Steve Goddard
Contents Topic DC Electrical Principles Complex Waveforms Non-Resonant and Resonant Single Phase R L C Circuits Transformers Bibliography
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Steve Goddard
Engineering Science – Assignment 3 DC And AC Theory DC Electrical Principles 1. With reference to the circuit below: R 1
A
R 2
B
C
R 3
+
E
D F
R 5
E = 300 V ,
R1 = 20 Ω,
E
R 4
R 2 = 30 Ω,
R3 = 40 Ω,
R 4 =10 Ω
R5 = 50 Ω
1.1 Draw a well-labeled diagram showing all voltages and currents
R 1
A
R 2
B
C
R 3
+
E
D F
R 5
E
R 4
1.2 Calculate the total resistance of the circuit R1+R2+R3+R4+R5 30+20+40+10+50 = 150 Ω 1.3 Calculate the total current from the battery
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Steve Goddard
Using Ohm's Law
I=
E 300 v =2A = RT 150 Ω
(To check total resistance = R =
E 300 v = = 150 Ω) I 2
1.4 Calculate the voltages across R1, R2, R3, R4 and R5
R1 →V1 = I × R1 = 2 A × 20 Ω = 40v R2 → V2 = I × R 2 = 2 A × 30Ω = 60v R3 →V3 = I × R3 = 2 A × 40 Ω = 80 v
R4 →V 4 = I × R 4 = 2 A × 10Ω = 20v R5 →V5 = I × R5 = 2 A × 50 Ω =100 v 300 v
1.5 Calculate the potential differences between the following points: AD, BE, CE & DF AD = V1 + V 2 + V3 = 40 + 60 + 80 = 180 v BE = V2 + V3 + V4 = 60 + 80 + 20 = 160 v CE = V3 +V4 = 80 + 20 = 100 v DF = V4 +V5 = 100 + 20 = 120 v 2. With reference to the circuit shown below: R 1 2 8 + 10v
R 2 2 0
-
R 3 3 0
R 4 1 0
2.1 Draw a well-labeled diagram showing all voltages and currents
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Steve Goddard R 1 2 8 + 10v
R 3 3 0
R 2 2 0
-
R 4 1 0 2.2 Calculate the total circuit resistance
Total resistance = 28 + 10 + 12 = 50 Ω
1 To calculate 12 I did this:
Rt
=1
30
+1
20
Rt = 12Ω
2.3 Calculate the total current from the battery Using Ohm's Law
I=
V 10 v = 0 .2 A = RT 50 Ω
(To check total resistance = R =
V 10v = = 50 Ω ) I 0.2
2.4 Calculate the current in R2
R3 iR2 = × i1 R 2 + R3 30 Ω iR2 = × 0.2 20 Ω + 30 Ω i R 2 = 0.12 Amps 2.5 Calculate the current in R3
R2 i R3 = × i1 R3 + R 2 20 Ω i R3 = × 0.2 30 Ω + 20 Ω i R 3 = 0.08 A
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Steve Goddard
Complex Waveforms You will need to carry out some research and it is suggested that you read Electrical Circuit Theory and Technology By Bird. In particular, see pages 252 and 470, 631 and 666 onwards. However there are other sources that you may wish to use including the web and Tooley and Dingle. You should list any references in a bibliography. 3. Explain how complex waveforms are produced from sinusoidal waveforms Sine waves can be mixed with DC signals, or with other sine waves to produce new waveforms. Here is one example of a complex waveform:
'Complex' doesn't mean difficult to understand. A waveform like this can be thought of as consisting of a DC component with a superimposed AC component. More dramatic results are obtained by mixing a sine wave of a particular frequency with exact multiples of the same frequency, in other words, by adding harmonics to the fundamental frequency. The V/t graphs below show what happens when a sine wave is mixed with its 3rd harmonic (3 times the fundamental frequency) at reduced amplitude, and subsequently with its 5th, 7th and 9th harmonics
As you can see, as more odd harmonics are added, the waveform begins to look more and more like a square wave. This surprising result illustrates a general principle first formulated by the French mathematician Joseph Fourier, namely that any complex waveform can be built up from a pure sine
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Steve Goddard waves plus particular harmonics of the fundamental frequency. Square waves, triangular waves and saw tooth waves can all be produced in this way. 4. Synthesise the following complex waveforms graphically using a spreadsheet:
v = 100 Sin ωt + 30 Sin 3ωt Plot a minimum of 100 points at intervals of
Periodic time ( t ) seconds 100
4.1 Produce a print out of the graph and data See Print out Equations used in the spreadsheet are: x = 100*sin(2*PI()*50*0.001*A2) y = 30*sin(2*PI()*150*0.001*A2) z = B2 + C2 4.2 Describe how electrical and electronic devices produce complex waveforms Complex waveforms can be produced through electrical and electronic generators. Otherwise known as an oscillator. An oscillator can be thought of as an amplifier that provides itself (through feedback) with an input signal. By definition, it is a non rotating device for producing alternating current, the output frequency of which is determined by the characteristics of the device. The primary purpose of an oscillator is to generate a given waveform at a constant peak amplitude and specific frequency and to maintain this waveform within certain limits of amplitude and frequency. An oscillator