# ELECTRIC CIRCUIT ANALYSIS - I

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Chapter 17 Methods of Analysis & Sel TopicsLecture 23 by Moeen Ghiyas**

Independent vs Dependent SourcesSource ConversionsMesh AnalysisBridge Networks**

Independent SourcesThe term independent specifies that the magnitude of the source is independent of the network to which it is applied and that the source displays its terminal characteristics even if completely isolated.

Dependent (Controlled) SourcesA dependent or controlled source is one whose magnitude is determined (or controlled) by a current or voltage of the system in which it appearsOld SymbolsNew Symbols

Dependent (Controlled) SourcesUnlike with the independent source, isolation such that V or I = 0 will result in short-circuit or open-circuit equivalent as indicated

Source conversion can be accomplished in much the same manner as for dc circuits, except now we shall be dealing with phasors and impedances instead of just real numbers and resistors

EXAMPLE - Convert the voltage source of fig to a current sourceSolution:

EXAMPLE - Convert the current source of fig to a voltage sourceSolution:

For dependent sources, the direct conversion can be applied if the controlling variable (V or I) is not determined by a portion of the network to which the conversion is to be appliedConversions of the other kind, where V and I are controlled by a portion of the network to be converted, are covered in chapter 18 (Not part of syllabus for 2nd semester)

EXAMPLE - Convert the voltage source of fig to a current sourceSolution:

StepsAssign a distinct current direction to each independent, closed loopIndicate the polarities within each loop for each resistor (impedance) . Note that the polarities be placed within each loop. Thus 4 resistor have two sets of polarities across it.

StepsApply Kirchhoffs voltage law around each closed loopthe total current through the element (impedance) is the assumed current of the loop plus the assumed currents of the other loops passing through in the same direction, minus the assumed currents through in the opposite directionThe polarity of a voltage source is unaffected by the direction of the assigned loop currentsSolve the resulting simultaneous linear equations for the assumed loop currents

The general approach to mesh includes the same sequence of steps as for dc except to substitute impedance for resistance and admittance for conductance in the general procedure with minor additional changes as mentioned below:Independent Voltage SourcesSame as dc analysis with impedances and admittance valuesDependent Voltage SourcesStep 3 is modified: Treat each dependent source like an independent source when KVL is applied. However, once the equation is written, substitute the equation for the controlling quantity (i.e an additional eqn is generated for controlling qty)

Independent Current SourcesTreat each current source as an open circuit (recall supermesh concept), and write mesh equations for remaining paths.Then relate the chosen mesh currents to the independent sources to ensure that the unknowns of the final equations are limited simply to the mesh currentsDependent Current SourcesThe procedure is same as for independent current sources (i.e. supermesh concept), except now the dependent sources have to be defined in terms of the chosen mesh currents so that the final equations have only mesh currents as the unknown qtys.

EXAMPLE - Using the general approach to mesh analysis, find the current I1 in FigSolution

The network is redrawn

Apply KVL,Loop 1:+E1 I1Z1 I1Z2 + I2Z2 = 0 I1Z1 + I1Z2 - I2Z2 = E1 I1(Z1 + Z2) I2Z2 = E1 ----- A

Loop 2: E2 I2Z2 + I1Z2 I2Z3 = 0 I1Z2 + I2Z2 + I2Z3 = E2 I1Z2 + I2(Z2 + Z3) = E2 ------ BSolve by determinants

Solve by determinants and then substitute values I1(Z1 + Z2) I2Z2 = E1 I1Z2 + I2(Z2 + Z3) = E2

EXAMPLE - Write the mesh currents for the network of fig having a dependent voltage source.Solution:

EXAMPLE - Write the mesh currents for the network of fig having a independent current source.Solution:Apply supermesh concept

EXAMPLE - Write the mesh currents for the network of fig having a dependent current source.Solution:

Problem # 9 Using mesh analysis, determine the current IL (in terms of V) for the network of figSolution:

Problem # 9 Using mesh analysis, determine the current IL (in terms of V) for the network of figSolution:

Solving by determinants

Bridge networks with reactive components & ac voltage or currentMaxwells Bridge (V source & RC parallel)

From dc we remember, for IZ5 = 0, the following condition must be met

or

Hay Bridge (I source & RC series)

From dc we remember, for VZ5 = 0, the following condition must be met

or

or

Hay bridge when Z5 is replaced by a sensitive galvanometer is used for measuring the resistance and inductance of coils in which the resistance is a small fraction of the reactance XL.Maxwell bridge when Z5 is replaced by a sensitive galvanometer is used for inductance measurements when the resistance of the coil is large enough not to require a Hay bridge.

Another popular bridge is the capacitance comparison bridge of fig. An unknown capacitance and its associated resistance can be determined using this bridge.

Independent vs Dependent SourcesSource ConversionsMesh AnalysisBridge Networks

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