Week 2: Measurement of Voltages and Currents

Wk 2

Measurement of Voltages and Currents

Introduction

Sine waves

Square waves

Measuring Voltages and Currents

Analogue Ammeters and Voltmeters

Digital Multimeters

Oscilloscopes

Introduction

Alternating currents and voltages vary with time and periodically change their direction.
Equation of a Sine Wave
– The angular frequency $\omega$ can be thought of as the rate at which the angle of the sine wave changes.

– at any time.

$\bbox[lightblue,6pt]{\theta = \omega t}$

– therefore
$\bbox[lightblue,6pt]{V = V_p \; sin \; \omega t}$ or $\bbox[lightblue,6pt]{V = V_p \; sin \; 2 \pi ft}$
– similarly
$\bbox[lightblue,6pt]{\; i = I _p \; sin \; \omega t}$ or $\bbox[lightblue,6pt]{i = I _p \; sin \; 2 \pi ft}$

Example - see Example 2.2 in the course text.Determine the equation of the following voltage signal.

– From the diagram:

Period is $50$ms = $0.05$ s

Thus $f = 1/ \text{T} = 1/0.05 = 20$ Hz

Peak voltage is $10$ V

Therefore,
\begin{align*} v =& V_p \text{ sin } 2\pi ft \\ =& 10 \text{ sin } 2\pi 20t \\ =& 10 \text{ sin } 126t \end{align*}

Sine Waves

🎥 Watch an Introduction and sine waves video - Adobe Flash video

Sine Waves

– by far the most important form of alternating quantity.

📷 important properties of a sine wave are shown in this diagram.

Instantaneous Value
– shape of the sine wave is defined by the sine function $ y = A \text{ sin } \theta $

– in a voltage waveform.

Angular Frequency

– frequency $f$ (in hertz) is a measure of the number of cycles per second

– each cycle consists of $2 \pi $ radians

– therefore there will be $2 \pi f$ radians per second

– this is the angular frequency $\omega$ (units are rad/s).

$\bbox[lightblue,6pt]{\omega = 2\pi f}$
Average value of a sine wave
– average value over one (or more) cycles is clearly zero.

– however, it is often useful to know the average magnitude of the waveform independent of its polarity.

we can think of this as the average value over half a cycle…

… or as the average value of the rectified signal

\begin{align*} \large V_{av} &= \frac{\large 1}{\large \pi} \large \int _{\large _0}^{\large \pi}\large V _p \; sin \theta \; d\theta \\\\ &= \frac {\large V_p}{\large \pi} [-cos\theta] _{\large _0}^{\large ^\pi} \\\\ &= \frac {\large 2 V_p}{\large \pi} = 0.637 \times \large V_p \end{align*}

Phase angles
– The expressions given above assume the angle of the sine wave is zero at $t = 0$

– If this is not the case the expression is modified by adding the angle at $t = 0$

Phase difference
– Two waveforms of the same frequency may have a constant phase difference.

We say that one is phase-shifted with respect to the other.
Form factor
– for any waveform the form factor is defined as

$\text{Form factor} = \frac{\large \text{r.m.s. value} }{ \large\text{average value}} $

– for a sine wave this gives

$\text{Form factor} = \frac{\large 0.707 \; V _p }{\large 0.637 \; V_p} = 1.11$

Peak factor
– for any waveform the peak factor is defined as

$\text{Peak factor} = \frac{\large \text{peak value} }{\large \text{r.m.s. value}} $

– for a sine wave this gives

$\text{Peak factor} = \frac{\large V _p }{\large 0.707 \; V_p} = 1.414$

r.m.s. value of a sine wave
– the instantaneous power (p) in a resistor is given by.

$p = \large \frac{v^2}{R} \\$

– therefore the average power is given by.

$P _{\large av} = \frac{\large [\text{average (or mean) of} \; v^2]}{\large R} = \frac {\overline{\large v^2}}{\large R} \\$
– where $\overline{ v^2}$ is the mean-square voltage.

