Performing Engineering Operations. Basic AC Theory Student Handout And Worksheet

Transcription

1 Basic AC Theory Student Handout And Worksheet Legh Richardson Page 1 04/04/2011

2 Introduction to AC Voltage and Current DC Voltage and Current flows in one direction and is either positive or negative in relation to a relatively fixed zero point. The current flow is in one direction only, from Positive to negative (Conventional Current theory) The supply is usually from a Chemical source (Batteries) or a Power supply either where the AC voltage has been dropped down by a Transformer and then rectified or by SMPS where the AC voltage is rectified pulsed with a high frequency oscillator and then rectified and smoothed to provide the required voltage + Amplitude DC current or voltage Time - AC voltage Alternating Voltage and current. A voltage which alternates or swings between positive and negative. If the voltage changes then the current flow will also change; In fact, in purely resistive circuits it tracks the voltage ( in phase ) All ac voltages have a frequency of alternation. This is known as the supply frequency or signal frequency. Measurement of this alternation is over a time period of 1 second. So the number of Cycles that an alternating current has in one second is known as its frequency. Its unit is the Hertz ( Hz ) and is cycles per second This can be shown using the formula:f 1 to find the time given the frequency we T just transform the formula f 1 T f 1 T 1 T T f T f f If the supply frequency is at 50Hz then the periodic time for each cycle will be: 1 1 T 0. 02s or 20mS if you think about it, there will be cycles every f 50 second. AC voltages and currents are produced by alternators and generators Legh Richardson Page 2 04/04/2011

3 The advantages of ac over DC voltage generation are many. (i) ac generators are more robust, financially cheaper to maintain and run; costs less to produce and can deliver higher voltages than a DC generators (ii) The power loss in transmission lines depends on the square of the current carried and the resistance of the copper or aluminium conductors P I 2. R So the higher the voltage the lower the current flow to deliver the same amount of power, and the smaller the cross section of cable, there are then lower power losses. Transformers are used to raise the voltage so that the electricity can be transmitted efficiently. Transformers will not transform using DC voltages. (iii) 3 - Phase induction motors are cheap, robust, and easily maintained (iv) Energy meters are much simpler and therefore much cheaper for ac supplies than DC supplies. (v) Discharge lamps operate more efficiently from AC supplies, particularly starting a discharge lamp, due to the very large back emfs generated by the choke, although filament lamps work equally as well in both DC and AC supplies (vi) DC supplies are subject to severe corrosion due to electrolytic action which is not so present in ac systems (vii) If a DC supply is required it is more cost effective to rectify AC voltage to DC voltage rather than using a DC supply. The amplitude of an electrical signal is always measured vertically and the time or periodic length of each cycle is measured horizontally. Legh Richardson Page 3 04/04/2011

4 The instantaneous value of any signal is the vertical value of amplitude at any instant in time. For any alternating wave, it could be zero, 2 times in one cycle or maximum either positive or negative once in each cycle. The peak value is the maximum amplitude a signal or wave can obtain in any cycle. The average or mean value is the average amplitude of the current or voltage over 1 cycle. If both halves of the cycle were averaged out then the resultant value would be zero, since for a sine wave both halves are identical and opposite in values. To find the average value or amplitude of a sine wave we take one half of the cycle only. The average or mean value appears not to be so important as other values but we will see that these other values can easily be obtained from the mean value. Definition of a sine wave: The wave form generated from a single turn coil rotating at a constant speed ( Angular Velocity ) in a uniform magnetic field. The relationship of the maximum voltage or current at any instant to the instantaneous voltage or current is:e. inst E.max Sin or I. inst I.max Sin The emf or current produced is a sinusoidal output. AC generators are designed to produce as near to a sinusoidal output as possible. Since the voltage and current is constantly varying with time the instantaneous values will be correct for any given instant. If we were to plot the position of each instantaneous value over a one cycle we would plot a circle with a circumference equal to twice the maximum or peak value of the voltage or current The average value can be found by adding all the instantaneous values together, and dividing by the number of values used. Example: Legh Richardson Page 4 04/04/2011

5 If the peak voltage of a sine wave is 110v and measurements were taken every 30 degrees then we would have instantaneous values corresponding to a change in a fixed time period over half a cycle of an ac sinusoidal wave of peak 110v Degrees V.max * Sin0 V. max * Sin0 V.Inst * Sin0 110 * * Sin * * Sin * * Sin * * Sin * * Sin * V. av V 6 6 A more classical method of observing a sine wave form is to measure the changes of angle not in degrees but in Radians. There are.radians in 180 and 2.Radians in 360. If there are.radians in 180 : 1 Rad = 180/ = / / /6 /3 /2 2/3 5/ / /2 This idea may seam obscure, however, we use when we look at frequency and calculate frequencies. Capacitors and Inductors are frequency dependant devices ( 2 fl or 1/2 fc ) Note: The represents the frequency of rotation which = 2 f. Since the heat dissipated is proportional to its square ( P I 2 R ), the average value of an alternating current is not the same as the direct current which produces the same heat or does the same work in the same time. The equivalent AC current to a DC current value we most use to describe ac systems is called the Effective value or the Root Legh Richardson Page 5 04/04/2011

