# How to Ensure Correct RMS Measurements

In power electronics, we deal almost exclusively with nonsinusoidal signals. The article describes the many obstacles and pitfalls on the way to correct RMS results.

## 1. Definitions

Signals can be characterized by a variety of numbers: peak, peak-to-peak, average, and, most important, by their root-mean-square (RMS) values. By definition an AC signal of 230V ACRMS creates the same amount of heat as 230 V DC, the shape may be any as long as its RMS value remains constant. While e.g. the peak value can be read directly from the signal display of a scope or the average value from an average reading instrument, the RMS value must be calculated from:

TITrueRMS = √ (1/T x ∫I2 dt)    (T = signal period)

### Root-Mean-Square (RMS)

Beware of the first pitfall: “RMS” is ambiguous and can mean “AC only” or “AC + DC” which is designated “True RMS”. Most measuring instruments measure the AC content only, so in most cases “RMS” means RMS AC. If an instrument is labeled “RMS” it may well measure True RMS, on the other hand, an inscription “True RMS” does not guarantee just that. It is advisable to always consult the manual before using an unknown instrument.

If the DC content of a signal is ignored gross errors may result. In practice, a DC content can be expected unless the signal is clearly accoupled via a capacitor or transformer.

If only an RMSAC instrument is available, the DC content can be measured separately from a true averaging instrument, the True RMS value can be calculated from the familiar formula

ITrueRMS  = √ (IAC2 + IDC2 ).

This is cumbersome, the extra expenditure for a True RMS instrument saves time and prevents errors. The above formula is generally applicable if there are e.g. RMS currents from several sources which may be correlated or not; their effects add up the same:

IRMS total  = √ (I12 + I22  + ... ).

### Example:

In an offline SMPS, the capacitor following the line rectifier sees 100 Hz line ripple current and the switching frequency ripple current. The output capacitor of an SMPS which drives a DC motor sees the switching frequency ripple current and the motor’s ripple current. In a multiple-output SMPS the ripple currents of the individual outputs are interrelated such that their waveforms and hence their RMS values depend on the loads on all outputs.

This example demonstrates that all other measuring instruments can not be used, because they are calibrated in RMS for sine waves, non-sinusoidal signals cause gross errors! See also the article “DC and AC Signals/Parameters and their Correct Representation by Measuring instruments” in “Bodo’s Power” July 2015.

## 2. Accuracy

First of all it is important to be realistic about the necessary accuracy of RMS measurements and also to have no illusions about the accuracy which can at all be achieved in a given situation. Before buying an instrument it is wise to seriously consider the truly required accuracy and to reckon with the many conditions to be met in order to achieve accurate results, see below.

RMS measuring instruments are available with basic measurement errors < 0.05 %, however, in order to verify such high accuracies special ultraclean and ultraprecise signal sources (calibrators) are used while in the real world such perfect signals are not encountered. Such specs like 0.1 or 0.05 % are likely to impress buyers and tend to make them believe that their measurement results can indeed be that precise.

In reality, in the field of power electronics, pulse trains with high repetition frequencies and short rise and fall times prevail. often superimposed by other signals, correlated or not. It is vital to understand that the basic error of an RMS meter is often inconsequential, that other specifications like crest factor, bandwidth and integration time as well as a correct signal pick-off will determine the measurement error which can easily surpass the basic error by one or two orders of magnitude!. Few applications will call for better accuracy than 1%.

### Example:

In SMPS it is mandatory to check the RMS currents in all capacitors; electrolytics typically have tolerances at best of + - 20%, and their capacitance is highly temperature and age-dependent. An RMS meter with a 5 % accuracy is by far adequate; high bandwidth and crest factor are much more important in order to catch the high-frequency components and signal peaks. Look at the ripple current in the output capacitor of a flyback converter which feeds a DC motor.

Because the flyback is a two-phase converter there will be no switching current from the transformer during the primary charging time, so the capacitor must supply the load during these intervals. During the other phase, the current will flow from the transformer into the capacitor and the load.