must provide amplification. Amplification of signal power occurs from input to output. In an oscillator, a portion of the output is fed back to sustain the input. Enough power must be fed back to the input circuit for the oscillator to drive itself, as does a signal generator. Wave generators can be classified into two broad categories according to their output wave shapes, Sinusoidal and Non - sinusoidal. Non sinusoidal oscillators generate complex waveforms, such as square, rectangular, trigger, saw tooth, or trapezoidal. Because their outputs are generally characterised by a sudden change, or relaxation, they are often referred to as Relaxation Oscillators. The signal frequency of these oscillators is usually governed by the charge or discharge time of a capacitor in series with a resistor. Some types, however, contain inductors that affect the output frequency. Thus, like sinusoidal oscillators, both RC and LC networks are used for determining the frequency of oscillation. 4.3 Describe the effects of complex waveforms on electrical and electronic systems. HARMONIC OVERLOADING OF CAPACITORS The impedance of a circuit dictates the current flow in that circuit. As the supply impedance is generally considered to be inductive, the network impedance increases with frequency while the impedance of a capacitor decreases. This encourages a greater proportion of the currents circulating at frequencies above the fundamental supply frequency to be absorbed by the capacitor, and all equipment associated with the capacitor. In certain circumstances such currents can exceed the value of the fundamental (50Hz) capacitor current. These currents in turn cause increased voltage to be applied across the dielectric of the capacitor. The harmonic voltage due to each
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Steve Goddard harmonic current added arithmetically to the fundamental voltage dictates the voltage stress to be sustained by the capacitor dielectric and for which the capacitor must be designed. Capacitors of the correct dielectric voltage stress must always be used in conditions of harmonic distortion to avoid premature failure. HARMONIC RESONANCE As frequency varies, so reactance varies and a point can be reached when the capacitor reactance and the supply reactance are equal. This point is known as the circuit or selective resonant frequency. Whenever power factor correction is applied to a distribution network, bringing together capacitance and inductance, there will always be a frequency at which the capacitors are in parallel resonance with the supply. If this condition occurs at, or close to, one of the harmonics generated by any solid state control equipment, then large harmonic currents can circulate between the supply network and the capacitor equipment, limited only by the damping resistance in the circuit. Such currents will add to the harmonic voltage disturbance in the network causing an increased voltage distortion. This results in an unacceptably high voltage across the capacitor dielectric coupled with an excessive current through all the capacitor ancillary components. The most common order of harmonics are 5th, 7th, 11th and 13th but resonance can occur at any frequency.
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Steve Goddard
Non-Resonant and Resonant Single Phase R L C Circuits 5. A coil has a resistance of 40 Ω, and inductive reactance of 75 Ω. The current in the coil is 1.70<0°A. 5.1 Draw a well labelled diagram of the circuit.
5. 2 Calculate the value of the supply voltage in POLAR form If
I=
VR then: R
And If I =
VR = 1.7 × 40 = 68
VL then: VL =1.7 ×75 =127 .5 XL
So : V 2 = 68 2 +127 .5 2 V 2 = 20880 .25 V = 144 .5 Converting to Polar form
127 .5 68 −1 tan (1.875 ) =θ = 61 .9275 ° tan θ =
5.3 Calculate the p.d across the 40 Ω resistance in POLAR form 0°×40 ∠ 0° =68 ∠ 0°V V = IR =1.70 ∠ 5.4 Calculate the p.d across the inductive part of the COIL in POLAR form V L = i. jX
L
=1.70 ∠0°× j 75 =1.70 ×75 ∠90 ° =127 .5∠90 °V
6. A SERIES R-L-C circuit has a resonant frequency of 1kHz and, at resonance, a Q-factor of 20. The impedance of the circuit at resonance is 100 Ω 6.1 Draw a well labelled diagram and determine
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Steve Goddard
6.2 The value of inductance
Q=
2πf 0 L 2π ×1000 × L = 20 = R 100
∴L =
2000 = 0.318 H 2π ×1000
6.3 The value of capacitance
Q=
1 1 = 20 = 2πf 0 RC 2π ×1000 ×100 × C
∴C =
1 1 = 2π ×1000 ×100 × 20 12566370
= 7.96 ×10 −8
6.4 The bandwidth Bandwidth =
f 0 1000 = = 50 Hz . Q 20
7. A a.c network consists of a coil of inductance 10mH and series resistance 25 Ω in PARALLEL with a 50 µF capacitor. The a.c supply voltage is 120<0° V at 400 Hz. 7.1 Draw a well labelled diagram of the circuit.