6 Mean Square Value ( rms ) of the system. It is calculated just as it sounds The square Root of the Mean of the Squares of each instantaneous values Looking at the above example we can say that the effective DC equivalent or rms value of the 110V peak Sine wave is: V. rms V 6 6 To check this we know that the rms value of any sinusoidal wave is of its peak value: V. rms V.max V Amplitude V.peak Sine Wave V.rms Effective DC equivilent Signal Time Note if we were to use the negative half cycle we would get the same answer. It does not matter which way the current is flowing in a circuit it will always produce the same positive Heating effect as a DC circuit. Ohms Law will now be the same for ac systems as it is for DC systems as long as we use the rms voltage and current of an ac system. Another way of looking at this is: V.max V.max V. rms V.max and V.max V. rms 2 or I.max I.max I. rms I.max Conditions: (i) Unless otherwise stated an AC quantity is assumed to be sinusoidal (ii) Unless otherwise stated rms values are always used and stated and read from an instrument scale. ( mains = 240V peak to peak = V ) Legh Richardson Page 6 04/04/2011

7 Form factor of ac wave forms: Form factor indicates the shape of the wave form or the 'Peakiness' of a wave form and is determined by: Form Factor rms value Average Value So different wave forms have different form factors: A pure Sine wave has a form factor of 1.11 A pure triangular wave form has a form factor of 1.15 A pure square wave has a form factor of 1.0 Average value of sine wave = * Maximum Amplitude rms Value of a sine wave = * Maximum amplitude Form Factor of a sine wave = / = 1.11 Peak value of a sine wave = rms value * AC waveforms and Periodic Time The waveform below shows one periodic cycle of an example of either voltage or current. Amplitude Vp V rms V ave Waveform ( V or I ) time Periodic time ( 1 Cycle ) The horizontal line represents time. Time and Frequency are one of the same thing. A waveform can be expressed in the time it takes to reproduce itself. This is known as its periodic time Legh Richardson Page 7 04/04/2011

8 Frequency expresses the number of times a waveform reproduces itself in one second The formula is: F 1. Where F = the Frequency and T = the periodic Time T Note if this formula is transposed it shows us: are reciprocals of each other. T 1 where the time and the frequency F Examples 1/ The periodic time of a waveform is 20mS. What is its frequency? 2/ The frequency of a waveform is 50Hz what is its periodic time? 3/ A sinusoidal waveform has a frequency of 1kHz. What is its Periodic time? 4/ A fully rectified sinusoidal waveform has a periodic time of 10mS. What is its frequency? The amplitude of an alternating signal has a peak value. Its peak value is the highest it can reach before it changes direction and falls to zero. The waveform can peak in both a positive and a negative direction. Notes: 1. The rms of an electrical waveform is equivalent to the same DC heating effect 2. The average value of an electrical waveform does not have a specific usefulness unless its used to compare as a ratio with the rms value, then it tells us how peakiness of the waveform is. This is important when evaluating electrical waveform and their effect on other systems such as electrical noise 3. The peak value is the highest, or lowest that a waveform will reach in amplitude its importance becomes apparent when using capacitors which rely on the peak rating values. Hence a 250V capacitor will blow up on a 240vAC supply. Legh Richardson Page 8 04/04/2011

9 1 Cycle 2 Cycles 3 Cycles The diagram above shows 3 cycles of an ac sine wave. Most questions relating to wave forms will require that you are able to draw more than one cycle, and label the amplitude, time base, the average value, the rms value and sometimes the form factor. Questions: 1/ An alternating voltage completes one cycle in 1ms. What is its frequency? 2/ What is the periodic time of an ac system with a frequency of 60Hz? 3/ An alternating sinusoidal wave has a maximum peak voltage of 568V.It completes one half cycle in seconds. Sketch 2 complete cycles of this wave indicating: a/ The peaks of the voltage b/ The time period of the wave c/ The supply frequency d/ The average of the voltage values e/ The rms voltage f/ the form factor 4/ An alternating voltage is triangular in shape, rising at a constant rate to a maximum of 300v in 0.01s, and then falling to zero at a constant rate in 0.005s. The negative half cycle is identical in shape to the positive half cycle. Calculate: a/ the supply frequency b/ the average voltage c/ the rms voltage d/ the form factor 4/ A voltage is represented by a sine wave, and has a maximum of 200v. Calculate its rms and average or mean Value. Legh Richardson Page 9 04/04/2011

10 5/ The positive half cycle of an alternating current has the following values at 1ms intervals: t (ms) I (A) t (ms) I (A) Plot this curve on suitable graph paper and hence calculate form the graph: a/ the supply frequency b/ the mean value of the current c/ the effective value of the current d/ the form factor e/ the instantaneous value of current after 27ms. 6/ A sinusoidal current has an effective value of 10A. Calculate its average and maximum values. 7/ Draw two complete cycles of a square wave of 10v peak to peak, illustrate your answer showing: a/ the positive and negative peak voltages b/ the mean and rms values of the voltage Legh Richardson Page 10 04/04/2011