The DC motor draws a substantial uncorrelated low-frequency AC current with a DC content, such that during the positive excursions current is drawn while during the negative excursions the motor delivers current into the capacitor - which is often overlooked. If the RMS meter is not synchronized to the motor frequency or a very long integration time is chosen, the RMS reading will fluctuate.

## 3. Methods of RMS Measurement

### 3.1 Thermal Converters

A thermocouple generates a DC output which is proportional to the RMS value of the input current, several can be stacked and are called a thermopile. It is thus a true RMS-to-DC converter, conforming ideally to the definition of RMS. However, a thermocouple is impractical in its basic form — it can not differentiate between a temperature change by the ambient or the heater, the relationship is non-linear, and it is easily destroyed by overloads.

Already in the ’60s, these problems were solved, and the principle is still used in many of today’s RMS/DC - converters. Figure 1 shows the block diagram of the famous HP 3400 RMSAC meter which sported at that time 1 mV to 300V full-scale ranges, 10Hz to 10MHz bandwidth and a crest factor of 10 at full scale. The error was specified 1 % f.s., but the units were far better.

Figure 1: Block diagram of the HP 3400 RMS meter with two thermocouples.

Two identical and matched thermocouples were used. The input signal was amplified by a wideband amplifier and applied to one. The other one was driven by a DC current in a servo loop. The thermocouples were series-connected in opposition, i.e. if both were heated to the same temperature the difference output voltage was zero.

This signal was amplified by a chopper amplifier; the demodulator DC output drove the heater and the meter. The loop tracked thus the temperature of the first thermocouple until the second one attained the same temperature, and the difference voltage became zero. The RMS values of the AC signal and the DC were then identical. The ambient temperature was a common-mode signal and thus compensated.

High precision could also be achieved with one thermocouple by alternately driving it with the amplified input signal and the output of a servo loop.

Thermocouples are expensive, in the 80’s some semiconductor firms built RMS/DC converters according to the above principle driving the cost of RMS meters way down.

Figure 2 shows the construction of a variety of semiconductor equivalents made by firms like AD and LT. The thermocouples were replaced by two matched diodes. Due to the tiny structure of the associated heaters, the first of this kind already operated up to 100 MHz with 1 - 2 % error and a crest factor of 50; today such units reach 10GHz.

Special chip mounting materials are used in order to isolate the chips thermally from each other and from the package. Such bandwidths are presently unattainable by other measuring methods. Thanks to the tight matching possible with semiconductors these converters achieve higher precisions than their predecessors.

Figure 2: Principle of the semiconductor equivalents of the two-thermocouple method: the thermocouples are replaced by two matched diodes. A practical circuit just needs one op-amp and one transistor plus some passive components more.

A disadvantage of all thermal converters is their slow settling time around 1s. It can be shortened by compensation circuits.

All these converters have to be protected from destructive overloads.

## 3.2 Power Analyzers

Power Analyzers measure all voltages and currents applied, they calculate the RMS values, the three power values, the spectra, etc. Today’s instruments digitize the input signals synchronously and perform the calculations in the digital domain. Undersampling can be used to determine the RMS values which is all right as long as there appear no artifacts, if those are detected, the sampling frequency is changed. These are the highest precision instruments with errors down to 0.05% at low frequencies.

Their bandwidth extends only to a few hundred kHz, at best to a few MHz. The built-in shunts are special low inductance designs, external ones can be connected. Their use is limited to circuits in which voltages and currents can be connected to the terminals without detrimental effects. It is impossible to measure the drain voltage of an SMPS switching transistor, the long screened cable to the PA’s input terminals would load the test point and even disturb the function.

## 3.3 Oscilloscopes

Digital Storage Oscilloscopes (DSO’s) all feature mathematical functions including the calculation of RMS. Their accuracy is modest compared to (True) RMS meters or Power Analyzers but by far sufficient for most purposes. They are the measuring instruments of choice for switching circuits. They also speed up work because RMS values are automatically displayed without any additional connections.

