Resistance
In an ac system, the voltage across a resistor and the
current flowing thought it are said to be in phase, that is, their zero value
and their maximum values occur at the same times. There are two types of fields
associated with an ac electric system; electric fields and magnetic fields.
Electric fields relate to the voltage and magnetic fields relate to the
current.12 The waveforms of the voltage and current associated with both of
these characteristics are not in phase, that is, the times of the maximum and
zero values are not identical.
Induction and Inductive Reactance
When we discussed the operation of a generator, we noted
that an electric voltage is induced in a wire when a moving magnetic field
“cuts” that wire. Similarly, a current varying with time (an alternating
current) will produce a magnetic field around the wire carrying the current.
Since the current is
varying so will the magnetic field. This varying magnetic
field “cuts” the conductor and a voltage is induced in the wire which acts to
impede the originating current.
The relationship between the current and the induced voltage
is defined by a quantity called the inductance. One Henry is the amount of
inductance required to induce one volt when the current is changing at the rate
of one ampere per second. The letter L is used to represent the inductance in
Henries.
The inductance, L, of one phase of a transmission or
distribution line is calculated by considering the self-inductance of the
individual phase conductor and the mutual inductance between that phase and all
other nearby phases both of the same circuit/feeder and other nearby
circuits/feeders.These quantities
are calculated based on the physical dimensions of the wires
and the distances between them. The induced voltage across an inductor will be
a maximum when the rate of change of current is greatest.
Because of the sinusoidal shape of the current, this occurs
when the actual current is zero. Thus the induced voltage reaches its maximum
value a quarter-cycle before the current does—the voltage across an inductor is
said to lead the current by 90 degrees, or conversely, the current lags the
voltage by 90 degrees.
Inductive Reactance
The inductive reactance, XL is a term defined to enable us
to calculate the magnitude of the voltage drop across an inductor.The inductive
reactance is measured in Ohms and it is equal to 2*p* f* L, where 2pf is the
rotational speed in radians per second; p is called pi and its value is 3.1416,
f = frequency in hertz and L = inductance in Henries. Inductances consume
reactive power or VARs equal to I2XL.
Capacitance and Capacitive Reactance
An electric field around the conductor results from a
potential difference between the conductor and ground. There is also a
potential difference between each conductor in a three phase circuit and with
any other nearby transmission lines. The relationship between the charge and
the potential difference
is defined by a quantity called the capacitance. One Farad
is the amount of capacitance present when one coulomb produces a potential
difference of one volt. The letter F is used to represent the capacitance in
Farads.
The capacitance C, depends on the dimensions of the
conductor and the spacing between the adjacent lines and ground. Since the
charge on a capacitor varies directly with the voltage, when an alternating
voltage is impressed across a capacitor, the flow of charge (or current) will
be greatest when the rate of change of voltage is at a maximum.
This occurs when the voltage wave crosses the zero point.
Thus in an alternating current system, the current across a capacitor reaches
its maximum value a quarter-cycle before the voltage does—the voltage is said
to lag the current by 90 degrees, or conversely, the current leads the voltage
by 90 degrees.
Capacitive Reactance
The capacitive reactance, XC, is a term defined equal to
1/2*p * f*C, where C = capacitance in Farads. The unit of the capacitive
reactance is Ohms. In a power system the capacitive reactance is viewed as a
shunt connecting the conductor to ground. Capacitors supply reactive power or
VARs equal to I2Xc.
Reactance
Both inductive reactance and capacitive reactance have an
impact on the relationship between voltage and current in electric circuits.
Although they are both measured in Ohms, they cannot be added to the resistance
of the circuit since their impacts are quite different from that of resistance.
In fact, their impacts differ one from the other. The current
through an inductor leads the voltage by 90 degrees, while current through a
capacitor lags the voltage by 90 degrees. Because of this difference, their
effects will cancel one another.
The convention is to consider the effect associated with the
inductive reactance a positive value and that with the capacitive reactance a
negative value and VARs as consumed by inductive reactance and supplied by
capacitive reactance. A general term, reactance, is defined which represents
the net effect of the capacitive reactance and inductive reactance. It is
denoted by the capital letter X.
Impedance
Once determined, the reactance is combined with the
resistance of a circuit to form a new quantity called Impedance which is
denoted by the capital letter Z.To determine a single number representation of
the impedance, the concept known as complex numbers is employed. Simply
speaking, resistance and reactance are treated as both legs of a right triangle
separated by 90 degrees. A common way of representing the impedance term is:
Z=R+j(XL-XC)
where the letter j, is used as a convention to indicate that
the reactance is not to be directly added to the resistance. The magnitude of
the impedance is determined by Pythagoras’ Theorem, that is, the square of the
impedance is equal to the sum of the squares of the resistance and the
reactance.
where X=XL-XC
Z=(R2+X2)^0.5,
Z2=R2+X2,or
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