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LINEAR CIRCUIT ANALYSIS
(EED) – U.E.T. TAXILA 02
ENGR. M. MANSOOR ASHRAF
INTRODUCTION
To actually determine the values of voltage or current of elements in given circuit, requires study of basic laws. The basic laws: Ohm’s law and Kirchhoff’s laws, form the foundation of electric circuit analysis. In addition to these laws, some techniques are applied for circuit design and analysis. These techniques includes combining resistors in series or parallel, voltage division, current division, and delta-to- wye and wye-to-delta transformations.
OHM’S LAW
Ohm’s Law states that the voltage across a resistor is directly proportional to the current flowing through the resistor. Mathematically;
Where ‘R’ is known as resistance of element (Resistor).
RESISTANCE
Materials in general have a characteristic behavior of resisting the flow of electric charge. This physical property, or ability to resist current, is known as resistance. The Resistance of an element denotes its ability to resist the flow of electric current, measured in ohms (Ω). Mathematically;
PASSIVE SIGN CONVENTION
To apply Ohm’s law on resistor, the direction of current i and the polarity of voltage v must conform the passive sign convention.
The Passive Sign Convention implies that current flows from a higher potential to a lower potential, v=iR. If the current flows from a lower potential to higher potential, v=-iR.
RESISTOR
The circuit element used to model the current-resisting behavior of a material, is known as Resistor. Resistor is passive element. There are two major types of resistors: fixed resistor and variable resistor.
FIXED RESISTOR
Fixed Resistor is that which exhibits the constant value of resistance. There are two common types of fixed resistors: wire wound and composition. The wire wound resistors are used for large values of resistance.
VARIABLE RESISTOR
Variable Resistor is that which exhibits adjustable value of resistance. The variable resistor is also known as Potentiometer.
LINEAR RESISTOR
A resistor that obeys Ohm’s law is known as Linear Resistor. Its has a constant resistance and thus its current-voltage characteristic (i-v graph) is straight line passing through origin.
NON-LINEAR RESISTOR
A resistor that does not obey Ohm’s law is known as Non-Linear Resistor. Its resistance varies with current its current-voltage characteristic (i-v graph) is not straight line.
CONDUCTANCE
Conductance is the ability of an element to conduct electric current; measured in mhos (Ʊ) or siemens (S).
Conductance is the reciprocal of resistance and is a useful quantity in electric circuit analysis.
POWER DISSIPATION IN RESISTOR
The power dissipated by a resistor may be expressed in terms of resistance as well as conductance.
NODE
A Node is the point of connection between two or more branches. If a short circuit connects two nodes, the two nodes constitute a single node.
LOOP
A Loop is any closed path in a circuit. A loop is a closed path formed by starting at a node, passing through a set of nodes, and returning to the starting node without passing through any node more than once. A loop is said to be independent if it contains at least one branch which is not a part of any other loop.
LOOP
The loop abca with 2 Ω resistor and loop abca with 3 Ω resistor are examples of independent loops. A network with b branches, n nodes and l independent loops will satisfy the fundamental theorem of network topology.
SERIES CONNECTED ELEMENTS
Two or more elements are in Series if they exclusively share a single node and consequently carry the same current. Elements are in series when they are chain-connected.
KIRCHHOFF’S CURRENT LAW (KCL)
Kirchhoff’s Current Law (KCL) states that the algebraic sum of currents entering a node is zero. Mathematically;
The KCL is based on the law of conservation of charge, which requires that the algebraic sum of charges within a system cannot change. To prove KCL, assume set of currents into a node.
KIRCHHOFF’S CURRENT LAW (KCL)
Integrating both sides;
By this law, currents entering a node may be regarded as positive, while currents leaving the node may be taken as negative or vice versa.
KIRCHHOFF’S CURRENT LAW (KCL)
The sum of currents entering a node is equal to the sum of current leaving the node. A simple application of KCL is combining the current sources in parallel.
KIRCHHOFF’S VOLTAGE LAW (KVL)
Kirchhoff’s Voltage Law (KVL) states that the algebraic sum of all voltages around a closed path (loop) is zero. Mathematically;
KVL is based on the law of conservation of energy. KVL can be applied by taking clockwise or counter- clockwise trip around the loop.
PROBLEMS
Find the voltages v 1 and v 2? (8 V, -12 V)
Find the current io and voltage vo? (6 A, 24 V)
PROBLEMS
Find currents and voltages in the circuit?
(3 A, 2 A, 1 A, 24 V, 6 V, 6 V)
SERIES RESISTORS AND VOLTAGE DIVISION
Consider the series circuit; Applying Ohm’s law;
Applying KVL;
SERIES RESISTORS AND VOLTAGE DIVISION
Voltages across each resistor;
The equivalent resistance of any number of resistors connected in Series is the sum of their individual resistances.
PARALLEL RESISTORS AND CURRENT DIVISION
Current through each resistor;
The equivalent resistance of two Parallel resistors is equal to the product of their resistances divided by their sum.
PARALLEL RESISTORS AND CURRENT DIVISION
The equivalent conductance of resistors connected in Parallel is the sum of their individual conductances.
The principle of current division for N resistors is expressed as;
PROBLEMS
Find equivalent resistance? (14.4 Ω)
PROBLEMS
Find voltages and currents? (4 V, 4/3 A)
(5 V, 10 V,
416.7 mA, 250 mA)