While the mean-square voltage is useful, more often we use the square root of this quantity, namely the root-mean-square voltage $V$_rms
– where $V$_rms = $\sqrt{\overline{\large v^2}}$

– we can also define I_rms = $\sqrt{\overline{\large i^2}}$

– it is relatively easy to show that (see text for analysis).

\begin{align*} \large V_{rms} &= \frac{\large 1}{\sqrt{2}} \times \large V _p \normalsize = 0.707 \times \large V_p \end{align*}

\begin{align*} \large I_{rms} &= \frac{\large 1}{\sqrt{2}} \times \large I _p \normalsize = 0.707 \times \large I_p \end{align*}

r.m.s. values are useful because their relationship to average power is similar to the corresponding DC values.

\begin{align*} \large P_{av} &= \large V_{rms} \large \; I _{rms} \end{align*}

\begin{align*} \large P_{av} &= \frac{\large V_{rms} \;^2 }{R} \end{align*}

\begin{align*} \large P_{av} &= \large I_{rms} \large ^2 R \end{align*}
Average and r.m.s. values

– the average value of a symmetrical waveform is its average value over the positive half-cycle

– thus the average value of a symmetrical square wave is equal to its peak value \begin{align*}\large V _{av} = V_p \end{align*}

– similarly, since the instantaneous value of a square wave is either its peak positive or peak negative value, the square of this is the peak value squared, and
\begin{align*}\large V _{rms} = V_p \end{align*}

Form factor and peak factor
– from the earlier definitions, for a square wave.

$\text{Form factor} = \frac{\large \text{r.m.s. value} }{\large \text{average value}} = \frac{\large V_p}{\large V_p} = 1.0$

$\text{Peak factor} = \frac{\large \text{peak value} }{\large \text{r.m.s. value}} = \frac{\large V_p}{\large V_p} = 1.0$

Square Waves

Frequency, period, peak value and peak-to-peak value have the same meaning for all repetitive waveforms

Phase angle

– we can divide the period into 360° or 2π radians

– useful in defining phase relationship between signals

– 📷 in the waveforms shown here, B lags A by 90°

– we could alternatively give the time delay of one with respect to the other.
Loading effects – voltage measurement
– our measuring instrument will have an effective resistance ($R_M$)

– when measuring voltage we connect a resistance in parallel with the component concerned which changes the resistance in the circuit and therefore changes the voltage we are trying to measure

– this effect is known as loading

Measuring Voltages and Currents

🎥 Watch a video on triangular waves - Adobe Flash video

Measuring voltage and current in a circuit
– when measuring voltage we connect across the component.

– when measuring current we connect in series with the component.
Measuring alternating quantities

– Moving coil meters respond to both positive and negative voltages, each producing deflections in opposite directions.

– a symmetrical alternating waveform will produce zero deflection (the mean value of the waveform).

– therefore we use a rectifier to produce a unidirectional signal.

– meter then displays the average value of the waveform.

– meters are often calibrated to directly display r.m.s. of sine waves.

all readings are multiplied by 1.11 - the form factor for a sine wave.

– as a result waveforms of other forms will give incorrect readings.

for example when measuring a square wave (for which the form factor is 1.0, the meter will read 11% too high).

Analogue multimeters
– general purpose instruments use a combination of switches and resistors to give a number of voltage and current ranges.

– a rectifier allows the measurement of AC voltage and currents

– additional circuitry permits resistance measurement

– very versatile but relatively low input resistance on voltage ranges produces considerable loading in some situations

– 📷 View a typical analogue multimeter.

Analogue Ammeters and Voltmeters

Most modern analogue ammeters are based on moving coil meters
– see Chapter 13 of textbook.

Meters are characterised by their full-scale deflection (f.s.d.) and their effective resistance ($R_M$)
– typical meters produce a f.s.d. for a current of $50 \mu \text{A} - 1$ mA

– typical meters have an $R_M$ between a few ohms and a few kilohms

Measuring direct currents using a moving coil meter

– 📷 use a shunt resistor to adjust sensitivity (can be seen here)

– see Example 2.5 in the set text for numerical calculations.

Measuring direct voltages using a moving coil meter
– 📷 use a series resistor to adjust sensitivity

– see Example 2.6 in the set text for numerical calculations.
Digital oscilloscope

Digital oscilloscopes use an analogue-to-digital converter (ADC) and appropriate processing.
– 📷 analogue-to-digital converter - see a simplified block diagram here.

A typical digital oscilloscope

Measurement of phase difference

Digital Multimeters

Digital multimeters (DMMs) are often (inaccurately) referred to as digital voltmeters or DVMs.
– 📷 at their heart is an analogue-to-digital converter (ADC) - see here.

Measurement of voltage, current and resistance is achieved using appropriate circuits to produce a voltage proportional to the quantity to be measured.
– in simple DMMs alternating signals are rectified as in analogue multimeters to give its average value which is multiplied by 1.11 to directly display the r.m.s. value of sine waves

– more sophisticated devices use a true r.m.s. converter which accurately produced a voltage proportional to the r.m.s. value of an input waveform

– 📷 View a typical digital multimeter.

Oscilloscopes

An oscilloscope displays voltage waveforms
– 📷 See oscilloscope diagram here.

