btec hnc - electrical and electronic principles - investigate circuit theory

Investigate Circuit TheoryElectrical and Electronic Principles

By Brendan Burr

Brendan Burr BTEC Higher National Certificate in ElectronicsInvestigate Circuit Theory

Table of Contents

TABLE OF CONTENTS - 2 -

TASK 1 - 5 -

1.1 Transform the following star networks to their corresponding delta networks: - 5 -

Solution:- - 5 -Check:- - 6 -

1.2. Transform the following delta networks to their corresponding star networks: - 7 -

Solution:- - 7 -Check:- - 8 -

1.3 An AC source of 50 0 V and internal resistance 10 kΩ is matched to a load by a 10:1 ideal transformer. Determine for maximum power transfer :- - 9 -

(a) The value of the load resistance. - 9 -Solution:- - 9 -

(b) The power dissipated in the load. - 10 -Solution:- - 10 -

(c) Draw a circuit diagram - 11 -Solution:- - 11 -

1.4 Use Thevenin’s Theorem to determine the power dissipated in the 50 Ω resistor of the network shown below:- - 12 -

Solution:- - 12 -

1.5 For the following network obtain the Norton equivalent circuit at terminals AB. Hence determine the power dissipated in a 10 Ω resistor connected between A and B. - 14 -

Solution:- - 14 -

1.6 For the following network determine :- - 18 -

(a) The values of R and X for maximum power transfer across terminals AB. - 18 -Solution:- - 18 -

(b) The value of the maximum power. - 20 -Solution:- - 20 -

2


TASK 2 - 21 -

2.1. For the a.c network below draw a well labelled diagram and hence determine using Mesh current analysis :- - 21 -

(a) The mesh currents and . - 21 -Solution:- - 21 -

(b) The current flowing in the capacitor. - 22 -Solution:- - 22 -

2.2 Use Nodal analysis or the Super Position Theorem to determine the currents Ia, Ib and Ic in the following network : - 23 -

Solution:- - 23 -

TASK 3 - 28 -

3.1 A coil L1 produces a magnetic flux of 60 mWb. If a second coil L2 is wound over the first coil and 18 mWb of flux is linked to this coil determine the coefficient of coupling. - 28 -

Solution:- - 28 -

3.2. An emf of 20 mV is induced in a coil when the current in a second coil is changed at a rate of 1.5 A/s. Determine the mutual inductance that exists between the two coils. - 28 -

Solution:- - 28 -

3.3. Two coils have a mutual inductance of 40 mH. Determine the e.m.f induced in one coil when the current in the other coil is increased at a uniform rate from 100 mA to 600 mA in 500 μs. - 29 -

Solution:- - 29 -

3.4 Two coils have self inductances of 60 mH and 100 mH. If the two coils exhibit a mutual inductance of 20 mH determine the coefficient of coupling. - 29 -

Solution:- - 29 -

3.5. An ideal transformer has 900 primary turns and 225 secondary turns. If the primary winding is connected to 220 V ac supply and the secondary is connected to an 11 Ω resistive load : - 30 -

(a) Draw a circuit diagram. - 30 -Solution:- - 30 -

(b) Calculate the secondary voltage - 30 -Solution:- - 30 -

(c) Calculate the secondary current - 31 -Solution:- - 31 -

3


(d) The supply power. - 31 -Solution:- - 31 -Check:- - 31 -

TASK 4 - 32 -

4.1 An R-L-C series circuit has a maximum current of 2 mA flowing in it when the frequency of the 115 Volt supply is 4 kHz. Under these conditions the Q factor of the circuit is 90. - 32 -

(a) Draw a well labelled diagram. - 32 -Solution:- - 32 -

(b) Find the voltage across the capacitor. - 32 -Solution:- - 32 -

(c) Find the value of the resistance. - 32 -Solution:- - 32 -

(d) Find the value of the inductance. - 33 -Solution:- - 33 -

(e) Find the value of the capacitance. - 33 -Solution:- - 33 -

4.2 A 10 µF capacitor is connected in parallel with a coil of inductance 10mH. The coil has a small resistance of 1 Ω. The circuit is connected across a 100V variable frequency supply. - 34 -

