# Experiments suggest methods for CMRR measurement, Part 1

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Differential ports make possible the recovery of signals mixed with common-mode noise, and the use of such ports gives rise to the need for measuring common mode rejection ratio (CMRR). Circuit blocks with differential input ports include operational amplifiers, differential amplifiers, isolation amplifiers, instrumentation amplifiers, comparators, data-line receivers, and analog-to-digital converters.

The property for which a differential input port is named provides, at the corresponding output port, a function of the difference in electrical potential (voltage) across its two terminals. That voltage is called the differential-mode voltage. In contrast, a voltage applied simultaneously to both terminals (common to both terminals) with respect to the output reference point (a node with the traditional name of ground) is called a common-mode voltage (Figure 1).

**Figure 1. Differential and Common-mode voltages.**

For an ideal differential port, the common-mode voltage causes no response at the output port. (The term "port" in this article denotes a pair of terminals used as an input to which a voltage or current is fed (input port), or an output that produces an electrical signal (output port). In communication systems, and for historical reasons associated with the way these voltages were first observed, differential-mode voltages are called transversal voltages and common-mode voltages are called longitudinal voltages.

The ratio of a change in differential-mode dc input voltage to the resulting output change is called the dc differential gain. When measured with an ac signal, this quantity (the ac small-signal gain) is sometimes plotted in data sheets as a function of frequency (Figure 2).

**Figure 2. Modeling CMRR as Amplifier Gain Asymmetry.**

Actual devices always include some common-mode response, but as a useful abstraction you can consider a port that exhibits differential gain only, and a zero response for common-mode voltages. Then, common-mode gain is defined in the same way as for differential signals, as the relationship between an applied common-mode voltage and the resulting output-port response. Common-mode gain measures the departure of a differential port from the ideal, but is not very meaningful by itself.

**CMRR**

A figure of merit or a quality factor is needed to express the quality of a differential portī¾quality in this case defined as the ability to amplify differential-mode voltages while rejecting applied common-mode voltages. The quantity in question for a differential port is CMRR, which is formally defined as the ratio of the module of the differential gain to the module of the common-mode gain, for the same output amplitude. For an ideal differential port, the common-mode gain is zero and the CMRR infinite. Equations 1-3 show the mathematical expressions for differential gain, common-mode gain, and CMRR:

A: Differential-mode gain, Acm: Common-mode gain, Vout: Output voltage, Vdiff: Differential input voltage, Vcm: Common-mode input voltage, CMRR: Common-mode rejection ratio.

CMRR is much more useful than common-mode gain because it relates the ability of a differential input to process differential signals (which are desired) to its ability to reject common-mode signals (which are always considered detrimental noise). CMRR is usually expressed in dB:

Datasheets commonly list CMRR as a dc value, measured by determining the two voltage levels necessary to cause the same output change, first in differential mode and then in common mode. The amount of output change is arbitrary, provided only that the circuit remains within the limits specified for output voltage and output current, and within the ranges for common-mode and differential input voltage. You measure the output response after it settles, yielding a dc CMRR as the ratio of the two input voltage levels. Depending on the intended application, CMRR may be given at one or more frequencies, or plotted vs. frequency in the typical operating characteristics.

**Concepts and definitions**

When introducing the concepts of differential port and the CMRR factor of merit, other technical terms and concepts also become necessary. Among these are the output-swing range, the common-mode input range, and the differential-mode input range.

- Differential voltage is the voltage difference between the two inputs of a differential port.

- Common-mode voltage is the average of voltages at the two inputs of a differential port, with respect to the output reference point.

- Output swing is the range of output values from minimum negative to maximum positive, for which the amplifier's output value remains proportional to the input value.

- Common-mode input range is the range of common-mode input voltage, from minimum negative to maximum positive, for which the amplifier functions normally (maintains its differential gain and linearity). This range varies considerably, according to the application for which the circuit is designed. In the case of instrumentation for high-voltage lines it can range up to hundreds of kilovolts, but normally is much smallerī¾usually within the supply-voltage range for op amps. An amplifier with very large common-mode range is usually called an isolation amplifier.