7.2 Calculate the current in the capacitor in POLAR form
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Steve Goddard
V Xc
Ic =
Xc =
Ic =
1 1 = = 7.958 2πfc 2π × 400 × 5 ×10 −5
(
)
120 = 15 .08 @ 90 ° 7958
7.3 Calculate the current in the coil in POLAR form. Il =
V Xl
X = 2πfl = 2π × 400 × 0.01 = 25 .132 Ω
( z) =
Im pedance I =
(25
2
)
+ 25 .132 2 = 35 .449
V 120 = − 3.385 Z 35 .449
Converting to Polar Form:
tan θ = tan
−1
Xl R
(1.005 ) = 45 .152 °
I lR = 3.385 @ − 45 .15 7.4 Calculate the total current from the supply in POLAR form I 2 −3.385
2
+15 .08 2 −2 ×15 .08 ×3.385 ×Cos 44 .85
I 2 =166 .48 I =12 .903 Amps
8. A 100 µF capacitor is connected in PARALLEL with a coil of inductance 100mH. The coil has a small resistance of 10 Ω. The circuit is connected across a 100<0°V variable frequency supply. 8.1 Draw a well labelled diagram and hence determine:-
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Steve Goddard
8.2 The frequency of the supply when the current is a minimum The current will be at its minimum when X L + X C is minimum. (Please see attached pages).
f0 =
1 2π
1 R2 1 − 2 = LC 2 π L
1 10 2 − 2 100 m ×100 µ 100 m
f 0 = 0.159 × 300 = 47 .75 Hz 8.3 The dynamic resistance of the circuit
RD =
100 m = 100 Ω (100 µ ×10 )
8.4 The resonant Q-factor of the circuit
Q=
2πf 0 L 2π (47 .7) × (100 m) = =3 R 10
9. A current of (15 + j8) A flows in a circuit whose supply voltage is (120 + j200) V. 9.1 Calculate the circuit IMPEDANCE in POLAR form
V = (120 + J 200 ) Firstly I'll convert this to polar form by using modulus: R = 120 2 + 200 2 = 233 .238 V And argument:
Hence
θ = tan −1
200 = 59 .036 120
(1 + 2J2 0) =0 2 0.23 〈 35 .089° 3 6
And I = 15 + J 8 Using Modulus R = 15 2 + 8 2 =17 Amps −1 And argument: θ = tan
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8 = 28 .072 15
Steve Goddard
So, using ohms law
V 2 . 5.0 °〈 3 4 9 3 3 6 Z= 1.7 3.9 Ω〈= 3 2 0 3 I 1 2.0°〈 7 8 7
9.2 Calculate the ACTIVE power Active power is the same as true power therefore Vs ×I s ×Cos φ 233 .238 ×17 ×Cos 30 .96 = 3.4 Kw
9.3 Calculate the REACTIVE power V s × I s ×Sin φ 233 .238 ×17 ×Sin 30 .96 = 2.039 Kw
9.4 Calculate the APPARENT power
PA = V × I = 233 .238 ×17 = 3965 VA
Transformers 10. A 1k Ω resistor is connected across the secondary windings of an ideal transformer whose secondary voltage is 100v. The current in the primary windings is 10mA. 10.1 Draw a circuit diagram
10.2 Determine the secondary current
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Steve Goddard
Using Ohms law
Is =
V 100 = = 100 mA R 1000
10.3 Determine the primary voltage
V P = Vs ×
N2 100 = 100 × = 1000 V N1 10
To determine this: N 2 =V2 and N 1 =V1 I determined V1 by using ohms law. I knew the primary current was 10mA and this part of the circuit had no resistance so:
V =
I 10 = = 10 V R 0
10.4 Determine the transformer turns ratio
N=
N 1 100 = = 10 N2 10
Transformer turns ratio = 10 : 1 11. An ideal transformer has 1000 primary turns and 100 secondary turns. If the primary winding is connected to a 230V ac supply and the secondary is connected to an 100 Ω resistive load: 11.1 Draw a circuit diagram
11.2 Determine the secondary voltage
V 2 N 2 V2 100 = = = V1 N 1 230 1000
V2 = 23 V
11.3 Determine the secondary current
I2 =
V2 23 = = 230 mA R 100
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Steve Goddard 11.4 The power supplied in the primary circuit If Power = IV then I 1 ×V1 = 0.23 × 23 = 5.29 W
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Steve Goddard
Bibliography Higher Engineering Science – W. Bolton Course Notes – Roger Macey http://hep.physics.indiana.edu/~rickv/Complex_waveforms.html Waveforms http://www.doctronics.co.uk/signals.htm - Complex Waveforms
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