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Fundamental Theory
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INTRODUCTION
The Protection Engineer is concerned with limiting the
effects of disturbances in a power system. These
disturbances, if allowed to persist, may damage plant
and interrupt the supply of electric energy. They are
described as faults (short and open circuits) or power
swings, and result from natural hazards (for instance
lightning), plant failure or human error.
To facilitate rapid removal of a disturbance from a power
system, the system is divided into 'protection zones'.
Relays monitor the system quantities (current, voltage)
appearing in these zones; if a fault occurs inside a zone,
the relays operate to isolate the zone from the remainder
of the power system.
The operating characteristic of a relay depends on the
energizing quantities fed to it such as current or voltage,
or various combinations of these two quantities, and on
the manner in which the relay is designed to respond to
this information. For example, a directional relay
characteristic would be obtained by designing the relay
to compare the phase angle between voltage and current
at the relaying point. An impedance-measuring
characteristic, on the other hand, would be obtained by
designing the relay to divide voltage by current. Many
other more complex relay characteristics may be
obtained by supplying various combinations of current
and voltage to the relay. Relays may also be designed to
respond to other system quantities such as frequency,
power, etc.
In order to apply protection relays, it is usually necessary
to know the limiting values of current and voltage, and
their relative phase displacement at the relay location,
for various types of short circuit and their position in the
system. This normally requires some system analysis for
faults occurring a
Complex Numbers
A complex number may be defined as a constant that
represents the real and imaginary components of a
physical quantity. The impedance parameter of an
electric circuit is a complex number having real and
imaginary components, which are described as resistance
and reactance respectively.
Confusion often arises between vectors and complex
numbers. A vector, as previously defined, may be a
complex number. In this context, it is simply a physical
quantity of constant magnitude acting in a constant
direction. A complex number, which, being a physical
quantity relating stimulus and response in a given
operation, is known as a 'complex operator'. In this
context, it is distinguished from a vector by the fact that
it has no direction of its own.
Because complex numbers assume a passive role in any
calculation, the form taken by the variables in the
problem determines the method of re
CIRCUIT QUANTITIES AND CONVENTIONS
Circuit analysis may be described as the study of the
response of a circuit to an imposed condition, for
example a short circuit. The circuit variables are current
and voltage. Conventionally, current flow results from
the application of a driving voltage, but there is
complete duality between the variables and either may
be regarded as the cause of the other.
When a circuit exists, there is an interchange of energy;
a circuit may be described as being made up of 'sources'
and 'sinks' for energy. The parts of a circuit are described
as elements; a 'source' may be regarded as an 'active'
element and a 'sink' as a 'passive' element. Some circuit
elements are dissipative, that is, they are continuous
sinks for energy, for example resistance. Other circuit
elements may be alternately sources and sinks, for
example capacitance and inductance. The elements of a
circuit are connected together to form a network having
nodes (terminals or junctions) and branches (series
groups of elements) that form closed loops (meshes).
In steady state a.c. circuit theory, the ability of a circuit
to accept a current flow resulting from a given driving
voltage is called the impedance of the circuit. Since
current and voltage are duals the impedance parameter
must also have a dual, called admittanc
Sign Conventions
In describing the electrical state of a circuit, it is often
necessary to refer to the 'potential difference' existing
between two points in the circuit. Since wherever such
a potential difference exists, current will flow and energy
will either be transferred or absorbed, it is obviously
necessary to define a potential difference in more exact
terms. For this reason, the terms voltage rise and voltage
drop are used to define more accurately the nature of the
potential difference.
Voltage rise is a rise in potential measured in the
direction of current flow between two points in a circuit.
Voltage drop is the converse. A circuit element with a
voltage rise across it acts as a source of energy. A circuit
element with a voltage drop across it acts as a sink of
energy. Voltage sources are usually active circuit
elements, while sinks are usually passive circuit
elements. The positive direction of energy flow is from
sources to sinks.
Kirchhoff's first law states that the sum of the driving
voltages must equal the sum of the passive voltages in a
closed loop. This is illustrated by the fundamental
equation of an electric circuit:
BASIC CIRCUIT LAWS, THEOREMS AND NETWORK REDUCTION
Most practical power system problems are solved by
using steady state analytical methods. The assumptions
made are that the circuit parameters are linear and
bilateral and constant for constant frequency circuit
variables. In some problems, described as initial value
problems, it is necessary to study the behaviour of a
circuit in the transient state. Such problems can be
solved using operational methods. Again, in other
problems, which fortunately are few in number, the
assumption of linear, bilateral circuit parameters is no
longer valid. These problems are solved using advanced
mathematical techniques that are beyond the scope of
this book.