1. Measurement of Voltages and Currents
Chapter 11
Measurement of Voltages and Currents
Introduction
Sine waves
Square waves
Measuring Voltages and Currents
Analogue Ammeters and Voltmeters
Digital Multimeters
Oscilloscopes

2. 11.1
Introduction
Alternating currents and voltages vary with time and periodically change their direction

3. 11.2
Sine Waves
Sine waves
by far the most important form of alternating quantity
important properties are shown below

4. Instantaneous value
shape of the sine wave is defined by the sine function
y = A sin 
in a voltage waveform
v = Vp sin 

5. 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 2f radians per second
this is the angular frequency  (units are rad/s)
 = 2f

6. 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 2ft
similarly
i = Ip sin t or i = Ip sin 2ft

7. Example – see Example 11.2 in the course text
Determine 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

8. 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

9. 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

10. 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

11. Average value of a sine wave

12. 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

13. 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)

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

15. Form factor for any waveform the form factor is defined as
for a sine wave this gives

16. Peak factor for any waveform the peak factor is defined as
for a sine wave this gives

17. 11.3
Square Waves
Frequency, period, peak value and peak-to-peak value have the same meaning for all repetitive waveforms

18. 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 with respect to the other

19. 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

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

21. Measuring Voltages and Currents
11.4
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

22. Measuring Voltages and Currents
11.4
Measuring Voltages and Currents
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

23. Measuring Voltages and Currents
11.4
Measuring Voltages and Currents
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

24. Analogue Ammeters and Voltmeters
11.5
Analogue Ammeters and Voltmeters
Most modern analogue ammeters are based on moving-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

25. 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

26. 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

27. 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)

28. 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
A typical analogue multimeter

29. 11.6
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)
A simplified block diagram

30. 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

31. Oscilloscopes An oscilloscope displays voltage waveforms 11.7
A simplified block diagram

32. A typical analogue oscilloscope

33. Measurement of phase difference

34. 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.