A typical analogue oscilloscope
Further Study

🎥 Watch a further study video on power measurement with alternating signals - Adobe Flash video

The Further Study section at the end of Chapter 2 looks at the measurement of different forms of alternating waveform.

Have a look at the problem and then watch the video to see how you did.

Key Points

The magnitude of an alternating waveform can be described by its peak, peak-to-peak, average or r.m.s. value.

The root-mean-square value of a waveform is the value that will produce the same power as an equivalent direct quantity.

Simple analogue ammeter and voltmeters are based on moving coil meters.

Digital multimeters are easy to use and offer high accuracy.

Oscilloscopes display the waveform of a signal and allow quantities such as phase to be measured.
2.11 What are the frequency and peak amplitude of the waveform described by the following equation?
$v = 25 \text{ sin } 471 t$

2.12 Determine the equation of the following voltage signal.

2.13 A sine wave has a peak value of $10$. What is its average value?

2.14 A sinusoidal current signal has an average value of $5$ A. What is its peak value?

2.15 Explain what is meant by the mean‐square value of an alternating waveform. How is this related to the r.m.s. value?

2.16 Why is the r.m.s. value a more useful quantity than the average value?

2.17 A sinusoidal voltage signal of $10$ V peak is applied across a resistor of $25 \Omega$. What power is dissipated in the resistor?

⬇Download chapter 2 tutorial

Exercises

2.1 Sketch three common forms of alternating waveform.

2.2 A sine wave has a period of $10$ s. What is its frequency (in hertz)?

2.3 A square wave has a frequency of $25$ Hz. What is its period?

2.4 A triangular wave (see textbook Figure 2.1) has a peak amplitude of 2.5 V. What is its peak‐to-peak amplitude?

2.5 What is the peak‐to‐peak current of the waveform described by the following equation?
$i = 10 \text{ sin } \theta$

2.6 A signal has a frequency of $10$ Hz. What is its angular frequency?

2.7 A signal has an angular frequency of $157$ rad/s. What is its frequency in hertz?

2.8 Determine the peak voltage, the peak‐to‐peak voltage, the frequency (in hertz) and the angular frequency (in rad/s) of the following waveform.

2.9 Write an equation to describe a voltage waveform with an amplitude of $5$ V peak and a frequency of $50$ Hz.

2.10 Write an equation to describe a current waveform with an amplitude of $16$ A peak to peak and an angular frequency of $150$ rad/s.
2.27 How do some digital multimeters overcome the problem associated with different alternating waveforms having different form factors?

2.28 Explain briefly how an analogue oscilloscope displays the amplitude of a time-varying signal.

2.29 How is an analogue oscilloscope able to display two waveforms simultaneously?

2.30 What is the difference between the ALT and CHOP modes on an analogue oscilloscope?

2.31 What is the function of the trigger circuitry in an oscilloscope?

2.32 A sinusoidal waveform is displayed on an oscilloscope and has a peak-to-peak amplitude of $15$ V. At the same time, the signal is measured on an analogue multimeter that is set to measure alternating voltages. What value would you expect to be displayed on the multimeter?

2.33 Comment on the relative accuracies of the two measurement methods outlined in the last exercise.

2.34 What is the phase difference between waveforms A and B in the following oscilloscope display? Which waveform is leading and which lagging?

⬇ Tutorial Solutions

Exercises (cont.)

2.18 A sinusoidal voltage signal of $10$ V r.m.s. is applied across a resistor of $25 \Omega$. What power is dissipated in the resistor?

2.19 A sinusoidal waveform with an average voltage of 6 V is measured by an analogue multimeter. What voltage will be displayed?

2.20 A square‐wave voltage signal has a peak amplitude of 5 V. What is its average value?

2.21 A square wave of $5$ V peak is applied across a $25 \Omega$ resistor. What will be the power dissipated in the resistor?

2.22 A moving‐coil meter produces a full‐scale deflection for a current of $50 \mu$A and has a resistance of $10 \Omega$. Select a shunt resistor to turn this device into an ammeter with an f.s.d. of $250$ mA.

2.23 A moving‐coil meter produces a full‐scale deflection for a current of 50 μA and has a resistance of $10 \Omega$. Select a series resistor to turn this device into a volt‐ meter with an f.s.d. of $10$ V.

2.24 What percentage error is produced if we measure the voltage of a square wave using an analogue multimeter that has been calibrated to display the r.m.s. value of a sine wave?

2.25 A square wave of $10$ V peak is connected to an analogue multimeter that is set to measure alternating voltages. What voltage reading will this show?

2.26 Describe the basic operation of a digital multimeter.