1. C.K.PITHAWALA COLLEGE OF ENGINEERING & TECHNOLOGY, SURAT
ELECTRICAL ENGINEERING DEPARTMENT

2. ELECTRICAL POWER SYSTEM – I
SUBJECT
ELECTRICAL POWER SYSTEM – I
PRESENTED BY
PATEL KRUNAL
MURYA SHUBHENDOO
MEVAWALA KISHAN
PADSALA BHAVIK
PANCHAL BHAVIK

3. PRESENTATION TOPIC Inductance of composite conductor lines
Inductance of three phase lines
Double circuit three phase lines
Bundled conductors
Resistance, Skin effect and proximity effect
Magnetic field induction

4. Inductance of composite conductor lines
 We will consider a single phase 2 wire system. It consists of two conductors say P and Q which are composite conductors. The arrangement of conductors is shown in the Fig. 1.
Conductor P is consisting of x identical, parallel filaments. Each of the filament carries a current of I/x. Conductor Q consists of y filament with each filament carrying a current of - I/y. The conductor Y carries a current of I amps in opposite direction to the current in conductor X as it is forming return path.
       The flux linkages of filament say a due to all currents in all the filaments is given by

5. The inductance of filament a is given by,
The inductance of filament b is given by,
      The average inductance of the filaments of conductor P is
The conductor P consists of x number of parallel filaments. If all the filaments are equal inductances then inductance of the conductor would be 1/x times inductance of one filament. All the filaments have different inductances but the inductance of all of them in parallel is 1/x times the average inductance.
       Inductance of conductor P is given by,
 Substituting the values of La, Lb ..... Lx  in the equation and simplifying the expression we have,
   In the above expression the numerator of argument of logarithm is the xy, the root of xy terms. These terms are nothing but products of distance from all the x filaments of conductor P to all the y filaments of the conductor Q.

6.   For each filament in conductor P there are y distances to filaments in conductor Q and there are x filaments in conductor P. The xy terms are formed as a result of product of y distances for each of x filaments. The xyth root of the product of the xy distances is called the geometric mean distance between conductor P and Q. It is termed as Dm or GMD and is called mutual GMD between the conductors.
        The denominator of the above expression is the x2 root of x2 terms. There are x filaments and for each filament there are x terms consisting of r' (denoted by Daa, Dbbetc) for that filament times the distance from that filament to every other filament in conductor P.
       If we consider the distance Daa then it is the distance of the filament from itself which is also denoted as ra'. This r' of a separate filament is called the self GMD of the filament. It is also called geometric mean radius GMR and identified as Ds.
       Thus the above expression now becomes
 Comparing this equation with the expression obtained for inductance of a single phase two wire line. The distance between solid conductors of single conductors line is substituted by the GMD between conductors of the composite conductor line. Similarly the GMR (r') of the single conductor is replaced by GMR of composite conductor.
       The composite conductors are made up of number of strands which are in parallel. The inductance of composite conductor Q is obtained in a similar manner. Thus the inductance of the line is,
L = Lp + LQ

7. Inductance of three phase lines
To calculate the inductance of three phase line following are the two considerations :-
Conductors are at equal spacing
Conductors are at unequal spacing A B C D
Conductors are at equal spacing :- .
Fig (a) shows a 3 phase line whose conductors are at equal distance from each other.
Each conductor have radius r & distance between them is D.
There phase line carries a balanced current therefore Ia + Ib + Ic =0
Flux linkages to conductor A due to its own flux
Fig.(a) :- Equal conductor spacing for 3-Ø line

8. Flux linkages to conductor A due to current in conductor B is.
Flux linkages to conductor B due to current in conductor C
Total flux linkages with conductor A is
But

9. Inductance of conductors ‘A’ is given as
Due to the symmetry in the spacing between conductors and same radius of conductors, inductance for conductor B and C can be derived in similar way.

10. Fig.(b) :- 3-Ø line with unequal spacing
(b) Conductors are at unequal spacing:-
When the conductors are space at unequal distance from each other, then the flux linkage with each conductor is different which causes unequal inductance per phase. This results into unequal voltage drop in 3 phase line. Thus receiving end voltage becomes unbalance through the three line currents are same. To over come this difficulty conductors are transposed (exchange of their position after certain distance).
Fig. b shows a 3 phase line whose conductors are spaced at unequal distance.
The effect of transposition results same average inductance therefore let us calculate the flux linkages per conductor in position 1,2 and 3 respectively (as for the transposed line each conductor will occupy position 1, 2 and 3 each for one third of its length). a b c 1 2 3
D31
D12
D23
Fig.(b) :- 3-Ø line with unequal spacing

