You’ve seen it on datasheets, but what is common-mode rejection and why does it matter?

We recently took a long look at electromagnetic interference (EMI) and how to combat this type of noise. The solution—the so-called instrumentation amplifier—is yet to come. But before we can get there, we need to understand a bit more about differential amplifiers.

The differential amplifier is the precursor to an instrumentation amplifier and provides the same basic features. However, it does not provide an easily adjustable voltage gain without degrading the common-mode rejection.

In this article, we’ll explain differential voltage gain, common-mode voltage gain, and the common-mode rejection ratio (CMMR).

# Understanding Common-Mode Voltage Gain

The differential input voltage (v_{D} or V_{d}) is the difference between the non-inverting input voltage and the inverting input voltage (see Figure 1a). Ideally, a differential amplifier provides an output voltage that is proportional to the product of its differential input voltage and its open-circuit (no load) differential voltage gain A_{vd(oc)}.

If both voltage sources are equal, the differential input voltage is zero and the output voltage should also be zero (see Figure 1b). In this case, we can consider the non-inverting and inverting inputs to be tied together to a common* *signal source with a voltage called the common-mode input voltage (v_{CM} or V_{cm}; see Figure 1c). The voltage gain that is provided by the differential amplifier on the common-mode input voltage is called the open-circuit common-mode voltage gain A_{vcm(oc)}.

Ideally, the output voltage should be zero, which means that the ideal value of the common-mode voltage gain is zero. In a real differential amplifier, this value will be made as small as possible. Any imbalance between the left- and right-hand sides of the internal differential amplifier will result in a non-zero common-mode voltage gain. The degree to which this gain approaches zero depends on how well the internal resistors and transistors are matched, along with how well they track one another with temperature changes. The frequency of the common-mode voltage is also important. At high frequencies, internal stray capacitances and wire inductances can further alter the circuit balance.

# The Common-Mode Rejection Ratio (CMRR)

The common-mode rejection ratio (CMRR) is a parameter that describes the effectiveness of a differential amplifier. Since op amps and instrumentation amplifiers are differential amplifiers, the CMRR parameter is provided by many manufacturers on their datasheets.

The common-mode rejection ratio is the magnitude of the ratio of the differential voltage gain to the common-mode voltage gain:

Quite often, the common-mode rejection ratio is expressed in decibels:

# Modeling the Differential Signal and Common-Mode Noise

To understand common-mode noise, examine Figure 2a. The voltage at the non-inverting input (*V _{2}*) is 5.01 V, while the voltage at the inverting input (

*V*) is 4.99 V. The differential input voltage

_{1}*V*is determined easily as 0.02 V. The common-mode voltage is the average value of the input voltages, which is 5 V.

_{D}The differential input voltage can be represented by two equivalent voltage sources equal to *V _{D}*/2, with an equivalent common-mode voltage source connected between them as shown in Figure 2b. With this equivalent representation, the voltages at the non-inverting input and the inverting input with respect to ground are identical to the ones shown in Figure 2a. This is emphasized in Figure 2c.

We can apply this same approach to representing AC differential and common-mode voltages, as shown in Figure 2d. If the AC differential signal source has an average value of zero, one might suspect the common-mode voltage should also be zero. This is often *not* the case, however. Quite often, the common-mode voltage is produced by electrical noise coupling into input signal lines.

In North America, most of the common-mode noise is at the power-line frequency of 60 Hz. In Europe and many other countries, the common-mode noise is at their power-line frequency of 50 Hz. However, common-mode noise can also be present at other frequencies.

The equivalent circuit shown in Figure 2d offers a big advantage in that one can set the values of the common-mode voltage and the differential input signal independently. It also provides an approach to using electronic design automation (EDA) to analyze noise problems via simulation. We shall see that the common-mode rejection that is provided by the differential amplifier eliminates electrical noise.

Have you ever heard an annoying humming sound coming from a public address system or the loudspeakers of your stereo system? An amplifier with a differential input can reject that kind of noise pickup. This is illustrated in Figure 3.

# Using EDA to Examine a Differential Amplifier

Figure 4 shows a differential amplifier with voltage follower buffers on its non-inverting and inverting inputs, as implemented in EDA software Multisim. A two-channel virtual oscilloscope monitors the signal applied to the non-inverting input (channel 2) and the amplifier output (channel 1).

The differential voltage gain *A _{vd}* can be determined as follows:

The output resistance is established by U1A and is essentially zero. The equivalent input resistances are approximated as the equivalent resistances provided by the non-inverting inputs of U1B and U1C, which are effectively infinite.

Figure 4 shows us that the common-mode input voltage is 100 mV peak at 100 Hz. The differential input signal is 10 mV peak at 1 kHz. The low-frequency common-mode noise is 10 times greater in amplitude.

Figure 5 provides the input and output waveforms as monitored by the oscilloscope. The amplifier provides a voltage gain of 10, meaning the output will be 100 mV peak or 200 mV peak-to-peak.

Multisim includes a virtual spectrum analyzer that can be employed to determine the spectral (frequency) components, which we connected to measure the input signal as shown in Figure 6.

Figure 7 shows the results of the simulation for peak input voltages and frequencies.

Figure 8 shows how the differential input signal is amplified and the common-mode noise is suppressed when the spectrum analyzer is connected to the amplifier output.

We can use the spectrum analyzer values in Figures 7 and 8 to determine the differential voltage gain *A _{vd},* the common-mode voltage gain

*A*, and the common-mode rejection ratio.

_{vcm}This is essentially our ideal gain of 10. The ideal common-mode voltage gain is zero. In practice, it is a very small value:

The common-mode rejection ratio is the ratio of the differential voltage gain to the common-mode voltage gain. It should be large:

# Coming Up Next

Like the differential amplifier, the instrumentation amplifier rejects common-mode noise. However, it preserves common-mode rejection as its voltage gain is adjusted by changing the value of a single resistor. In the next part of this series, we will explore instrumentation amplifiers in greater detail.