- Differential input range is the range of differential input voltage for which the amplifier output follows changes at the input. For reasonably linear amplifiers, differential input range approaches the output swing divided by the differential gain.

**Modelingī¾CMRR seen as asymmetry in the differential amplifier**

Common-mode rejection for a differential amplifier can also be analyzed by modeling the amplifier as a pair of single-input amplifiers that feature inputs IN+ and INī, outputs connected to a summing amplifier, and complementary gains A+ and A- (Figure 2). Output for the model is the summing output (OUT). A simple analysis of the model yields

Substituting equations (5) and (6) in equation (3),

Inspection of equation (7) shows that for a fixed differential gain A, the value of CMRR is an inverse function of the difference between inverting and non-inverting gains. Equation (7) demonstrates that high CMRR depends on symmetry in the amplifier gain from each input to the output. The word symmetry indicates that to approach an ideal CMRR value, the signal paths to each input must be identical images not only electrically (with exception of the sign for gain), but also geometrically.

The concept of symmetry applies not only to static (dc) gain, but also to gain over the frequency range for which CMRR is of interest. Though not very useful when characterizing the CMRR of an existing amplifier, this concept is extremely useful when designing with differential amplifiers. If good CMRR requires almost perfect symmetry from inputs to output, any input asymmetry introduced by the application can produce errors comparable to those of poor CMRR, even if the amplifier CMRR is excellent. In particular, The CMRR-vs.-frequency response depends strongly on input-to-input symmetry of the unavoidable parasitic components (R,L,C) added by the circuit construction and printed circuit board (PCB) layout.

**CMRR as an offset induced by the common-mode voltage**

Another way to analyze CMRR is to model the common-mode response to a differential voltage Vcm_diff, induced internally in the amplifier by the application of common-mode voltage Vcm to the differential port (Figure 3).

**Figure 3. Modeling CMRR as a Vcm induced offset.**

Because Vcm_diff is in differential mode,

Rearranging equation (7),

As an expression independent of the amplifier's gain and frequency dependence, equation (12) lends insight into the CMRR specification. CMRR is clearly defined as the ratio between a common-mode input voltage and its undesirable effect, referred to the amplifier input.

**CMRR for differential analog inputs of analog/digital devices**

The discussion above on CMRR and its quantification applies only to circuits in which the differential inputs receive analog voltages and produce analog outputsī¾even in cases for which the transfer relation or gain is nonlinear, where small signal excursions are used to measure CMRR. For comparators and differential data-line receivers the situation is different, because the outputs are digital, and because the analog differential inputs define a point at which the output or outputs change their digital state. That point is (ideally) a differential input of 0V, but some applications introduce an intentional offset.

The analysis of errors induced by common-mode voltage in these circuits can proceed in terms of the offset induced by the common-mode voltage, as in circuits with analog inputs and outputs. In both cases the error-inducing mechanism is modeled as an offset proportional to the common-mode voltage appearing in series with the input signal, which distorts the timing of digital data and the input voltage at which the output switches. These differential inputs are not well characterized by a single number such as CMRR.

**Why differential input ports?**

As discussed above, CMRR measures the quality of a differential port. To understand the need for good CMRR, we must first discuss the need for differential ports. The most familiar differential port is that found at the input of the ubiquitous operational amplifier. A differential port provides the essential feature of op ampsī¾their design flexibility. It allows inverting and non-inverting configurations with the same device, and all combinations of negative and positive feedback.

The op amp's low CMRR, however, causes gain error in the non-inverting configurations, and affects the CMRR of more complex op-amp circuits such as the basic differential amplifier (Figure 4) and the instrumentation amplifier (Figure 5). Instrumentation amplifiers are used to process and transport signals, and demand a very high CMRR.

**Figure 4. Basic differential amplifier.**

**Figure 5. Instrumentation amplifier.**

To further appreciate the differential port, consider that voltage-signal sources generally appear in one of three formats: single-ended, differential, or floating. A single-ended signal is carried on one line only, with the signal voltage taken between that line and "ground." A differential signal appears on two lines as the difference between those lines, but the common mode voltage is fixed, or known, or at least bounded. A floating signal source is also a differential source (two lines), but neither one is referred to ground. In technical parlance they are "floating," which means their potential to ground is not defined and can vary without bound, as influenced by neighboring charged objects or electric fields.