Most DSO’s are still 8-bit types, 10-bit and 12-bit types are presently extremely expensive. Whenever using DSO’s it is mandatory to check whether the display is valid in order not to fall for aliases and grossly false results: if the display is an alias all numbers derived and displayed are also false! Many if not most DSO’s presently in use, even with 4-digit price tickets, have unacceptable 1K to 10K memories! At last, competition is forcing the established DSO manufacturers to provide larger memories. If the memory is at least 1MB, the danger of false displays is low unless very slow time scales are chosen.

Figure 3: CF and duty cycle D are related as depicted.

Scopes are seldom used without probes so their errors contribute to the total error which is in the area of 5%. “DC accuracy 0.5%”: This accuracy holds only up to the transition frequency of passive probes, that is the transition from resistive to capacitive division, which is in the kHz area; i.e. all frequencies above the transition frequency depend on the accuracy of the capacitive division, the precision resistors are irrelevant. Just bending of the probe cable or the probe can generate percent errors. Scopes excel by their high bandwidth, most sport > 100MHz, which assures that all relevant high-frequency components of the signals are included in the measurement, but it is necessary to check whether the whole signal is on screen because signal portions outside the screen are clipped, leading to false measurements. Hence the crest factor of DSO’s is low.

Figure 4: Additional measurement error as a function of the highest harmonic which an RMS instrument can measure with the crest factor as the parameter.

The use of scopes and their accessories like probes and current probes assures the least loading of the circuit under test and is, in general, the only practical measuring method applicable to SMPS and similar circuitry. It is no problem to measure the RMS current in the drain lead of a switching transistor with a DC/AC current probe in spite of the fact that the drain voltage may rise within some ten nanoseconds up to 800Vp. The 8-bit resolution problems of a DSO can be circumvented by connecting the DC/AC current probe’s output, properly terminated, to a (True) RMS instrument. The accuracy of this measurement can be better. By measuring the voltage and the current in a circuit DSO’s can also calculate the three types of power.

## 4. Pitfalls

### 4.1 Crest Factor and Bandwidth

The crest factor (CF) of a signal waveform is defined as the ratio peak/RMS, the CF of a sine wave is hence 1.41. With non-sinusoidal signals like pulse trains, it can be very high. An RMS meter must have a correspondingly wide linear dynamic input range, the CF is, therefore, one of the most important specifications. Bandwidth is also of major importance because pulse trains contain very high frequencies.

The CF is specified for a full-scale reading and increases proportionally when the reading is lower; at 1/10 of a range, the CF is hence ten times higher than at full scale. There is a simple check for an adequate crest factor: just switch to a higher range: if the reading does not change the crest factor was sufficient, i.e. the reading was correct as far as the crest factor is concerned. If the readings differ, the instrument was first overdriven. It is therefore recommended to start measurement in the high-est range and downrange until the reading changes or goes off-scale and then back off one range.

Relationship between crest factor and duty cycle of a pulse train:

CF = √ [(1 - D)/D] = √(1/D - 1)

Figure 4 shows the additional measurement error as a function of the highest harmonic in a pulse train with the crest factor as the parameter. The number of the highest harmonic which an instrument still recognizes is given by the bandwidth divided by the repetition frequency.

Note e.g. that for the additional error to remain e.g. below 1% at a CF = 10 the 400th harmonic must still be measured by the instrument! Thus a basic measurement error of 0.1% is fast ruined by insufficient bandwidth. This is especially to be reckoned with when using a Power Analyzer because their bandwidth is only some hundred kHz to a few MHz.

This diagram dramatically illustrates the superiority of scopes also for RMS measurements in switching circuits. RMS meters also feature wide bandwidths, and, as mentioned, the 50-ohm outputs of current probes can be accepted, but scope voltage probes can not be adjusted to meter inputs.

