# Designing with a complete simulation test bench for op amps: Input-referred errors

#### Article By : Ian Williams Measure some of the most common input-referred error sources that affect both DC and AC accuracy: common-mode rejection ratio, power-supply rejection ratio, input offset voltage, input bias current, and input offset current.

Previous installments of this series discussed the need to verify SPICE model accuracy and how to measure the output impedance and small-signal bandwidth of operational amplifier (op amp) models. In part 3, I’ll show how to measure some of the most common input-referred error sources that affect both DC and AC accuracy: common-mode rejection ratio (CMRR), power-supply rejection ratio (PSRR), input offset voltage (Vos), input bias current (Ib), and input offset current (Ios).

Common-mode rejection ratio
An op amp is designed to amplify the differential signal applied across its input pins while rejecting any common-mode signal present. Put simply, a differential signal is created when a voltage difference exists between nodes of a circuit. A common-mode signal, on the other hand, describes a signal that is common to two or more nodes of a circuit. The input common-mode voltage of an op amp is formally defined as the average voltage present across its two inputs, since either input may change over time.

Figure 1 gives an example of differential versus common-mode signals. In this case, the differential signal (Vdiff) is the difference between the voltage present at IN+ and the voltage present at IN−. The common-mode signal (Vcm) is the 2.5V DC voltage present on both inputs. Figure 1
Differential signal vs. common-mode signal

Some typical examples of common-mode signals in op amp circuits are DC bias voltages and coupled noise from electromagnetic fields or parasitic circuit paths. You want an op amp to reject these common-mode input signals, because if they were amplified instead, the resulting output could cause major issues in the circuit’s operation. The amplification of a large DC bias voltage could cause the amplifier output to exhibit a large offset or saturate at either of the power-supply rails. If noise was amplified, the true output signal could deteriorate or even be lost in the resulting noise at the amplifier output.

CMRR quantifies how well an op amp rejects these common-mode signals. When defined as a rejection in this manner, a higher value is better, since that means more of the common-mode signal is rejected and it will therefore have less effect on the op amp. However, since no real op amp has infinite CMRR, common-mode signals do have some measurable effect on an op amp’s behavior.

Let’s revisit the simplified small-signal model of an op amp, shown in Figure 2. CMRR is modeled as an error voltage source (Vcmrr) connected in series with the noninverting input. This error voltage changes with the applied common-mode voltage according to the CMRR specification of the op amp. Since CMRR is input-referred, Vcmrr is amplified by the closed-loop gain of the op amp circuit along with the differential input signal (Ve) to create the total output voltage (Vout). Figure 2
Simplified input-referred CMRR model

An op amp’s common-mode rejection also changes over frequency. CMRR is highest at low frequencies, with most op amps exhibiting between 80dB (100µV of input-referred error per 1V of Vcm) and 160dB (10nV of input-referred error per 1V of Vcm) of common-mode rejection. The level of rejection rolls off at higher frequencies as the op amp runs out of bandwidth, so take care when selecting a device to ensure that CMRR performance is sufficient at the frequencies of interest. Figure 3 is a typical CMRR vs. frequency curve. Figure 3
Typical CMRR vs. frequency curve

Let’s use an example calculation to show how the curve in Figure 3 translates to an input-referred error voltage. If a common-mode input signal is present with a frequency of 100kHz (1E+05Hz), you can see from the curve that the op amp has approximately 100dB of CMRR at that frequency. Equation 1 converts 100dB into an attenuation factor given in linear voltage gain units (V/V): Now it’s possible to compute the input-referred error voltage induced by the common-mode signal at 100kHz. Simply multiply the amplitude of the common-mode signal by the linear CMRR attenuation factor of 10µV/V to determine the input-referred error voltage. Equation 2 is an example of a common-mode signal with an amplitude equal to 1Vpp. Therefore, a common-mode input signal with an amplitude of 1Vpp and a frequency of 100kHz generates an input-referred error signal of approximately 10µVpp. Keep in mind that this calculation is only valid at 100kHz, and would need to be repeated for different frequencies according to the CMRR characteristic given in Figure 3.

From a measurement perspective, CMRR is defined as the ratio of an op amp’s open-loop differential gain to its open-loop common-mode gain. In the real world, these two gain characteristics can be tricky to isolate from one another, but the power of simulation allows you to do this effectively. Figure 4 shows the recommended test circuit. Figure 4
CMRR test circuit

This unique test circuit uses two identical copies of the op amp under test to measure the open-loop differential gain and open-loop common-mode gain separately. In the top circuit, AC source Vin is applied equally to both inputs of op amp U1 to create a purely common-mode input signal. Inductor L1 acts as a short circuit at DC, enabling SPICE to compute a valid DC operating point. At AC, L1 acts as an open circuit, placing U1 in an open-loop configuration to measure its open-loop common-mode gain (ACM).

In the bottom circuit, AC source Vin is routed to a single-ended-to-differential conversion circuit made up of voltage-controlled voltage sources E1 and E2. This generates a differential version of Vin biased around 0V, which is then applied to the inputs of op amp U2. Like in the top circuit, inductor L2 acts as a short circuit at DC and an open circuit at AC to allow for both a valid DC operating point and measurement of the open-loop differential gain (ADM).

As usual, I recommend verifying that the op amp is operating in its linear region by running a DC operating point test. Make sure to match the specified data-sheet conditions for power-supply voltage and input common-mode voltage.

To measure CMRR, run an AC transfer function over the desired frequency range and plot the magnitude in decibels of ACM and ADM. Use your simulator’s post-processing function to generate a curve for ADM/ACM, the definition of CMRR.

Let’s use this circuit to test the CMRR of the OPA2187 SPICE model. The OPA2187 is a zero-drift, low-power precision op amp from Texas Instruments. Figure 5 shows the results. Figure 5
OPA2187 CMRR test results

Based on this test, the op amps CMRR is modeled very closely to the data-sheet curve throughout the majority of the measured frequency range. At frequencies approaching and above the unity-gain bandwidth of the amplifier – around 1MHz in this example – measured CMRR is dominated by parasitic components and higher-order behavior and becomes difficult to model. For more information on CMRR, watch the TI Precision Labs Op Amps: Common Mode Rejection video.

[Continue reading on EDN US: Power supply rejection ratio]

Ian Williams is an applications engineer and SPICE model developer for the Precision Amplifiers group at Texas Instruments.

Related articles: 