**What is "ground"?**

In the signal-source descriptions above there appears the concept of "ground." Sometimes referred to as "earth" but more correctly as "common," ground is the reference point in a circuit with respect to which all the circuit's node voltages are measured (voltage is a potential difference). Thus, a point (node) voltage is more formally called the potential difference to ground of that point in the circuit.

The names "ground" and "earth" come from the early use of electricity, when a reference point (earth as the zero voltage point) was found very useful in modeling the phenomena caused by electric charges. It was also useful in experiments, where a common-point reference for all voltages produced much more reproducible results. Originally, the reference point was an actual connection to earth. That concept was then extended to accommodate the practical needs of an equipotential surface, sometimes called the ground plane. The terms earth and ground are often considered interchangeable, but they are not. Many large electrical and electronic systems (aircraft and spacecraft, for example) have a ground that is not connected to earth.

Limitations of the ground concept were found later, in measuring small changes in voltage. When the measuring apparatus was positioned at some distance from the voltage source, engineers realized that the idea of an equipotential surface cannot be extended beyond a short distance. That distance becomes shorter as measurements specify higher levels of precision, resolution, speed, or frequency. Ignoring the minimum distance may allow ground points with voltage differences larger than the minimum signal increment (resolution), thereby introducing unacceptable uncertainties in the signal value, even within the same board or box.

**Where do common-mode noise voltages come from? **

Ground-voltage differences are caused by currents called vagabond or ground currents (sometimes very large), which can circulate between any two ground points. Providing alternative paths of very low resistance can reduce such dc ground differences. AC ground-voltage differences are hard to reduce or even mitigate, because their source impedance is already very low, and because the unavoidable inductance of any alternative path produces a rapidly diminishing return for the effort.

Other sources of common-mode voltage can appear, even without ground differences, as the result of uncertainties introduced into signal-connection lines passing through electric or magnetic fields. Shielding can reduce the magnitude of such induced voltages. The type of shield depends on the type of interference generating the fields, including its frequency and intensity.

**Differential inputs solve the common-mode problem**

The solution to common-mode problems is to sense a signal with two wires, and send it to the differential input of a measuring device for which neither input terminal is connected to ground. But, the reference point of the measuring instrument (instrument ground) and the reference point of the signal source (signal ground) usually differ in voltage, and that difference appears as common-mode voltage (See Figure 1).

If the layout of signal lines and the construction of signal-connecting cables are symmetrical, then voltages picked up by each line are identical and appear to the receiving differential port as common-mode noise. If the port has good CMRR, it rejects the common-mode voltage while processing the true signal value. Note that common-mode voltage is inherent in some applications. These include electron guns biased at high negative acceleration voltages, and the instrumentation used to measure current, temperature, and other variables in high-voltage transformers or transmission lines.

**Low CMRR introduces error in op-amp circuits**

The type and magnitude of error introduced by poor common-mode rejection (CMR) depends on the type of semiconductor device or circuit being considered. A case-by-case analysis is necessary. For example, CMRR errors appear in every linear circuit including an op amp, when both inputs vary as a function of the input signal (see Basic Differential Amplifier, below).

Op amps used in the inverting configuration have no common-mode voltage applied to the inputs, and hence no CMRR-induced errors. The non-inverting configuration is different. Poor CMRR can appear as a gain error, which can be modeled as an additional offset originating as a function of the input voltage. It appears as a gain error instead of SNR degradation, because "noise" in this case is indistinguishable from the signal. Signal and common-mode voltages are the same.

Because CMRR is a consequence of uncontrolled residual imbalance in the amplifier circuit, the phase and magnitude of any offset (relative to the input signal) is unknown at design time, and therefore remains an uncertainty that cannot be compensated. In the best caseī¾CMRR constant through the range of input common-mode voltageī¾the CMRR-induced errors can be eliminated by resistor trimming or adjustment. Moreover, any environmental or aging changes in the trimming components must match any CMRR changes in the amplifier being trimmed.

