Chap11.ppt

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 11.1

Measurement of Voltages and Currents

Introduction Sine waves Square waves Measuring Voltages and Currents Analogue Ammeters and Voltmeters Digital Multimeters Oscilloscopes

Chapter 11


Introduction

Alternating currents and voltages vary with time and periodically change their direction

11.1


Sine Waves

Sine waves– by far the most important form of alternating quantity

important properties are shown below

11.2


Instantaneous value– shape of the sine wave is defined by the sine function

y = A sin – in a voltage waveform

v = Vp sin


Angular frequency– frequency f (in hertz) is a measure of the number of

cycles per second– each cycle consists of 2 radians– therefore there will be 2f radians per second– this is the angular frequency (units are rad/s)

= 2f


Equation of a sine wave– the angular frequency can be thought of as the rate at

which the angle of the sine wave changes– at any time

= t– therefore

v = Vp sin t or v = Vp sin 2ft

– similarly

i = Ip sin t or i = Ip sin 2ft


Example – see Example 11.2 in the course textDetermine the equation of the following voltage signal.

From diagram: Period is 50 ms = 0.05 s Thus f = 1/T =1/0.05 = 20 Hz Peak voltage is 10 V Therefore

tt

ftpVv

126sin10

202sin10

2sin


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


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 valueof the rectified signal

pp

p

pav

VV

V

VV

637.02

cos

dsin1

0

0

θ


Average value of a sine wave


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

– therefore the average power is given by

– where is the mean-square voltage

R

vp

2

2v

R

v

R

vavP

2] of mean) (or average[ 2


While the mean-square voltage is useful, more often we use the square root of this quantity, namely the root-mean-square voltage Vrms

– where Vrms =

– we can also define Irms =

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

2v

2i

pprms VVV 707.02

1pprms III 707.0

21


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

rmsrmsavIVP

RIPrmsav

2

RV

P rmsav

2


Form factor– for any waveform the form factor is defined as

– for a sine wave this gives

value averagevalue r.m.s.factor Form

11.1 0.637

0.707factor Form

pVpV


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

– for a sine wave this gives

value r.m.s.value peakfactor Peak

414.1 0.707factor Peak pV

pV


Square Waves

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

11.3


Phase angle– we can divide the period

into 360 or 2 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 withrespect to the other


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

– 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

pVavV

pVrmsV


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

0.1value averagevalue r.m.s.factor Form

pVpV

0.1value r.m.s.value peakfactor Peak

pVpV


Measuring Voltages and Currents

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

11.4



Loading effects – voltage measurement– our measuring instrument will have

an effective resistance (RM)

– 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

11.4



Loading effects – current measurement– our measuring instrument will have an

effective resistance (RM) – when measuring current we connect a

resistance in series with the component concerned which again changes the resistance in the circuit and therefore changes the current we are trying to measure

– this is again a loading effect

11.4


Analogue Ammeters and Voltmeters

Most modern analogueammeters are based onmoving-coil meters– see Chapter 4 of textbook

Meters are characterised by their full-scale deflection (f.s.d.) and their effective resistance (RM)– typical meters produce a f.s.d. for a current of 50 A – 1 mA

– typical meters have an RM between a few ohms and a few kilohms

11.5


Measuring direct currents using a moving coil meter– use a shunt resistor

to adjust sensitivity– see Example 11.5 in

set text for numerical calculations


Measuring direct voltages using a moving coil meter– use a series resistor

to adjust sensitivity– see Example 11.6 in

set text for numerical calculations


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 resistorsto 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

A typical analogue multimeter


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)

11.6

A simplified block diagram


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 A typical digital multimeter


Oscilloscopes

An oscilloscope displays voltage waveforms

11.7

A simplified block diagram


A typical analogue oscilloscope


Measurement of phase difference


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.

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