CHAPTER-2 NETWORK THEOREMS

CONTENT 1. Kirchhoff’s laws, voltage sources and current sources. 2. Source conversion, simple problems in source conversion. 3. Superposition theorem, simple problems in super position theorem. 4. Thevenin’s theorem, Norton’s theorem, simple problems. 5.Reciprocity theorem, Maximum power transfer theorem, simple problems. 6. Delta/star and star/delta transformation.

Gustav Robert Kirchhoff

Definitions Circuit – It is an interconnection of electrical elements in a closed path by conductors(wires). Node – Any point where two or more circuit elements are connected together Branch –A circuit element between two nodes Loop – A collection of branches that form a closed path returning to the same node without going through any other nodes or branches twice

Example How many nodes, branches & loops? R1 + Vo - Is + - Vs R2 R3

Example-Answer Three nodes R1 + Vo - Is + - Vs R2 R3

Example-Answer 5 Branches R1 + Vo - Is + - Vs R2 R3

Example-Answer A B C Three Loops, if starting at node A R1 + Vo - Is + Vs R2 R3 C

Example 9 How many nodes, branches & loops? 5 5

Kirchhoff's Current Law (KCL) Also called Kirchhoff's Point law and Kirchhoff's First rule..

Kirchhoff's Current Law (KCL) Total volume of water flowing through pipe 1 = (total volume of water flowing through pipe 2 + total volume of water flowing through pipe 3)

Kirchhoff's Current Law (KCL) Total current entering the node through the wire 1 = (total current leaving the node through the wire 2 + total current leaving the node through the wire 3)

Kirchhoff's Current Law (KCL) "The algebraic sum of all currents entering and leaving a node must equal zero" ∑ (Entering Currents) = ∑ (Leaving Currents) Established in 1847 by Gustav R. Kirchhoff

Kirchhoff's Current Law (KCL) It states that, in any linear network the algebraic sum of the current meeting at a point (junction) is zero. ∑ I (Junction) = 0

Kirchhoff's Current Law (KCL) ∑ I (Entering) = ∑ I (Leaving) ∑ I (Entering) - ∑ I (Leaving) =0

Kirchhoff's Current Law (KCL) Assign positive signs to the currents entering the node and negative signs to the currents leaving the node, the KCL can be re- formulated as: S (All currents at the node) = 0

Kirchhoff's Current Law (KCL)

Kirchhoff's Current Law (KCL)

Kirchhoff's Current Law (KCL)

Example I1= 1 A I2= 3 A I3= 0.5 A Find the current I4 in A

Kirchhoff's Voltage Law (KVL) Also called Kirchhoff's loop rule and Kirchhoff's second rule..

Kirchhoff's Voltage Law (KVL) “The algebraic sum of voltages around each loop is zero”. Σ voltage rise - Σ voltage drop = 0 Or Σ voltage rise = Σ voltage drop

Kirchhoff's Voltage Law (KVL) It states that, in any linear bilateral active closed network the algebraic sum of the product of the current and resistance in each of the conductors in any closed path (mesh) in the network plus the algebraic sum of e.m.f in the path is zero. ∑ IR + ∑ e.m.f = 0

Kirchhoff's Voltage Law (KVL)

Sign Convention The sign of each voltage is the polarity of the terminal first encountered in traveling around the loop. The direction of travel is arbitrary. Clockwise: Counter-clockwise:

Example A B C Kirchoff’s Voltage Law around 1st Loop I1 + I1R1 - R1 + Is + - Vs R2 R3 C Assign current variables and directions Use Ohm’s law to assign voltages and polarities consistent with passive devices (current enters at the + side)

Example A B C Kirchoff’s Voltage Law around 1st Loop I1 + I1R1 - R1 + Is + - Vs R2 R3 C Starting at node A, add the 1st voltage drop: + I1R1

Example A B C Kirchoff’s Voltage Law around 1st Loop I1 + I1R1 - R1 + Is + - Vs R2 R3 C Add the voltage drop from B to C through R2: + I1R1 + I2R2

Example A B C Kirchoff’s Voltage Law around 1st Loop I1 + I1R1 - R1 + Is + - Vs R2 R3 C Subtract the voltage rise from C to A through Vs: + I1R1 + I2R2 – Vs = 0 Notice that the sign of each term matches the polarity encountered 1st

Source Conversion

Voltage Source to Current Source

Current Source to Voltage Source

Proof

Convert to Current Source

Answer-1

Convert to Voltage Source

Answer-2

Superposition Theorem STATEMENT- In a network of linear resistances containing more than one generator (or source of e.m.f.), the current which flows at any point is the sum of all the currents which would flow at that point if each generator were considered separately and all the other generators replaced for the time being by resistances equal to their internal resistances.

