NETWORK ANALYSIS

UNIT – I INTRODUCTION TO ELECTRICAL CIRCUITS: Circuit concept – R-L-C parameters Voltage and current sources Independent and dependent sources Source transformations Kirchhoff’s laws network reduction techniques series, parallel, series parallel star – to –delta and or delta – to – star transformation Mesh Analysis Nodal analysis Super mesh super node concept

Network: The interconnection of two or more circuit elements (voltage sources,resistors, inductors and capacitors) is called an electrical network. If the network contains at least one closed path is called circuit. Every circuit is a network, but all the networks are not circuit.

4 Active Components (have directionality) Voltage and current sources Passive Components (Have no directionality) Resistors, capacitors, inductors (with all the initial conditions are zero)

Ohm’s Law I = V / R Georg Simon Ohm ( ) I= Current (Amperes) (amps) V= Voltage (Volts) R= Resistance (ohms)

How you should be thinking about electric circuits: Voltage: a force that pushes the current through the circuit (in this picture it would be equivalent to gravity)

Resistance: friction that impedes flow of current through the circuit (rocks in the river) How you should be thinking about electric circuits:

Current: the actual “substance” that is flowing through the wires of the circuit (electrons!) How you should be thinking about electric circuits:

Lect1EEE 2029 Basic Electrical Quantities Basic quantities: current, voltage and power – Current: time rate of change of electric charge I = dq/dt 1 Amp = 1 Coulomb/sec – Voltage: electromotive force or potential, V 1 Volt = 1 Joule/Coulomb = 1 N·m/coulomb – Power:P = I V 1 Watt = 1 Volt·Amp = 1 Joule/sec

Overview of Circuit Theory Power is the rate at which energy is being absorbed or supplied. Power is computed as the product of voltage and current: Sign convention: positive power means that energy is being absorbed ; negative power means that power is being supplied.

Lect1EEE Active vs. Passive Elements Active elements can generate energy – Voltage and current sources – Batteries Passive elements cannot generate energy – Resistors – Capacitors and Inductors (but CAN store energy)

Energy Storage Elements Capacitors store energy in an electric field. Inductors store energy in a magnetic field. Capacitors and inductors are passive elements: – Can store energy supplied by circuit – Can return stored energy to circuit – Cannot supply more energy to circuit than is stored.

Independent sources : 1.Voltage source 2.Current source Dependent sources: 1.Voltage dependent voltage source 2.Voltage dependent current source 3.current dependent voltage source 4.current dependent current source Types of sources

Ideal voltage source : An ideal voltage source has zero internal resistance so that changes in load resistance will not change the voltage supplied. An ideal voltage source gives a constant voltage, whatever the current is. A simple example is a 10V battery. For example, a 1ohm resistor or a 10ohm resistor could be connected to it; the voltage across both resistors would be 10V but the currents would be different.

Practical voltage source: Practical voltage source has an internal resistance (greater than zero), but we treat this internal resistance as being connected in series with an ideal voltage source. An ideal voltage source has zero internal resistance

Ideal current source: An ideal current source is a circuit element that maintains a prescribed current through its terminals regardless of the voltage across those terminals. A ideal current source gives a constant current whatever the load is. If you have a 2A current source for example: -with a 3 ohm resistor it would automatically change the voltage to 6V -with a 30 ohm resistor it would automatically change the voltage to 60V but the current would be 2A whichever resistor was connected.

Practical current source: Practical current source has an internal resistance, but we treat this internal resistance as being connected in parallel with an ideal current source. An ideal current source has infinite internal resistance.

Dependent sources : Dependent sources behave just like independent voltage and current sources, except their values are dependent in some way on another voltage or current in the circuit.

A dependent source has a value that depends on another voltage or current in the circuit.

Source transformation Another circuit simplifying technique It is the process of replacing a voltage source v S in series with a resistor R by a current source i S in parallel with a resistor R, or vice versa ++ R vsvs a b Terminal a-b sees: Open circuit voltage: v s Short circuit current: v s /R For this circuit to be equivalent, it must have the same terminal charateristics R isis a b

Source Transformations A method called Source Transformations will allow the transformations of a voltage source in series with a resistor to a current source in parallel with resistor. The double arrow indicate that the transformation is bilateral, that we can start with either configuration and drive the other

Equating we have,

Simple Circuits Series circuit – All in a row – 1 path for electricity – 1 light goes out and the circuit is broken Parallel circuit – Many paths for electricity – 1 light goes out and the others stay on

Resistors in Series A single loop circuit is one which has only a single loop. The same current flows through each element of the circuit - the elements are in series.