## 4.2 DC Content

Unless the circuit configuration clearly excludes a DC content it is wise to assume there is one. It is difficult to judge from a scope display of a pulse train whether the areas above and below zero are equal, but there is a simple test: just switch from “AC” to “DC” coupling and check for any movement of the display in vertical direction: if it does move there is a DC content. The True RMS value of an asymmetrical square wave is:

ITrue RMS = 1/√2 x √ (I+2 + I-2 ),

where I+ and I- are the positive and negative-going portions of the signal. If e.g. I+ = 10 A and I- = 5A, the result is ITrue RMS = 7.9A. A DC-coupled scope would show a 15 App square wave with a 7. A DC content, an average reading instrument 7.5A, and an RMS AC instrument would only see the 15

App AC signal and show 7.5 ARMS AC. In general, most of the power in a non-sinusoidal signal is in the DC content, the fundamental frequency, and the lower harmonics.

## 4.3 Extraneous Signals and Ground Loops

The wide bandwidth of RMS instruments implies that all signals at the input contribute to the reading whether desired or not: hum, noise, hf pickup, distortions. This is, in contrast, to e.g. an average measuring instrument which suppresses high frequencies and does not react to signals with zero average which applies to most extraneous signals.

Even with a pure sinusoid, an RMS instrument will indicate a higher reading if the signal is corrupted; it does not matter whether the undesired contributions’ average may be zero because the RMS value is derived from the square which is always positive. Imagine a high power noise source with an average of zero: the average responding instrument will indicate zero while the RMS instrument will show the true power because by squaring the negative signal portions will be treated alike the positive ones.

### Example:

The output of a CD player will also contain residual hf above the audio band. The output of an FM receiver will contain residual 19 and 38kHz and more. The output of an analog tape recorder during recording will contain some bias signal above 100 kHz. Frequency response measurements are performed - 20dB to 26dB below reference level, i.e. the signal-to-noise will deteriorate Due to the noise the readings will be too high.

This problem becomes the more acute the smaller the useful signal is. Unless a signal has been looked at with a scope and found to be clean, any measurement of small signals with an RMS instrument is highly uncertain. It is good practice anyway to check all signals with a scope before applying them to any other instrument.

Some RMS instruments use BNC connectors mounted to the case, this is an invitation to ground loops because the case is connected to safety earth.

## 4.4 Integration Interval

A stable display can only be achieved if the RMS meter integrates over complete periods. For example, the standby power consumption of a small SMPS is to be measured. In order to meet norms, such SMPS go into burst modes when idling. Depending on the integration interval chosen different results will be obtained. In such cases, it is necessary to select long integration times.

## 4.5 Current Probes

If higher accuracy is desired than available from a scope, scope DC/AC current probes can be connected to RMS meters if they are properly terminated into 50ohms. Their errors in the percent area have to be added to the meters. In principle also AC current probes can be used, but the danger of false results is higher because a DC content will saturate them. It is advisable to connect current probes to an RMS meter only after the probe was first connected to a scope, and it was checked that the signal was undistorted.

DC/AC current probes with their own amplifiers usually have a vertical position control on that amplifier. First, the scope’s position control has to be set exactly to zero, then the probe amplifier is connected to the scope, and from then on only the position control on the amplifier must be used for positioning the trace vertically which is necessary if the signal has a DC content which would drive the signal off-screen.

This means that the vertical position control of the current probe amplifier can add an arbitrary DC level to the output, if this is disregarded and the output applied to a True RMS meter, absurd results will be obtained, because the True RMS meter can not differentiate between the true signal DC content and the DC added by the amplifier.

In order to achieve correct results, it is thus necessary to follow this procedure:

1. With the probe disconnected set the scope’s position control exactly to zero.

2. Then connect the current probe und adjust the probe’s position control until the trace is again exactly in zero position.

3. Now clamp the probe around the conductor to be measured. The scope will display the True RMS value as long as the signal remains on-screen. If it goes off-screen the True RMS value can not be measured by the scope. Do not touch any vertical position control.