(a) Draw a well labelled diagram. - 34 -Solution:- - 34 -

(b) Determine the dynamic resistance of the circuit. - 34 -Solution:- - 34 -

(c) Determine the Q factor of the circuit. - 35 -Solution:- - 35 -

EVALUATION - 36 -

CONCLUSION - 36 -

BIBLIOGRAPHY - 37 -

Books - 37 -

Catalogues - 37 -

Websites - 37 -

4


5


Task 1

1.1 Transform the following star networks to their corresponding delta networks:

Solution:-

Ohms Ohms

Ohms

6


Check:-

Ohms Ohms

Ohms

7


1.2. Transform the following delta networks to their corresponding star networks:

Solution:-

milli-Ohms milli-Ohms

milli-Ohms

8


Check:-

Ohms Ohms

Ohms

9


1.3 An AC source of 50 0 V and internal resistance 10 kΩ is matched to a load by a 10:1 ideal transformer. Determine for maximum power transfer :-

(a) The value of the load resistance.

Solution:-

By maximum power transfer theorem:

Ohms

10


(b) The power dissipated in the load.

Solution:-

Therefore the Primary Current is:

Therefore the Secondary Current is:

So Power delivered to the Load is given by:

11


(c) Draw a circuit diagram

Solution:-

What the Source sees:

12


1.4 Use Thevenin’s Theorem to determine the power dissipated in the 50 Ω resistor of the network shown below:-

Solution:-

(i) First disconnect the load and calculate the voltage at .

(ii) Find Z looking into ‘AB’ (Load still disconnected).

Thevenin’s Equivalent circuit is now:

13


Using Ohms Law:

mA

So Power (P) dissipated in the Load is given by:

14


1.5 For the following network obtain the Norton equivalent circuit at terminals AB. Hence determine the power dissipated in a 10 Ω resistor connected between A and B.

Solution:-

(i) Find the current across AB.

Therefore Total Circuit Current is given by:

15


Amps

However:

Amps

16


(ii) Remove the Short Circuit. Look into ‘AB’ and find .

17


Norton’s Equivalent Circuit:

Therefore, Power in is given by:

Watts

18


1.6 For the following network determine :-

(a) The values of R and X for maximum power transfer across terminals AB.

Solution:-

(a) Find Thevenin’s Equivalent Circuit for AB:

19


Thevenin’s equivalent circuit is as follows:

Therefore:

Ohms

So:

20


(b) The value of the maximum power.

Solution:-

Watts

21


Task 2

2.1. For the a.c network below draw a well labelled diagram and hence determine using Mesh current analysis :-

(a) The mesh currents and .

Solution:-

Loop 1:

Loop 2:

So:

22


We can now drop the equation for into Equation 1, as follows:

Amps

Amps

(b) The current flowing in the capacitor.

Solution:-

Convert to Rectangular:

Current in the Capacitor:

Amps

23


2.2 Use Nodal analysis or the Super Position Theorem to determine the currents Ia, Ib and Ic in the following network :

Solution:-

Redraw the circuit with one EMF:

Ohms

24


Ohms Law then states that:

Amps

Using the Current Division Laws leaves:

Amps

Amps

25


Produce a second circuit with the other EMF:

Ohms

Amps

26


Amps

Amps

27


Redrawing the original circuit gives:

So:

Amps

Amps

Amps

28


Task 3

3.1 A coil L1 produces a magnetic flux of 60 mWb. If a second coil L2 is wound over the first coil and 18 mWb of flux is linked to this coil determine the coefficient of coupling.

Solution:-

3.2. An emf of 20 mV is induced in a coil when the current in a second coil is changed at a rate of 1.5 A/s. Determine the mutual inductance that exists between the two coils.

Solution:-

29


3.3. Two coils have a mutual inductance of 40 mH. Determine the e.m.f induced in one coil when the current in the other coil is increased at a uniform rate from 100 mA to 600 mA in 500 μs.

Solution:-

Volts (Back EMF)

3.4 Two coils have self inductances of 60 mH and 100 mH. If the two coils exhibit a mutual inductance of 20 mH determine the coefficient of coupling.