CMRR can be expressed as the ratio between the common-mode voltage (input signal in this case) and the offset voltage generated at the input. The value of the CMRR-induced offset error equals the signal magnitude divided by the CMRR value. The resulting gain is the ratio of the signal plus offset error divided by the signal, and the gain error is the ratio of the CMRR-induced offset to the input signal. That error can be calculated simply as the inverse of CMRR. A 60dB CMRR generates a 0.1% contribution to gain error in the output of an op amp in the non-inverting configuration. Note that this error alone exhausts the error budget for a 10-bit application (1 LSB = 1/210 = 1/1024 = 0.0976%), which must also include contributions such as gain error and resistor ratio error. If CMRR is not constant through the common-mode voltage range, the nonlinear offset can (in extreme cases) generate distortion and undesired harmonics in the output signal.

**Circuit blocks for which CMRR is important**

Basic differential amplifiers Instrumentation amplifiers Isolation amplifiers Current-sensing amplifiers

A basic differential amplifier is a simple op-amp application (Figure 4). The resistor network connected between output and inverting input sets the gain of the differential amplifier, and the other network (connected to the non-inverting input) makes the gain from non-inverting input to output the same (but for sign) as from the inverting input.

The major advantage of this circuit is simplicity. The major drawbacks are asymmetry of the input impedances and interdependence of the gain and CMRR adjustments. Input impedance for the inverting input is R1, and for the non-inverting input is R3 + R4.

Frequency responses also differ for the inverting and non-inverting inputs. Differential gain of the amplifier is R2/R1. For perfect gain symmetry and CMRR (assuming ideal CMRR in the op amp), the ratio R/R2 must equal R3/R4. Equation (13) shows the CMRR degradation introduced by resistor tolerances only in a basic differential amplifier circuit, assuming an ideal op amp with infinite CMRR. Equation (14) shows the combined effect of contributions from resistor tolerances and the op-amp CMRR. Asymmetry of input impedances is a drawback, because the signal-source impedance forms a divider with each input impedance for common-mode noise. If those dividers don't have exactly the same ratio, the input-to-output gain is different from each input, reducing the CMRR even further.

CMRRres = (1+R2/R1)/4K (13)

CMRRtot = (CMRRres * CMRRamp)/(CMRRres + CMRRamp), (14)

where components R1-R2 refer to Figure 4, and:

CMRRtot = CMRR error for the basic differential amplifier circuit CMRRamp = Op amp CMRR from the data sheet CMRRres = Error introduced by tolerance of the network resistors K = Resistor tolerance (for 0.1% resistors, K = 0.001).

Instrumentation amplifiers (IAs) are used for precision signal processing. They feature precise and stable gain, low noise, high CMRR, differential inputs, high input impedance, and very stable dc characteristics (current and voltage offsets are stable with time, temperature coefficients are low, etc.). IAs are used as the input stages for instruments of all kinds, and as the interface for sensors and transducers. Poor CMRR in an instrumentation amplifier degrades the signal-to-noise ratio (SNR) of the system in which it resides, and with SNR also goes the quality of the processed signal.

An isolation amplifier is a special kind of IA in which no galvanic connection exists between the input and output. The common-mode range is usually much larger than for any other type of differential amplifier, and it accommodates much larger common-mode voltages. Because of their larger common-mode voltages, isolation amplifiers usually specify a higher CMRR than do other amplifiers. The consequence of poor CMRR in an isolation amplifier is the same as for IAs.

A current-sense amplifier has its inputs connected across a current-sense resistor that is in series with the high-side line of a power supply, and produces an output proportional to the current through that resistor, referred to the system common reference or ground. The potential on that line can vary with respect to the system common, depending on the time (battery discharge), load, and other influences, but the current reading must remain true. That capability depends on the amplifier's CMRR, because the line potential is the common-mode voltage for the inputs. A sensing amplifier with poor CMRR gives a function of the current being sensed plus or minus an error, which depends on the power-supply voltage.