Calculations for parallel circuits
A number of examples will be discussed to combine all these aspects in parallel RCL circuits. These examples will be used as revision to expand the way one thinks about RCL circuits.
Question 1:
A parallel RC circuit consists of a resistor of 10 Ω and a capacitor of 450 µF is connected to a 240 V 50 Hz supply.
Calculate the following:
a) capacitive reactance
b) current through the resistor and the capacitor.
c) the current drawn from the supply IL IC
IL
d) the power factor.
e) Represent the current values by means of a neat phasor diagram.
f) What will happen to the current flowing in the circuit if the frequency is decreased?
Answers to Question 1:
a) XC = 1 = 1 = 7,07 Ω 2 πfC 2 π(50)(450 μF)
b) VT = VR = VC = 240 V IR = = 240 = 24 A R 10
IC = = 240 = 33,93 A XC 7,07
c) IT = √ IR2 + IC2 = √ 242 + 33,932 = 41,56 A
d) Cos θ = = 24 = 0,58 leading IT 41,56
e)
f) f ↓ XC ↑ IC ↓ IT ↓ Question 2
The phasor diagram for an RLC circuit is shown. Interpret the information and answer the questions that follow.
IC = 33,93 mA IT = 41,56 A
IR = 24 A
VT = 240 V θ
VR
VC
IR
a) Draw the circuit that is represented by the above phasor diagram.
b) Is the circuit more inductive or capacitive? Motivate your answer.
c) Calculate the total current in the circuit.
d) Calculate the power factor in the circuit.
e) What will happen to the phase angle if the frequency should decrease?
Explain your answer.
Answers to question 2:
a)
A parallel RCL circuit diagram
b) Capacitive because IL > IC which means that XL < XC c) IT = √ I2R + (IL – IC)2
IT = √ 102 + (10,61 – 4,72)2
IT = 11,61 A
d) Cos θ = IR
___
IT = 10 11,61 = 0,86 lagging
e) θ will increase because Xc ∞ 1 if F decreases then Xc will increase which in
F
turn will let IC decrease. Therefore the difference between IL and IC will become larger which means that IT will tend more towards IL on the phasor diagram.
Question 3:
A parallel RLC network has the following components connected across a 230 V/
50 Hz alternating current supply:
resistor = 12 Ω capacitor = 70 µF inductor = 100 mH Determine the:
a) inductive reactance of the coil b) capacitive reactance of the capacitor
c) current flowing through each of the branches d) total current flowing in the network
e) phase angle between the supply voltage and the current f) impedance of the network.
g) Draw a fully labelled phasor diagram that represents all the current and the voltage values.
Answers to question 3:
a) XL = (2 πfL) = 2 π(50)(100 × 10-3) = 31,42 Ω b) XC = 1 = 1 = 45,47 Ω 2 πfC 2 π(50)(70 × 10-6)
c) IL = = 230 = 7,32 A XL 31,42
IR = = 230 = 19,17 A R 12
IC = = 230 = 5,06 A XC 45,47
d) IT = √ IR2 + (IL – IC)2
= √ 19,172 + (7,32 – 6,06))2 = 19,3 A e) Cos θ = = 19,17 = 0,993 lagging IT 19,3
f) Z = = 230 = 11,92 Ω IT 19,3
g)
IC = 5,06 A
IR = 19,17
IL = 7,32 A
IT = 19,3 A
VT = 230 V VT
VT
VT
IR
VT
Practical: Simulate an RLC circuit and display waveform on oscilloscope To investigate the relationship between the voltage across the resistor, the voltage across the capacitor, the voltage across the coil and the supply voltage in a series RCL – Alternating Current Circuit.
Apparatus
1. 1,0 kΩ resistor (R1)
2. 0,1 micro farad capacitor (C1) 3. 26 mH coil
3. Matrix board with hook-up wires 4. Audio signal generator (50 Hz 10 kHz) 5. Dual trace oscilloscope
Method
1. Draw a series circuit with the resistor, coil and capacitor connected to the signal generator. Show on your sketch how the oscilloscope is connected to the circuit to comply with the following.
