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Instructor : Dr. Engr. A. K. M. Baki Room # 4A08

Ahsanullah University of Science and Technology Fall 2014

COURSE: POWER SYSTEM I (EEE 3205)

LECTURE 2: INDUCTANCE

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DEFINITION OF INDUCTANCE

The voltage induced in a circuit due to change in flux linkage can be expressed as:

Where, e is the induced voltage; is the flux linkage of the circuit in wbt (weber-turn);

In a circuit, when current changes its associated magnetic field (flux linkage) changes.

(1)

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DEFINITION OF INDUCTANCE For a constant permeability, no. of flux linkage is proportional to the current. Therefore,

L= constant of proportionality (inductance of the circuit), H (Henry) e = induced voltage, V

= rate of change of current, A/s

(2)

From (1) and (2) (H) (3)

The inductance of a circuit is defined as the flux linkages per unit current.

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If the flux linkages of the circuit vary linearly with current, which means that the magnetic circuit has a constant permeability

(H)

(wbt)

Where and i are the instantaneous values.

In phasor form,

(wbt)

Since and I are in phase, L is real

(4)

(5)

(6)

DEFINITION OF INDUCTANCE

[Note: A phasor contains the amplitude and phase information but is

independent of time variable t.]

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The phasor voltage drop due to flux linkage is

(7)

DEFINITION OF INDUCTANCE

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MUTUAL INDUCTANCE

Mutual inductance between two circuits is defined as the flux linkages of one circuit due to current in the second circuit per ampere of current in the second circuit. If the current I2 in the second circuit produces flux linkages with circuit 1, the mutual inductance is,

Where, M12: Mutual inductance between circuit 1 and circuit 2; 12: Flux linkage with circuit 1 for a current I2 in circuit 2; I2: Current in circuit 2;

(8)

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AMPERES CIRCUIT LAW

THE LINE INTEGRAL OF THE MAGNETIC FLUX AROUND A CLOSED CURVE IS EXACTLY EQUAL

TO THE ALGEBRAIC SUM OF ELECTRIC CURRENTS ENCLOSED BY THAT CURVE,

= encIdlH.

Andr-Marie Ampre

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AMPERES CIRCUIT LAW

AN INFINITELY LONG STRAIGHT FILAMENT CARRYING A DIRECT CURRENT I

ACCORDING TO ACL,

r

IH

IrH

drHrdHd

enc

enc

pi

pi

pi

pi

2

2

.H2

0

2

0

=

==

== l

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AMPERES CIRCUIT LAW

THE STREAMLINES OF THE MAGNETIC FIELD INTENSITY AROUND AN INFINITELY LONG

STRAIGHT FILAMENT CARRYING A DIRECT CURRENT I. THE DIRECTION OF I IS INTO THE

SLIDE.

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INDUCTANCE OF A CONDUCTOR DUE TO INTERNAL FLUX

Consider a conductor of radius r carrying a current I. At a

distance x from the center of this conductor, the magnetic field

intensity Hx can be found from Amperes law:

(9)

Figure 5. Cross section of a cylindrical conductor.

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Where Hx is the magnetic field intensity (At/m) at each point along a closed path, ds is a unit vector (m) along that path and Ix is the net current (A) enclosed in the path. For the homogeneous materials and a circular path of radius x, the magnitude of Hx is constant, and ds is always parallel to Hx.

INDUCTANCE OF A CONDUCTOR DUE TO INTERNAL FLUX

Therefore,

(10)

Assuming uniform current density,

(11)

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Where I is the total current in the conductor. Then substituting

(11) in (10) and solving for Hx, we obtain,

INDUCTANCE OF A CONDUCTOR DUE TO INTERNAL FLUX

(At/m) (12)

The flux density x meters from the center of the conductor is

(Wb/m2) (13)

Where is the permeability of the conductor.

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In the tubular element of thickness dx the flux d is Bx times the cross-sectional are of the element normal to the flux lines, the area being dx times the axial length. The flux per meter of length is

INDUCTANCE OF A CONDUCTOR DUE TO INTERNAL FLUX

(Wb/m) (14)

The flux linkage/meter d, which are caused by the flux in the tubular element, are the product of flux/meter and the fraction of the current linked. Thus,

(15) (Wbt/m)

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Total flux linkage inside the conductor is,

INDUCTANCE OF A CONDUCTOR DUE TO INTERNAL FLUX

(16) (Wbt/m)

For a relative permeability of 1, H/m.

(17) (Wbt/m)

(H/m) (18)

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REFERENCES

1. Power System Analysis

John J. Grainger, W. D. Stevenson, Jr.

2. Class notes provided