AC POWER MEASUREMENT: YOU PROBABLY don't think about it very much. Most engineers and technicians normally don't even give it a second thought. Nonetheless, it's an extremely important and pervasive area of electronics, used in a wide variety of fields, include determining RF transmitter power and field strength, radar, motor and generator testing, sonar, stereo, and acoustics, and one area that affects all our daily lives, commercial AC power distribution.

Wow this article is something else ! It was pulled from an electronic magazine in August 1990. Thats 20 years ago. Boy am I getting old ! At any rate it sure shows the principles of power and the complexities of the measurements that need to be observed. Now days it is really simple we just hook up two leads to the voltage and our microprocessor displays on the LCD screen everything we will ever need. Some of the higher end Fluke Meters have phaze angle and all built into them. This article doesn't mention the complexities of 3 phaze measurement either which is important these days with the cost of Industrial Power. It is common to have data loggers set up and in some cases there is automatic Power Factor Correction built into the Factory Power Cosumption.

But what is good about this and other articles I am archieving is that the principles are the same, and if you can appreciate the complexity of your measurement or circuit you will appreciate the technology we have today. Also it should give one better grounding in understanding the instrument or the black box components that we are getting so used to using these days.

We're all billed using a kilowatt- hour (kWh) meter, and obviously want it to be accurate. Utilities need to measure true usage, because it affects generator loading and fuel consumption. Consumers likewise need to control operating efficiency, peak load usage, and billing. With rising fuel costs and the need for conservation, the subject has become increasingly relevant. AC power isn't always easy to measure, because it's affected by such factors as phase shift and waveform distortion. This article discusses techniques of measuring AC power, using some special IC's designed for this purpose.

True power and RMS voltage

Let's review DC power; The power law
P= E x 1
Ohm's law
E = I x R

Produce the 12 equations in Fig. 1.

They're for DC only, and omit capacitive and inductive reactances

They're valid for AC if the load is purely resistive (causing no phase shift), and if E and I are true Root Mean Square (RMS) values. RMS means the square root of the average of the squares of a series of voltages or currents. The RMS concept is necessary, because the time-average value of a sinusoidal voltage or current with no DC component is zero.

Let's see why, and how this relates to power.

Figure 2-a shows a sinusoidal voltage sampled 16 times, applied across RI in Fig. 2-b. As each sample varies, the dissipated power is P(t)= e2(t)/ RI. To find the approximate RMS coefficient for time-average power, the power is found for each sample, and the average of the values is determined. Increasing the number of intervals increases resolution, and improves accuracy.


Table 1 shows the calculations; the average sampled instantaneous power is

P = 0.5 0 3 x V 2PK/R 1 = (0.709 x VPK)2)/RI .

The letters AP in Fig. 2 correspond to the same letters in parentheses in the left-hand column of Table 1. The RMS value for sinusoidal current is found similarly. If the RMS value for a different wave- shape were desired, the process would be repeated using new samples. This crude approximation gives an RMS value of

0.709 VpK versus a correct value of

VpK/V2 =0.7071VpK, an error of only 0.27461%.

The average or mean value isn't just the average of the actual sinusoidal samples, since this is 0 volts, as mentioned earlier. To find the average value, the average of the absolute value of the samples in Table 1 (the full-wave rectified case) is 0.643V. The correct value is 0.6366VpK, an error of only 1.01%.

Fig. 3 shows the peak, RMS, and average values for several waveforms, with sketches of each.One of the more difficult concepts to understand about power is that there are really two separate types of power, instantaneous and time-average. Since DC values are time-invariant, instantaneous and time-average DC powers are always equal. The term used when referring to an RMS value is that of "equivalency." RMS power, in an AC context, is considered "equivalent" to DC power. The major factor for judging this equivalency is what is referred to as the "PR" effect, from the power formula.

RMS power yields the same heat dissipation in a resistive load as an equal value of DC power. Reactive loads (those with inductance or capacitance) cause a phase angle difference between voltage and current equal to the phase angle of the load impedance. Such a load may have both resistive (real) and reactive (imaginary) components; real "12R" power (heat) dissipation occurs only in the real part, never the imaginary part. The imaginary part refers to the energy stored in capacitive electric fields, and inductive magnetic fields.