• Channel 1 (or Y1) of the oscilloscope must be connected across the resistor to measure VR, and
• Channel 2 (or Y2) across the signal generator to measure VT . The triggering of the scope should be across the resistor. (NB Ensure that all the earth
connections are connected on one side of the resistor.)
2. Build the circuit exactly according to the circuit diagram. (NB Check that all the earth connections are connected on one side of the resistor. In other words, all black croc clips should be together.)
3. Follow these steps.
• Set the signal generator to 500 Hz.
• Go to channel 2 and set the supply voltage to 0,6 V.
• Go to CH 1 and make sure that the wave starts left centre. (The trigger must be CH 1.)
• Make the necessary adjustments on the oscilloscope so that one complete cycle is displayed over 10 cm on the screen.
• Set the V/div setting so that the wave is as high (big) as possible.
• Select the ALT or CHOP mode so that both waves are displayed simultaneously.
4. Draw the oscillograms as displayed on the screen. All labels must be clearly indicated. (Remember that the scales of CH 1 and CH 2 on the oscilloscope may possibly be different.)
Take the following readings:
a) VR = V/div × no. blocks × 0,707 = __________________________________
b) VT = V/div × no. blocks × 0,707 = __________________________________
c) The phase angle = _____________________________________________
(State which wave is leading.)
d) The supply current = = _____________________________________
R
e) What does it mean if the triggering is set across a certain component?
5. Convert the oscillograms in Step 4 to phasors. (They must be drawn to scale.) Follow these steps:
• Start the phasor diagram in the middle of the page.
• Plot the reference phasor.
• Draw in VR.
• Draw in VT with respect to VR.
• Graphically subtract VR from VT (This will determine the voltage drop which represents the difference between VL and VC.)
• At this stage, one should have good understanding of RL and RC circuits.
Estimate the sizes of VL and VC and indicate them on your phasor diagram.
• All labels (including the scale and phase angle) must clearly be indicated.
6. Change the frequency to 10 kHz and repeat points 3 to 5 above.
7. Without doing ANY CALCULATIONS, adjust the frequency to resonance.
Explain why you selected this particular frequency.
Draw the waveforms. (Remember that the scales of CH 1 and CH 2 on the oscilloscope may possibly be different.)
Take the following readings:
a) VR = V/div × no. blocks × 0,707 = _________________________________
b) VT = V/div × no. blocks × 0,707 = _________________________________
VR
c) The phase angle = ______________________________________________
(State which wave is leading.)
d) The supply current = = _____________________________________
R
e) What does it mean if the triggering is set across a certain component?
8. Draw a neat phasor diagram, not to scale, but representing all the voltage drops in the circuit as accurately as possible.
Observations:
Answer the questions below by comparing the phasor diagrams. Establish how a change in frequency will influence the relationship between VR and VT . 1. What is the phase relationship between VR and VT at 500 Hz?
(Refer to angle, as well as leading or lagging.)
2. What is the phase relationship between VR and VT at 10 kHz?
(Refer to angle, as well as leading or lagging.)
3. Did the change in frequency influence the phase relationship (angle) between the two phasors VR and VT?
4. What effect does the change in frequency have on the magnitude of VT , VR, VL and VC?
5. What influence does the change in frequency have on the current flowing in the circuit and thus on the impedance of the circuit? Draw a neat curve that represents frequency versus current.
6. What is the reason for all the changes that occurred as a result of the change in frequency (increase in frequency)?
VR
Activity 1
1. The sketch below depicts the reactance of an inductor and a capacitor and the resistance of a resistor versus frequency. Interpret the information and answer the questions that follow.
a) Draw TWO approximate impedance triangles of an RLC series circuit next to each other, with the frequency at point A and C respectively.
b) The circuit represented by the above graph consists of a 3kΩ resistor, a 50µF capacitor and a 0,1H coil. Calculate the frequency at point B.