Superposition Theorem STATEMENT- In a linear circuit with several sources the voltage and current responses in any branch is the algebraic sum of the voltage and current responses due to each source acting independently with all other sources replaced by their internal impedance.

Superposition Theorem Replace a voltage source with a short circuit.

Superposition Theorem Replace a current source with an open circuit.

Superposition Theorem Step-1: Select a single source acting alone. Short the other voltage source and open the current sources, if internal impedances are not known. If known, replace them by their internal resistances.

Superposition Theorem Step-2: Find the current through or the voltage across the required element, due to the source under consideration, using a suitable network simplification technique.

Superposition Theorem Step-3: Repeat the above two steps far all sources.

Superposition Theorem Step-4: Add all the individual effects produced by individual sources, to obtain the total current in or voltage across the element.

Explanation

Superposition Theorem Consider a network, having two voltage sources V1 and V2. Let us calculate, the current in branch A-B of network, using superposition theorem. Step-1: According to Superposition theorem, consider each source independently. Let source V1 is acting independently. At this time, other sources must be replaced by internal resistances.

Superposition Theorem But as internal impedance of V2 is not given, the source V2 must be replaced by short circuit. Hence circuit becomes, as shown. Using any of the network reduction techniques, obtain the current through branch A-B i.e. IAB due to source V1 alone.

Superposition Theorem Step 2: Now Consider Source V2 volts alone, with V1 replaced by short circuit, to obtain the current through branch A-B. Hence circuit becomes, as shown. Using any of the network reduction techniques, obtain the current through branch A-B i.e. IAB due to source V2 alone.

Superposition Theorem Step 3: According to the Superposition theorem, the total current through branch A-B is sum of the currents through branch A-B produced by each source acting independently. Total IAB = IAB due to V1 + IAB due to V2

Example Find the current in the 6 Ω resistor using the principle of superposition for the circuit.

Solution Step-1:Replace Current Source with open circuit

Step-2:Replace Voltage Source with Short circuit

Step-3:Current through 6 Ω resistor is

Thevenin’s theorem

Thevenin’s theorem Statement “Any linear circuit containing several voltages and resistances can be replaced by just a Single Voltage VTH in series with a Single Resistor RTH “.

Thevenin’s theorem Thevenin’s Equivalent Circuit Req or RTH VTH

Thevenin’s theorem

Steps to be followed for Thevenin’s Theorem Step 1: Remove the branch resistance through which current is to be calculated. Step 2: Calculate the voltage across these open circuited terminals, by using any of the network simplification techniques. This is VTH.

Steps to be followed for Thevenin’s Theorem Step 3: Calculate Req as viewed through the two terminals of the branch from which current is to be calculated by removing that branch resistance and replacing all independent sources by their internal resistances. If the internal resistance are not known, then replace independent voltage sources by short circuits and independent current sources by open circuits.

Steps to be followed for Thevenin’s Theorem Step 4: Draw the Thevenin’s equivalent showing source VTH, with the resistance Req in series with it, across the terminals of branch of interest. Step 5: Reconnect the branch resistance. Let it be RL. The required current through the branch is given by,

Example- Find VTH, RTH and the load current flowing through and load voltage across the load resistor in fig by using Thevenin’s Theorem.

Step 1- Open the 5kΩ load resistor

Step 2-Calculate / measure the Open Circuit Voltage Step 2-Calculate / measure the Open Circuit Voltage. This is the Thevenin's Voltage (VTH).