4. Then disconnect the probe amplifier output with its termination from the scope and connect it to the True RMS meter.

Due to the fact that even small DC contents will affect the result, don’t expect a much more precise True RMS result than from the scope. If only the AC RMS value is to be measured, the meter will yield indeed a more accurate result. Alternatively, one can measure the AC RMS content and the DC content with an average responding meter separately and calculate the total RMS current.

All DC/AC current probes are based on the Tektronix invention to place a Hall a sensor in the air gap of an AC probe’s core; the Hall sensor output is amplified, and a current is sent through the winding of the probe such that the DC content of the signal current will be canceled, and the DC flux in the core returns to zero. This amplified Hall sensor output signal extends from DC to about 3kHz, it is combined with the output from the probe winding which has a lower cut-off frequency such that a total true DC + AC signal results.

This is amplified by a wideband amplifier, the standard probe bandwidth is 50 or 100 MHz. The core is cut and lapped, a slider allows to retract one half so that the conductor to be measured can be inserted, the other core half is then pushed in place so that the air gap is minimized. Remanence in the core is another cause of DC shifts; all such probes feature a demagnetizing circuit, and it is necessary to demagnetize the core each time before low currents are to be measured and also after a high current was applied.

These probes are delicate, exerting undue mechanical stress on the probe or the core can cause a DC shift, they are also sensitive to temperature. This is explained in order to show that only modest DC accuracies can be expected.

## 4.6 Shunts

The alternative to a current probe is a shunt. Inductance-free precision shunts with 4-pole terminals offer the advantage of much higher precision like 0.01 % and complete freedom from parasitic DC contents, so they are ideal for True RMS measurements, but it is difficult to avoid ground loops. Shunts are always placed in the ground return path.

A coaxial cable is used to connect the shunt from the 2 inner terminals to the scope or RMS meter; the inner terminal which is close to the outer one at ground potential is connected to the shield. In fast circuits, the cable is terminated at the scope resp. RMS meter side into 50 ohms, using a feed-through 50ohm termination.

Termination at one side is usually sufficient to prevent reflections, in very fast circuits a 50ohm resistor is inserted in series with the center conductor at the shunt side. It is important to avoid loops; in many SMPS and similar circuits cost reasons dictate the use of low-cost E and similar cores which radiate strongly. If there is a loop between shunt and cable, the stray magnetic field can induce a sizeable false signal which adds to the true shunt signal

## 4.7 Voltage Probes

### 4.7.1 Active Voltage Probes

Active voltage scope probes can be used because of their minimal load on the test object and their 50ohm output. Most of today’s active probes use a 5:1 or 10:1 internal attenuator so they are not as easily damaged as earlier models, but the ground connection is always done first und removed last. Plug-on attenuators extend the voltage range x 10 and x 100. Note that their time constants are much shorter than those of passive probes, so higher square wave frequencies must be used for adjustment which requires a scope.

## 4.7.2 Passive Probes

Passive scope probes can only be used with scopes. Scope inputs are standardized: 1 MOhm in parallel with a capacitance usually in the range 10pF to 50 pF. All passive probes constitute RC dividers together with the scope input and are of no use in any other configuration. Also all passive probes can only be adjusted to scope inputs within a small capacitance range, i.e. not any probe can be adjusted to any scope input.

Every divider has to be adjusted. With scopes this is done by applying a 1 KHz square wave with perfect tops and of sufficiently short rise time and by adjusting a variable capacitor until the tops are perfectly flat. In addition to this basic adjustment capacitor hf probes have another up to 6 adjustment elements in the compensation box at the scope connector which must also be adjusted to the scope input.

For these adjustments a 50 ohm pulse generator with a rise time < 1 ns,and a flat top is necessary; the probe is inserted into a special 50 ohm termination with a probe socket. Unless these adjustments are performed gross over- or undershoots at high frequencies will occur. Even if a RMS meter has the standard scope input, no adjustment can be done because the test pulse can not be displayed.