You should know the meaning of the word "imaginary" in this context; it refers only to the phase difference between voltage and current, never that these quantities aren't physically present. As you've probably heard before, "imaginary" voltages and currents are quite capable of causing real shocks.

Measuring true RMS voltage

You may have heard of "average- responding" and "true RMS" AC meters. Average-responding meters, shown in Fig. 4, are simpler; they rectify and filter AC to give the average value as a DC voltage, that's then scaled to display RMS. Thus, a sine wave with VpK =1 volt produces 0.6366 volt going into the display, but the display is calibrated to read 0.7071 volt. Because the relationship between RMS and average depends on the waveform, the reading is accurate only for sinusoids.

For example, average-responding meters aren't useful for measuring RMS power in most modern circuits, since nonsinusoidal waveforms are very often used in SCR controllers and variable-speed motor drives, necessitating true RMS measurements. The most common methods are thermal, direct (explicit) computation, and implicit computation. In implementing these approaches, we'll also examine two RMS-to-DC converter IC's that take the work out of true RMS measurement.

Thermal RMS measurement

Thermal techniques find RMS voltage by exploiting the fact that the real component of power is dissipated as heat in a resistor, as mentioned earlier. In Fig. 5, R1= R2; the measured voltage is amplified, buffered, and/or scaled by Al, and passed to RI. Similarly, R2 is connected to A2, a high- gain differential amplifier producing DC. Matched series-opposing temperature sensors, usually thermocouples, are in physical contact with R1 and R2, and A2 responds to their temperature difference. If RI is hotter than R2, the output of A2 increases until the temperatures of R1 and R2 are equal. The two temperatures will be equal when the DC output of A2 equals the RMS output of Al.

Although this is a simple approach, accurate measurements using this method are difficult. The main problem is the need for highly accurate thermal matching. Any difference in size, shape, contact, or insulation causes errors. Also, thermal inertia has to be optimized. It has to be slow, to average the AC waveform over its whole cycle. However, excessive inertia causes sluggish measurements.

The Linear Technology LT1088 RMS-to-DC converter has matched pairs of 50- and 250-ohm resistors and diode sensors in a thermal package. The diode has a forward-bias voltage drop temperature coefficient of 1.75 mV per C, and the two sets of internal resistors provide flexibility.

Figure 6 shows the LT1088 in a practical circuit. VIN comes from an amplifier stage, heating the 50-ohm AC resistor while Al heats the 50- ohm DC resistor via emitter-follower Ql. The level at pin 9 is 1 volt DC per volt of RMS input, while A2 provides gain adjustment. The maximum input is 4.25 volts for 50 ohms, and 9.5 volts for 250 ohms. Accuracy is 1% up to 50 MHz, with a 3-dB bandwidth for Al of 300 MHz, and input crest factors (peak-to-RMS ratio) up to 50:1, making this method highly suitable for RF RMS measurement.

Direct RMS computation

Direct or explicit RMS computation solves the problem of RMS power measurement electronically. As shown in Fig. 7, the amplified or buffered AC is rectified, squared, averaged, and presented to a circuit that determines the square root of the average, giving the RMS level. The problem here is the dynamic range of the squared signal.

The squaring multiplier produces full-scale output with both X and Y at full scale. Thus, 100% inputs produce 100% output, 10% inputs produce 1% output, and 5% inputs produce 0.25% output, and so on. Thus, this method isn't suited to voltages varying over a wide range, or waveforms with high crest factors.

Implicit computation

Implicit computation as shown in Fig. 8 may seem less direct than explicit computation, but it's simpler. There's no square-root needed, and this approach has better dynamic range than is the case in direct RMS computation. Here, A and B in the multiplier/divider are VA, and C is VouT.

Thus V1= (ViN)2/VouT, This voltage is filtered (averaged) to become the output. so
VOUT = VAvg[(VIN)2].

Analog Devices makes several IC's for implicit RMS-to-DC conversion, such as the general-purpose AD536A, the low-power/low-level AD636,the high-performance AD637 and the low-cost/low-power AD736/7, all used by several manufacturers of true-RMS DVM's.