Solution:-

(No Units)

30


3.5. An ideal transformer has 900 primary turns and 225 secondary turns. If the primary winding is connected to 220 V ac supply and the secondary is connected to an 11 Ω resistive load :

(a) Draw a circuit diagram.

Solution:-

(b) Calculate the secondary voltage

Solution:-

Volts

31


(c) Calculate the secondary current

Solution:-

Amps

(d) The supply power.

Solution:-

Amps

Therefore Power in is given by:

Watts

Check:-

Watts

32


Task 4

4.1 An R-L-C series circuit has a maximum current of 2 mA flowing in it when the frequency of the 115 Volt supply is 4 kHz. Under these conditions the Q factor of the circuit is 90.

(a) Draw a well labelled diagram.

Solution:-

(b) Find the voltage across the capacitor.

Solution:-

Kilo-Volts

(c) Find the value of the resistance.

Solution:-

Kilo-Ohms

33


(d) Find the value of the inductance.

Solution:-

H

(e) Find the value of the capacitance.

Solution:-

pF

34


4.2 A 10 µF capacitor is connected in parallel with a coil of inductance 10mH. The coil has a small resistance of 1 Ω. The circuit is connected across a 100V variable frequency supply.

(a) Draw a well labelled diagram.

Solution:-

(b) Determine the dynamic resistance of the circuit.

Solution:-

Ohms

35


(c) Determine the Q factor of the circuit.

Solution:-

Hz

36


EvaluationI began this assignment with the knowledge that it was going to take a long time, due to the fact that the theory took a long time to go through in class. I had other assignments to work on at the time so focused on these, but as soon as I was available to do so I started working my way through the tasks in this assignment.I found Tasks 1.1 and 1.2 relatively simple, all that had to be done was an input of resistor values into a formula. I then checked it by reversing then using the formula to change from star networks to delta and vice versa. Task 1.3 was slightly more difficult as I needed to use addition and subtraction of indices to get the value of RL. I could then use this value to work out the Power in the Load which was the solution to section (b).Until we received the theory work for Task 1.4, the question looked very difficult. However when you start manipulating the circuit using Thevenin’s Theorem, I began to realise that all I had to do was work on small sections at a time, and eventually piece it all together to get the power dissipated in the 50 ohm resistor.Norton’s equivalent circuit was a little bit more difficult as it involved similar principles as Thevenin’s Theorem, however I also had to produce an alternative / equivalent circuit for the original circuit.For 1.6 I used Thevenin’s Theorem to work out the equivalent circuit and could then determine the Load, which could then be split to determine the values of the capacitor and resistor. Using this value of R, I could then determine the maximum power.I found Task 2 relatively difficult due to having to work out the two equations, solve one of them and then drop all of the information back into the first equation. There was easily room for error, but fortunately I think I have managed to avoid it.Tasks 3 and 4 took barely any time at all, compared to Tasks 1 and 2. I found these much easier and didn’t encounter any difficulty with working them out.

ConclusionOverall I am glad this assignment is out of the way, but also enjoyed working out the tasks. I found that the majority of the time I spent on this assignment was spent on typing it up. This was because it was almost entirely typed in Equation Editor which is slow to use when typing formula. This led me to learn the hot keys such as “Ctrl + F” to make a fraction and “Ctrl + R” to produce a root sign, which enabled me to type slightly quicker.I also spent a lot of time on the Images in Microsoft Paint. I have found it to be a very handy drawing software as it is simple to use and gives the desired affect quite easily.It took me around one and a half days to work out all of the tasks but a further two days to put the workings into a presentable format.I am pleased that my commitment to the course can be seen through the effort I put into my work.

37


Bibliography

Through guidance from my lecturer, the following text books, catalogues and websites I was able to complete this assignment:

Books

BTEC National Engineering (Mike Tooley & Lloyd Dingle) ISBN: 978-0-7506-8521-4Success in Electronics (Tom Duncan & John Murray)ISBN: 0-7195-4015-1Higher Engineering Mathematics (John Bird) ISBN: 0-7506-8152-7

Catalogues

N/A

Websites

N/A

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btec hnc - electrical and electronic principles - investigate circuit theory

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