Step 3-Open Current Sources and Short Voltage Sources

Step 4-Calculate /measure the Open Circuit Resistance Step 4-Calculate /measure the Open Circuit Resistance. This is the Thevenin's Resistance (RTH)

Step 5-Connect the RTH in series with Voltage Source VTH and re-connect the load resistor. i.e. Thevenin's circuit with load resistor. This the Thevenin’s equivalent circuit. RTH = Thevenin’s Equivalent Circuit =VTH

Step 6- Calculate the total load current & load voltage  

Norton’s theorem

Norton’s theorem STATEMENT- Any Linear Electric Network or complex circuit with Current and Voltage sources can be replaced by an equivalent circuit containing of a single independent Current Source IN and a Parallel Resistance RN.

Norton’s theorem Norton’s Equivalent Circuit IN RN

Norton’s theorem

Steps to be followed for Norton’s Theorem Short the load resistor Step 2: Calculate / measure the Short Circuit Current. This is the Norton Current (IN)

Steps to be followed for Norton’s Theorem Step 3: Open Current Sources, Short Voltage Sources and Open Load Resistor. Calculate /Measure the Open Circuit Resistance. This is the Norton Resistance (RN)

Steps to be followed for Norton’s Theorem Step 4 Now, Redraw the circuit with measured short circuit Current (IN) in Step (2) as current Source and measured open circuit resistance (RN) in step (4) as a parallel resistance and connect the load resistor which we had removed in Step (3). This is the Equivalent Norton Circuit.

Steps to be followed for Norton’s Theorem  

Example 1-Find RN, IN, the current flowing through and Load Voltage across the load resistor in fig (1) by using Norton’s Theorem.

Step 1-Short the 1.5Ω load resistor

Step 2-Calculate / measure the Short Circuit Current Step 2-Calculate / measure the Short Circuit Current. This is the Norton Current (IN).

Step 3-Open Current Sources, Short Voltage Sources and Open Load Resistor.

Step 4-Calculate /measure the Open Circuit Resistance Step 4-Calculate /measure the Open Circuit Resistance. This is the Norton Resistance (RN)

Step 5- Connect the RN in Parallel with Current Source INand re-connect the load resistor.

Step 6-Now apply the last step i. e Step 6-Now apply the last step i.e. calculate the load current through and Load voltage across load resistor  

Maximum Power Transfer Theorem

Maximum Power Transfer Theorem Statement: In an active resistive network, maximum power transfer to the load resistance takes place when the load resistance equals the equivalent resistance of the network as viewed from the terminals of the load.

Steps to be followed for MPTT  

Maximum Power Transfer Theorem

Example-In the network shown, find the value of RL such that maximum possible power will be transferred to RL. Find also the value of the maximum power.

Step 1- Remove RL and Calculate Thevenin’s Voltage VTH and RTH

Thevenin’s Equivalent Circuit

Reciprocity Theorem Statement: In any linear bilateral network, if a source of e.m.f E in any branch produces a current I in any other branch, then the same e.m.f E acting in second branch would produce the same current I in the first branch.

Reciprocity Theorem

Example-In the network given below, find (a) ammeter current when battery is at A and ammeter at B and (b) when battery is at B and ammeter at point A.

What is STAR Connection? If the three resistances are connected in such a manner that one end of each is connected together to form a junction point called STAR point, the resistances are said to be connected in STAR.

Star or Y or T Network

What is DELTA Connection? If the three resistances are connected in such a manner that one end of first is connected to first end of second, the second end of second to first end of third and so on to complete a loop then the resistances are said to be connected in DELTA.

Delta or π Network

STAR to DELTA

DELTA to STAR

To convert a STAR to DELTA

To convert a DELTA to STAR

Example 1-Convert given DELTA into STAR

Answer  

Example 2-Convert given STAR into DELTA

Answer R31 = =R12 =R23

Example 3-Calculate the effective resistance between points A & B

Answer-Step 1

Answer-Step 2

Answer-Step 3

Answer-Step 4 & 5 RAB = 3.69 Ω

Example 4-Find the equivalent resistance between P & Q in the ckt

Solution

Solution

Solution

Solution

Req =14.571Ω

Example 5-In the circuit shown, find the resistance between M and N.

Solution- Step 1

Solution- Step 2

Solution- Step 3

Solution- Step 4 & 5