2. A series circuit with a 250µF capacitor and a 120mH coil and a resistor of 15Ω is connected across a 240V/50Hz supply.
Calculate:
a) The total impedance in the circuit.
b) The voltage drops across the coil.
c) The phase angle.
d) Is the circuit more inductive or capacitive? Motivate your answer.
e) Must the frequency increase or decrease for the circuit to resonate?
Motivate your answer.
3. Name THREE measurements we can take with the scope.
4. Use the oscillograms below to answer the questions that follow.
Determine the following if both V/div = 0,5V and the T/div = 10µs a) Frequency
b) V1 (RMS)
c) Would the circuit be inductive or capacitive if V1 is the voltage across a supply and V2 is the resistor voltage? Explain your answer.
5. A 15V/10kHz supply is connected to the following components connected in series: R = 2kΩ, L = 60mH and C = 8nF.
a) Draw a neat diagrammatic circuit diagram for the circuit.
b) Calculate the inductive and capacitive reactances and then the impedance of the circuit.
c) Represent all the calculated ohm values on a neat phasor diagram.
(not to scale but in good proportion).
d) What will happen to capacitance if the frequency is decreased?
Give a reason for your answer.
e) Which factors will determine the resonant frequency.
6.
Use the phasor diagram to answer the following questions:
a) Calculate the supply voltage.
b) Is the power factor leading or lagging? Explain your answer.
c) Calculate the reactive value of the coil.
d) Calculate the actual value of the coil in mH.
V2
IT = 2 mA f = 7 kHz VL = 8 V
VR = 3 V
VC = 12 V
V1
e) Name 5 characteristics of a series RCL circuit if it is set to resonant frequency.
f) Draw a neat curve that shows the relationship between current and frequency in a series RCL circuit
7. A parallel circuit consisting of a capacitor of 18 nF is connected in parallel to a resistor of 2 kΩ. The circuit is connected to a 5 V/8 kHz.
Calculate the following:
a) The capacitive reactance.
b) The branch currents.
c) Draw a neat phasor diagram to represent all the current values.
8. An alternating current circuit of an inductor of 200 mH, a capacitor of 300 μF and a resistor of 15 Ω is connected in parallel to a 120 V/50 Hz supply. Answer the following questions.
a) Draw the circuit with labels.
b) Calculate the capacitive and inductive reactance respectively.
c) Calculate the current through each component d) Calculate the total current drawn from the supply.
e) If the total current through the circuit is 5,63 A, calculate the phase angle for the circuit.
f) Is the circuit more inductive or capacitive?
Motivate your answer.
9. A parallel RCL circuit consists of a resistor of 4 kΩ, a coil of 36 mH and a capacitor of 9,1 nF. The circuit is connected to a 12 V AC supply voltage of which the frequency can be changed. All the required calculations were done and recorded in the table shown below.
Freq XL (Ω) XC(Ω) IR(mA) IL(mA) IC(mA) IT(mA) Cos θ θ
Use the information provided in the table to answer the following questions:
What observation can be made regarding the following as frequency is increased?
a) Inductive reactance (XL) b) Capacitive reactance (XC) c) Resistance
i) What is the relationship between XL and IL? j) What is the relationship between XC and IC? k) What is the relationship between f and IR? l) What is significant about the circuit at 8 793 Hz?
m) What are the characteristics of the circuit (as seen on the table) at this frequency?
n) Draw a graph to represent IT vs frequency and Z vs frequency.
o) What can we deduce from the graphs drawn above?
Practical Activity 1
AimThis module can be done in class without the use of any equipment, and it will test if the learners has mastered all the basic concepts before they do real experiments using real equipment.
To investigate the relationship between the voltage across the resistor, the voltage across the capacitor and the supply voltage in a series rc-alternating current circuit.