11. Average flux linkages of conductor a is
But
Inductance of conductor a is,

12. DOUBLE CIRCUIT THREE PHASE LINES
Mostly in transmission system more than one transmission line (double circuit or more) and run in parallel on the same tower.
Consider the fig. c in which conductors 1, 2 and 3 form one circuit and conductors 1’, 2’ and 3’ forms another circuit.
Conductors 1 and 1’, 2 and 2’ and 3 and 3’ are in parallel. 1 2 3 3’ 2’ 1’ D 2D
Fig.(c) :- 3 phase double circuit line

13. Let us calculate the flux linkages to conductor 1 :-
But
Inductance of phase 1

14. Bundled Conductors
The high voltage surface gradient is reducedconsiderably by having two or more conductors per phase in close proximity. This is called conductor bundling .
The conductors of abundle are separated at regular intervals with spacerdampers that prevent clashing of the conductors andprevent them from swaying in the wind. They alsoconnect the conductors in parallel.

15. Fig. of two, three and four bundled conductors are as follow.
two conductors four conductors
three conductors

16. The conductors are bundled in groups of two,three or four as shown in Fig.
Fig. Bundled conductors: (a) 2-conductor,
(b) 3-conductor (c) 4-conductor bundles

17. The geometric mean radius (GMR) of two-conductor bundle is given by
where D s is the GMR of conductor
The GMR for three-conductor and four-conductor bundles are given respectively by

18. The inductance of the bundled conductor is then given by
where the geometric mean distance is calculated assuming that the center of a round conductor is the same as that of the center of the bundle.

19. Resistance

20. Skin Effect
However,an alternatin current flowing through the conductor does not distribute uniformly.
The tendency of alternating current to concentrate near the surface of a conductor is known as skin effect.
Due to this, the effective area of cros-section of the conductor through which current flows is reduced.
factor affecting on skin effect :-
1) frequency
2)Diameter of wire
3)Shape of wire
4)Nature of material

21. Proximity Effect
In a conductor carrying alternating current, if currents are flowing through one or more other nearby conductors, such as within a closely wound coil of wire, the distribution of current within the first conductor will be constrained to smaller regions.
The resulting current crowding is termed as the proximity effect.
This crowding gives an increase in the effective resistance of the circuit, which increases with frequency.

22. Magnetic field Induction

23. What is Electro magnetic Induction?
Electromagnetic Induction is the process of using magnetic fields to produce voltage, and in a complete circuit, a current.
It turns out that electromagnetic induction is created by just that - the moving of a conductive substance through a magnetic field.

24. Magnetic Induction
As the magnet moves back and forth a current is said to be induce in the wire.

25. Magnetic Flux
The first step to understanding the complex nature of electromagnetic induction is to understand the idea of magnetic flux. B A
Flux is a general term associated with a FIELD that is bound by a certain AREA. So MAGNETIC FLUX is any AREA that has a MAGNETIC FIELD passing through it.
We generally define an AREA vector as one that is perpendicular to the surface of the material. Therefore, you can see in the figure that the AREA vector and the Magnetic Field vector are PARALLEL. This then produces a DOT PRODUCT between the 2 variables that then define flux.

26. Magnetic Flux – The DOT product
How could we CHANGE the flux over a period of time?
We could move the magnet away or towards (or the wire)
We could increase or decrease the area
We could ROTATE the wire along an axis that is PERPENDICULAR to the field thus changing the angle between the area and magnetic field vectors.

27. Useful Applications
AC Generators use Faraday’s law to produce rotation and thus convert electrical and magnetic energy into rotational kinetic energy. This idea can be used to run all kinds of motors. Since the current in the coil is AC, it is turning on and off thus creating a CHANGING magnetic field of its own. Its own magnetic field interferes with the shown magnetic field to produce rotation.

28. Transformers
Probably one of the greatest inventions of all time is the transformer. AC Current from the primary coil moves quickly BACK and FORTH (thus the idea of changing!) across the secondary coil. The moving magnetic field caused by the changing field (flux) induces a current in the secondary coil.
If the secondary coil has MORE turns than the primary you can step up the voltage and runs devices that would normally need MORE voltage than what you have coming in. We call this a STEP UP transformer.
We can use this idea in reverse as well to create a STEP DOWN transformer.

29. Microphones
A microphone works when sound waves enter the filter of a microphone. Inside the filter, a diaphragm is vibrated by the sound waves which in turn moves a coil of wire wrapped around a magnet. The movement of the wire in the magnetic field induces a current in the wire. Thus sound waves can be turned into electronic signals and then amplified through a speaker.

30. THANK YOU