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

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

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

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

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

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

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 asthe average value over half a cycle… • … or as the average valueof the rectified signal

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

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)

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

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

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

11.3 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 2 radians • useful in defining phase relationship between signals • in the waveforms shownhere, B lags A by 90 • we could alternatively givethe 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

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

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

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

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

11.5 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

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

11.6 A simplified block diagram 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)

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

11.7 Oscilloscopes • An oscilloscope displays voltage waveforms 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.

Measurement of Voltages and Currents