Apparatus
1. 1,0 kohm resistor (R1)
2. 0,1 micro Farad capacitor (C1)
3. Matrix Board with hookup wires (intermediate-board) 4. Audio signal generator (50Hz 10kHz)
5. Dual trace oscilloscope Method
1. Draw a series circuit by making use of the resistor and capacitor that is connected to a signal generator. Indicate how the oscilloscope is connected to the circuit. The one input (Y2) of the oscilloscope is connected across the signal generator and the other input (Y1) across the resistor. The triggering must be set across Y1 (the resistor). Ensure that all the earth connections are connected on one side of the resistor.
2. Assume that this circuit was built correctly.
3. The necessary adjustments were done on the oscilloscope so that one complete cycle of the reference signal (Vr) is displayed over 10 cm on the screen. The V/div setting were adjusted so that the waves are as high (big) as possible.
T/div = 0,2ms V/div = 0,2V
4. These were the oscillograms (waves) observed on the oscilloscope. Interpret the information and do the necessary calculations for the following:
V = (V/div)(No blocks)(0,707)
Vs = ________________________________________________________
Vr = ________________________________________________________
f = __________________________________________________________
Phase angle = _________________________________________________
The supply current = Vr/R = _______________________________________
5. Convert the oscillograms to phasors by drawing the phasor diagram to scale (1:10) on the same page.
Graphically subtract Vr from Vs to determine the voltage drop across the capacitor.
Measure this voltage.
All labels (including the scales and phase angle) must clearly be indicated.
6. The frequency is now changed.
T/div = 10μs V/div = 0,2V
These were the oscillograms (waves) observed on the oscilloscope at the new frequency. Interpret the information and do the necessary calculations for the following:
V = (V/div)(No blocks)(0,707)
Vs = ________________________________________________________
Vr = ________________________________________________________
f = _________________________==_______________________________
Phase angle = _________________________________________________
The supply current = Vr/R = _______________________________________
7. Convert the oscillograms to phasors by drawing the phasor diagram to scale (1:10) on the same page.
Graphically subtract Vr from Vs to determine the voltage drop across the capacitor.
Measure this voltage (NB – This must be done at both frequencies).
All labels (including the scales and phase angle) must clearly be indicated.
8. Observation
Answer the questions below by comparing the phasor diagrams.
Establish how a change in frequency will influence the relationship
between Vr and Vs:
1. What is the phase relationship between Vr and Vs at frequency 1.
(refer to angle as well as leading or lagging)
2. What is the phase relationship between Vr and Vs at frequency 2.
(refer to angle as well as leading or lagging)
3. Did a change in frequency influence the phase relationship (angle) between Vr and Vs ?
4. What effect does the change in frequency have on the magnitude of Vs, Vr and Vc?
5. What influence does the change in frequency have on the current flowing in the circuit, and hence the impedance of the circuit?
(NO calculations allowed to answer this question).
6. What is the reason for all the changes that occurred due to the change in frequency?
Chapter 6 Logic
Logic circuits Boolean algebra
A B
Ladder diagrams PLCs
A B
Introduction
This chapter is only about an introduction into programmable logic control devices (PLCs). Aspects that will be covered in this chapter range from the advantages of PLCs, introduction to ladder logic, conversion from hard-wired schematics to ladder logic, to writing programs for various motor controlled circuits as real time application of the PLC.
A PLC can be defined as a specialised computer system that can be used to monitor different inputs, perform a certain function based on the conditions of such inputs (i.e. make decisions e.g. logic, sequencing, timing, counting and arithmetic and data handling) and then control various types of machines or processes. PLCs can be used in a multitude of industries:
• manufacturing/machining
PLCs can vary in size, from very small to very large. The smaller units can have up to 128 Input/Outputs ( I/Os) and a memory capacity of about 2 KB, and the larger units can have up to 8192 I/Os and a 750 KB memory capacity.