Figure 9 shows a practical circuit for a low-cost true-RMS meter using an AD536A as IC1. The input is scaled for 200 millivolts RMS at full scale, applied to a buffer amplifier on pin 7 (BUF IN) of IC1, and fed from pin 6 (BUF Out) to an absolute value precision rectifier on pin 1 (VIN). It then passes through a squarer/divider that provides a log output scaled in dB on pin 5.

The final output comes from a current mirror that produces a current of 40 uA DC per RMS volt of input. This current passes through an internal 25K resistor, to produce 1 volt DC/ RMS volt of input. There's no offset or gain adjustment, since the AD2026 Digital Panel Meter (DPM) module includes them. If you use your own meter, you can add an output amplifier with gain and offset adjustments.

Current transformers

Electronic current measurement usually begins by passing a current through a dropping resistor to create a proportional voltage. For currents of tens, hundreds, or thousands of amps, this is impractical, requiring resistors in the milliohm range, or below. Current transformers reduce such high currents to practical levels, using a single winding of N turns on a toroidal core.

The wire carrying the current is threaded through the center of the toroid and becomes a single-turn primary. The secondary current is 1/N times the primary current, and is passed through a dropping resistor to create the proportional voltage. Current transformers generally produce 5 amps out at full-scale primary current.

Power and phase shifts

Up to this point, we've considered only true RMS measurement. Next, we'll examine the effect of phase shifts on power measurement. Our discussion will deal only with pure sinusoids. The concept of phase shift has meaning for sinusoids as well as other periodic waveforms of the general form:

V(t)= VpK x sin[(2 x pi x f x t) + theta]

Phase shifts are caused by capacitive or inductive reactances. In a pure capacitance, voltage lags current by 90 degrees, while in a pure inductance, current lags voltage 90 degrees. RC or RL circuits produce phase shifts other than 0 degrees or 90 degrees, while pure resistances cause no phase shift at all.

In Fig. 10-a, a charging current starts to flow into the capacitor as the voltage rises, stops when the voltage reaches its peak, and reverses as the voltage falls. If there were no resistances and thus no power loss, the capacitor would store energy from the inductor part of each cycle, and feed it back the rest of each cycle, the generator acting like a motor. Looking at the waveforms, instantaneous power E x I is positive for half a cycle and negative the other half, so no net power is gained or lost.

Figure 10-b illustrates pure inductance. Any change in inductor current induces an EMF opposing the change, so current lags voltage by 90. Energy is alternately stored in the magnetic field and fed back to the AC source. As in the case of the capacitor, no net power is gained or lost in an ideal inductive circuit with no resistance.

Resistances convert electrical current to heat; Fig. 10-c shows a pure resistance, and Fig. 10-d an RL circuit of under 90 phase shift. In general, power usage for purely sinusoidal circuits is P=E x I x cos(Theta), where E and I are RMS voltage and current, Theta is the relative phase, cos(Theta) is the Power Factor (PF), and E x I is measured in Volt-Amps (VA). Also, at Theta = 0 degrees, PF =1, while at Theta = 90 degrees, PF = O.

Ideal reactive loads don't consume power, but they increase total current drawn from a source, causing extra loss in the resistance of generators, transformers and power lines. Phase shifts are minimized by adding capacitance to balance inductive loads (motors and transformers). As mentioned earlier, the imaginary component of a joint resistive-reactive load corresponds to the reactive part. The fraction of the total power going into the reactive part of the load is measured in Volt-Amps Reactive (VAR), given by VAR =E x I x sin(Theta). Also, at Theta = 0 degrees, VAR = 0 VA, while at Theta = 90 degrees, the VAR is maximized, with no power being resistively dissipated as heat.

Measuring phase shift

Phase shift provides a measure of the time difference between two waveforms, by detecting the time between zero crossings; Fig. 11 gives the general layout of a circuit that can be used to measure a time difference. Each comparator flips when its input crosses 0 volts; their polarities are opposite one another. Comparator B sets the flip-flop, while comparator A resets it, both on positive-going transitions.

