# Volt-Ampere Measurement Circuit

Quote of the Day

The beginnings and endings of all human undertakings are untidy.

— John Galsworthy

## Introduction

Figure 1: Woodward Circuit for Measuring Power and VAs.

This is a circuit designed by Stephen Woodward that I saw years ago in EDN. I originally was interested in the right-hand side of the circuit, which measures the real power usage of the load. I now have an interest in the left-hand side of the circuit, which measures the Volt-Ampere (VA) usage of the load. I will present an abbreviated analysis of the circuit operation along with some design equations.

• It measures both VA (i.e. apparent power) and real power usage of the load.
• It uses the idea of the differential resistance (another example), which is a nice illustration of the mathematical concept of a differential.
• It has an interesting output circuit for converting an AC waveform into a proportional DC level. I say interesting because it has aspects of a voltage doubler circuit.

My analysis will show that the output voltage is linearly related to the VA usage of the load by Equation 1.

 Eq. 1 $\displaystyle {{V}_{{Output}}}=\frac{{CTR}}{{2\cdot {{V}_{T}}\cdot \eta }}\cdot \frac{{{{R}_{{Sense}}}}}{{{{R}_{{5.1\text{ k }\!\!\Omega\!\!\text{ }}}}+\pi \cdot {{R}_{{100\text{ k }\!\!\Omega\!\!\text{ }}}}}}\cdot {{P}_{{VA}}}$

where

The circuit and my analysis does have some weaknesses:

• There is one potentiometer required to calibrate the VA-portion of the circuit (two pots are required to calibrate the real power-portion of the circuit).

While I use potentiometers in my personal designs, I make every effort not to have them in production designs. Calibration using a mechanical potentiometer is just too much trouble.

• My analysis assumes the optocouplers and transistors are matched.

In real life, optocouplers and transistors vary widely. This is one reason why a potentiometer is needed to calibrate the circuit. While not discussed in this post, there is a potentiometer in the right-hand side of the circuit (i.e. real power measurement) that is used to calibrate-out the optocoupler differences for that function.

I have included links to my LTSpice circuit and Mathcad file in Appendix A.

## Background

### Why Use Volt-Amperes?

The VA usage of a load is simply the product of the load's actual RMS voltage and current. The load's power (P) and VA usage are related by formula $P = VA \cdot PF$, where PF is the power factor, PF = cos(θ),  and θ is the phase shift between the load's voltage and current. While the power and volt-ampere usage of the load are equal for resistive loads, like heaters, the two numbers can be dramatically different for motors and certain lighting loads (example). Knowing both VA and P usage of a load allows one to compute the load's PF.

### Basic Optocoupler Operation

The Wikipedia has an excellent description of an optocoupler (aka optisolator) and I will refer you there for more information on their operation. Figure 2 shows a basic optocoupler circuit. There are numerous uses for optocouplers – I often use them to provide galvanic isolation.

Figure 2: Basic Optocoupler Circuit.

One item not covered is the current transfer ratio of the optocoupler, which is defined as follows (source).

Current Transfer Ratio (CTR)
Current Transfer Ratio (CTR) is the ratio of the phototransistor's collector current compared to the infrared emitting diode (IRED) forward current expressed as a percentage (%).

CTR is not a constant – it varies with the LED's forward current. Figure 3 shows an typical example.

Figure 3: A Typical CTR Example.

For my analysis, I will be assuming that we can use a single, average CTR value. I will assume that a sine wave passed through the CTR function will remain approximately sinusoidal. While a variable CTR clearly makes this circuit nonlinear, the results are close enough that I consider the model useful.

## Analysis

For my simulations, I assumed a 11 Ω load. This is near the maximum power level that the circuit supports. I did not worry about setting the VA calibration potentiometer to an optimal point – I just picked an arbitrary value.

### Reference Circuit

Figure 4 shows a slightly marked-up version of Woodward's original circuit.  The red mark-ups show points in the circuit where I will derive formulas for determining component values.

Figure 4: Circuit Diagram with Analysis Points Marked.

### Functional Overview

The overall circuit function can be described as follows:

• The voltage at Point A is proportional to the AC voltage amplitude.

This a simple half-wave rectifier circuit and we can easily determine a formula for the voltage at Point A.

• The two optocoupler LEDs are biased slightly differently – D3 is driven directly with the rectified voltage and D4 is driven with the rectified voltage minus the voltage drop across the 0.001 Ω sense resistor.

The voltage drop across the sense resistor is proportional to the current draw of the load. The voltage across the sense resistor is just a few millivolts. However, this small voltage is enough to generate a small current difference between the current sourced by Q3 and sunk by Q4. This current difference will be proportional to product of the the AC voltage and current values.

• The current difference output from Q3 and Q4 will be amplified by A2 with a constant diode voltage drop (actually a base-emitter junction) added to the amplifier's output voltage (Point C).

The diode voltage drop will compensate for a diode drop in the output circuit.

• The output circuit (Point D) consists of a simple low-pass filter circuit that is driven by current from A2.

The current from A2 is steered by the BE junctions of Q7 and Q8. The current drawn through the BE junction of Q7 charges the 4.7 µF capacitor on the output of A2. The discharge current through the 4.7 µF capacitor is steered to the output through the BE junction of Q7.

### Voltage at Point A

Figure 5 show my derivation for the voltage at Point A. I also compare my result with an LTSpice simulation. The formula accuracy is reasonable considering the accuracy of the modeling that I am using. I use the fact that the average value of a half-wave rectified sinusoid equals the peak of the signal divided by π.

Figure 5: Derivation of Formula for the Voltage At Point A.

### Current Through Capacitor at Point B

Figure 6 shows my expression for the current through the capacitor at Point B. Here is where I make of the differential resistance of the diodes gm = ID /(VT · η).

In Figure 6, the variable PVA refers to the measured VA value.

Figure 6: Current Through Capacitor at Point B.

### Voltage at Point C

I derive an expression for the voltage from amplifier A2 (Point C). The agreement with the LTSpice simulation is reasonable.

Figure 7: Output Voltage from Amplifier A2.

### Voltage at Point D

Figure 8 show my final expression, which is for the voltage at Point D. This is the output voltage from this circuit. My expression is giving me good agreement with the LTSpice simulation.

The key result is the formula for the circuit's output voltage (VPointD in Figure 8). This formula shows that the output voltage is linearly related to the VA usage of the load.

Figure 8: Expression for the Voltage at Point D.

Note the level of ripple on the output.

## Conclusion

I have presented an analysis of the VA measurement portion of the circuit that provides sufficient detail for a designer to select component values tailored to their specific application.

LTSpice Circuit