Resistors in Series Two elements are in series if the current that flows through one must also flow through the other. R1R1 R2R2 Series

Resistors in Series Consider two resistors in series with a voltage v(t) across them: v 1 (t) v 2 (t) R1R1 R2R v(t) i(t) Voltage division:

Resistors in Series If we wish to replace the two series resistors with a single equivalent resistor whose voltage- current relationship is the same, the equivalent resistor has a value given by

Resistors in Series For N resistors in series, the equivalent resistor has a value given by R1R1 R3R3 R2R2 R eq

Resistors in Parallel When the terminals of two or more circuit elements are connected to the same two nodes, the circuit elements are said to be in parallel.

Resistors in Parallel Consider two resistors in parallel with a voltage v(t) across them: R1R1 R2R2 + - v(t) i(t) Current division: i 1 (t)i 2 (t)

Resistors in Parallel If we wish to replace the two parallel resistors with a single equivalent resistor whose voltage- current relationship is the same, the equivalent resistor has a value given by

Resistors in Parallel For N resistors in parallel, the equivalent resistor has a value given by R eq R3R3 R2R2 R1R1

Lect1EEE Parallel Two elements are in parallel if they are connected between (share) the same two (distinct) end nodes. ParallelNot Parallel R1R1 R2R2 R1R1 R2R2

ECE 201 Circuit Theory I34 Series-Parallel Combinations of Inductance and Capacitance Inductors in Series – All have the same current

ECE 201 Circuit Theory I35 To Determine the Equivalent Inductance

ECE 201 Circuit Theory I36 The Equivalent Inductance

ECE 201 Circuit Theory I37 Inductors in Parallel All Inductors have the same voltage across their terminals.

ECE 201 Circuit Theory I38

ECE 201 Circuit Theory I39

ECE 201 Circuit Theory I40 Summary for Inductors in Parallel

ECE 201 Circuit Theory I41 Capacitors in Series Problem # 6.30

ECE 201 Circuit Theory I42 Capacitors in Parallel Problem # 6.31

Ch06 Capacitors and Inductors Series and Parallel Capacitors The equivalent capacitance of N parallel- connected capacitors is the sum of the individual capacitance.

Ch06 Capacitors and Inductors44 Fig 6.15

Ch06 Capacitors and Inductors45 Series Capacitors The equivalent capacitance of series- connected capacitors is the reciprocal of the sum of the reciprocals of the individual capacitances.

Ch06 Capacitors and Inductors46 Table 6.1

Delta -> StarStar -> Delta Y  transformation Star  delta transformation

Circuit in Diagram Form _ + battery bulb In a closed circuit, current flows around the loop Current flowing through the filament makes it glow. No Loop  No Current  No Light current electrons flow opposite the indicated current direction! (repelled by negative terminal)

Lect1EEE Kirchhoff’s Laws Kirchhoff’s Current Law (KCL) – sum of all currents entering a node is zero – sum of currents entering node is equal to sum of currents leaving node Kirchhoff’s Voltage Law (KVL) – sum of voltages around any loop in a circuit is zero

Lect1EEE KCL (Kirchhoff’s Current Law) The sum of currents entering the node is zero: Analogy: mass flow at pipe junction i1(t)i1(t) i2(t)i2(t)i4(t)i4(t) i5(t)i5(t) i3(t)i3(t)

Lect1EEE Open Circuit What if R =  ? i(t) = v(t)/R = 0 v(t)v(t) The Rest of the Circuit i(t)=0 + –

Lect1EEE Short Circuit What if R = 0 ? v(t) = R i(t) = 0 The Rest of the Circuit v(t)=0 i(t)i(t) + –

Lect1EEE Resistors A resistor is a circuit element that dissipates electrical energy (usually as heat) Real-world devices that are modeled by resistors: incandescent light bulbs, heating elements (stoves, heaters, etc.), long wires Resistance is measured in Ohms (Ω)

Overview of Circuit Theory Basic quantities are voltage, current, and power. The sign convention is important in computing power supplied by or absorbed by a circuit element. Circuit elements can be active or passive; active elements are sources.

KCL and KVL Kirchhoff’s Current Law ( KCL ) and Kirchhoff’s Voltage Law ( KVL ) are the fundamental laws of circuit analysis. KCL is the basis of nodal analysis – in which the unknowns are the voltages at each of the nodes of the circuit. KVL is the basis of mesh analysis – in which the unknowns are the currents flowing in each of the meshes of the circuit.

KCL and KVL KCL – The sum of all currents entering a node is zero, or – The sum of currents entering node is equal to sum of currents leaving node. i 1 (t) i 2 (t)i 4 (t) i 5 (t) i 3 (t)

KCL and KVL KVL – The sum of voltages around any loop in a circuit is zero. + - v1(t)v1(t) v2(t)v2(t) v3(t)v3(t)

KCL and KVL In KVL: – A voltage encountered + to - is positive. – A voltage encountered - to + is negative. Arrows are sometimes used to represent voltage differences; they point from low to high voltage. + - v(t)v(t) v(t)v(t) ≡