If the voltage and current are in phase, the transitions will be 1/2-cycle apart, and the flip-flop will be set 50% of the time. Lagging current delays the set, producing longer low and shorter high intervals, while leading current does the opposite. The filter averages the flip-flop output, producing a voltage proportional to phase shift, and rises or falls indicating lead or lag.

This isn't a complete working circuit, as component values depend on input levels, supply voltages, and desired output. Gain and offset adjustments will be needed for calibration; the output stage is a good place to add them. One or both inputs should be transformer-coupled for safety, but remember that transformers add phase shift. Also, the output represents theta, not sin(theta).

Measuring true AC power

Now that we've covered true RMS measurement and phase shifts, we could consider a computational circuit for
E x I x cos(Theta), but that would be complex, and inaccurate for non- sinusoidal waveforms. It's easier to go back to the basic definition of power; at any given instant, power is P = E x I. Power is found by integrating (averaging) the E x I product over one or more cycles, as shown in Fig. 12-a.

Figure 12-a shows what takes place in an electromechanical wattmeter movement, while Fig. 12-b shows the movement itself. In the meter, one field is created by E, the other by I, and the attractive force between them is proportional to E x I. Filtering is achieved by the inertia of the meter movement plus mechanical damping; otherwise, the pointer would vibrate rapidly with each cycle.

Three types of electronic multipliers are used; log/antilog amplifiers, duty-cycle (pulse-width) modulators, and Hall-effect devices.


Figure 13 shows the first, that performs multiplication by adding logarithms:
log(E x I) = log(E) + log(l)
and then takes the antilog:
antilog[log(E x I)] = E x I

The final result is integrated to find average power.

The object here isn't to discuss log/ antilog amplifiers; that's an article in itself. They all use the log relation between voltage and current in the Base-Emitter (BE) junction of a bipolar transistor. IC multipliers are available that perform the complete log-sum-anti log function internally. Examples include the Analog Devices AD530/630/830 , Burr-Brown MPY100 and 4203, and Raytheon RC4200. Their prime advantages are simplicity, typical frequency response of 400 kHz or higher, accuracy of 0.25-1.0%, and temperature stability of 0.02-0.05%/deg C.

The duty-cycle (pulse-width) modulator

When extreme accuracy is needed at low frequencies such as 50-60 Hz, the duty-cycle (pulse-width) modulator shown in Fig. 14 gives superior performance. At 0 volts in, the pulse-width modulator produces a 50% duty cycle; its output controls an analog switch that alternately connects non- inverted or inverted inputs to the filter. At 50% duty cycle, the signal fed to the filter spends equal times positive and negative, so the filtered output is 0 volts regardless of the current input.

As the input goes positive, the duty cycle exceeds 50%. The switch spends more time in the normal position and less in the inverted, so the filtered output polarity is the same as that of the input current. Similarly, when the input voltage goes negative, the switch spends more time in the inverted position and the output polarity becomes opposite to that of the input current. The higher the voltage, the higher the duty cycle, and the higher the filtered output. The output is proportional to the product of duty cycle x I, or e(t) x i(t).

Unlike the log/antilog circuit, this one isn't available as an off-the-shelf IC, and is more complex to design and build. It's frequency response is also lower, since the duty cycle frequency must be many times higher than that of the inputs. Properly designed, however, it's capable of accuracies lower than 0.1%, limited more by the ability to obtain precise AC calibration sources than the circuitry itself.

Hall effect multipliers

Finally, we'll examine Hall effect multipliers, as shown in Fig. 15. A Hall-effect device uses a shaped and doped semiconductor in a magnetic field. As the holes or electrons flow at right angles to the field, they're deflected in a curved path toward the semiconductor edges, inducing an EMF between two edges proportional to the product of flux density B and current i(t); reversing either reverses the EMF.

In AC power measurement, the power current controls the magnetic field, and the power voltage controls the semiconductor current. The induced EMF is amplified by a differential amplifier, filtered, and scaled. Hall effect devices are very linear, and have a wide bandwidth. Their main drawback is they need temperature compensation to achieve high stability.

Clearly, AC power measurement involves much more than multiplying voltage and current. The techniques shown here make possible precise measurement of RMS voltage/ current, phase shift, and true power in electronic circuits.

Revised 2013 by Larry Gentleman