unity-gain bandwidth (UGBW). At this frequency, the gain curve has rolled off to 0dB, or 1V/V. Since the gain curve continues to roll off above the UGBW, frequencies higher than this cannot pass through the op amp without some degree of attenuation. At the UGBW, the amount of phase shift remaining before reaching a total of -180 degrees is called the phase margin and is a primary indicator of the general stability response of the op amp for a closed-loop gain of 1V/V.

Let’s call this third region the high-frequency region, highlighted in blue. In the high-frequency region, higher-order poles and zeroes act on the op amp’s small-signal response, causing the phase to shift rapidly and making the overall system more complex to characterize. It’s common in this region for Zo, input capacitance (Cin), PCB parasitics and other higher-order characteristics to start to significantly influence the op amp’s AC response. For all of these reasons, I do not recommend attempting to operate an op amp in this region.

For a review on poles, zeroes and how they occur in op amps, watch the TI Precision Labs – Op Amps video series on bandwidth. For a deeper discussion about op amp stability, watch the TI Precision Labs – Op Amps video series on stability.

It should now be evident that it’s important to verify the Aol behavior of your op amp SPICE models. **Figure 4** shows the recommended test circuit.

Figure 4

The test circuit is very similar to the one used to measure open-loop output impedance. Inductor L1 creates closed-loop feedback at DC while allowing for open-loop AC analysis, while R1 provides a small amount of series resistance in the feedback loop to make the circuit more “real” and prevent mathematical errors in the analysis. Capacitor C1 shorts the op amp’s inverting input to signal source Vin at AC in order to receive the appropriate AC stimulus but acts as an open circuit at DC.

As explained by Bruce Trump in his classic blog post, “Offset Voltage and Open-Loop Gain – they’re cousins,” you can think of Aol as an offset voltage that changes with output voltage. Therefore, to measure Aol, run an AC transfer function over the desired frequency range and plot the gain and phase of Vout/Vos. The op amp must be in its linear operating region (as shown in **Figure 4**), where Vout is equal to a small offset voltage. Make sure to match the specified data-sheet conditions for the power-supply voltage, input common-mode voltage, load resistance and load capacitance.

Most simulators default to showing the results for gain in decibels and phase in degrees. Since these are the units you wish to see, you only need to make minimal adjustments to the results window. Let’s use this circuit to test the Aol of the OPA1678 SPICE model. The OPA1678 is a low-distortion, low-noise, general-purpose audio op amp from Texas Instruments. **Figure 5** gives the results.

Figure 5

In this case, the op amp’s Aol models very closely to the data-sheet curve and can be used for small-signal analysis to achieve results that will match the real world.

Continue reading on EDN: Closed-loop gain

*Ian Williams is an applications engineer and SPICE model developer for the precision amplifiers group at Texas Instruments.*

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