Chapter 9 – Network Theorems

9.1 - Introduction This chapter introduces important fundamental

theorems of network analysis. They aresuperposition, Thévenin’s, Norton’s, maximumpower transfer, substitution, Millman’s, andreciprocity theorems

9.2 - Superposition Theorem Used to find the solution to networks with two or more

sources that are not in series or parallel The current through, or voltage across, an element in a linear

bilateral network is equal to the algebraic sum of the currentsor voltages produced independently by each source

For a two-source network, if the current produced by onesource is in one direction, while that produced by the other isin the opposite direction through the same resistor, theresulting current is the difference of the two and has thedirection of the larger

If the individual currents are in the same direction, theresulting current is the sum of two in the direction of eithercurrent

Superposition TheoremThe total power delivered to a resistive element must

be determined using the total current through or thetotal voltage across the element and cannot bedetermined by a simple sum of the power levelsestablished by each source

9.3 - Thévenin’s TheoremAny two-terminal, linear bilateral dc network can be

replaced by an equivalent circuit consisting of a voltagesource and a series resistor

Thévenin’s Theorem The Thévenin equivalent circuit provides an

equivalence at the terminals only – the internalconstruction and characteristics of the original networkand the Thévenin equivalent are usually quite different

This theorem achieves two important objectives: Provides a way to find any particular voltage or current in a

linear network with one, two, or any other number ofsources

We can concentration on a specific portion of a network byreplacing the remaining network with an equivalent circuit

Thévenin’s Theorem Sequence to proper value of RTh and ETh

Preliminary 1. Remove that portion of the network across which the

Thévenin equation circuit is to be found. In the figure below,this requires that the load resistor R L be temporarily removedfrom the network.

Thévenin’s Theorem 2. Mark the terminals of the remaining two -terminal

network. (The importance of this step will become obviousas we progress through some complex networks)

RTh: 3. Calculate RTh by first setting all sources to zero (voltage

sources are replaced by short circuits, and current sourcesby open circuits) and then finding the resultant resistancebetween the two marked terminals. (If the internal resistanceof the voltage and/or current sources is included in theoriginal network, it must remain when the sources are set tozero)

Thévenin’s Theorem ETh:

4. Calculate ETh by first returning all sources to theiroriginal position and finding the open -circuit voltagebetween the marked terminals. (This step is invariably theone that will lead to the most confusion and errors. In allcases, keep in mind that it is the open -circuit potentialbetween the two terminals marked in step 2)

Thévenin’s Theorem Conclusion:

5. Draw the Théveninequivalent circuit with theportion of the circuitpreviously removed replacedbetween the terminals of theequivalent circuit. This step isindicated by the placement ofthe resistor RL between theterminals of the Théveninequivalent circuit

Insert Figure 9.26(b)Insert Figure 9.26(b)

Thévenin’s Theorem Experimental Procedures Two popular experimental procedures for

determining the parameters of the Théveninequivalent network: Direct Measurement of ETh and RTh

For any physical network, the value of ETh can be determinedexperimentally by measuring the open -circuit voltage across theload terminals

The value of RTh can then be determined by completing thenetwork with a variable resistance R L

Thévenin’s Theorem Measuring VOC and ISC

The Thévenin voltage is again determined by measuringthe open-circuit voltage across the terminals of interest; thatis, ETh = VOC. To determine RTh, a short-circuit condition isestablished across the terminals of interest and the currentthrough the short circuit Isc is measured with an ammeter

Using Ohm’s law:

RTh = Voc / Isc

9.4 - Norton’s Theorem Norton’s theorem states the following:

Any two linear bilateral dc network can be replaced by anequivalent circuit consisting of a current and a parallelresistor.

The steps leading to the proper values of I N and RN

Preliminary 1. Remove that portion of the network across which the

Norton equivalent circuit is found 2. Mark the terminals of the remaining two -terminal

network

Norton’s Theorem RN:

3. Calculate RN by first setting all sources to zero (voltagesources are replaced with short circuits, and current sourceswith open circuits) and then finding the resultant resistancebetween the two marked terminals. (If the internal resistanceof the voltage and/or current sources is included in theoriginal network, it must remain when the sources are set tozero.) Since RN = RTh the procedure and value obtainedusing the approach described for Thévenin’s theorem willdetermine the proper value of R N

Norton’s Theorem IN :

4. Calculate IN by first returning all the sources to theiroriginal position and then finding the short -circuit currentbetween the marked terminals. It is the same current thatwould be measured by an ammeter placed between themarked terminals.

Conclusion: 5. Draw the Norton equivalent circuit with the portion of

the circuit previously removed replaced between theterminals of the equivalent circuit

9.5 - Maximum Power TransferTheorem

The maximum power transfer theorem statesthe following: A load will receive maximum power from a linearbilateral dc network when its total resistive value isexactly equal to the Thévenin resistance of thenetwork as “seen” by the load

RL = RTh

Maximum Power Transfer Theorem For loads connected directly to a dc voltage

supply, maximum power will be delivered to theload when the load resistance is equal to theinternal resistance of the source; that is, when:

RL = Rint

9.6 - Millman’s Theorem Any number of parallel voltage sources can be

reduced to one This permits finding the current through or voltage across

RL without having to apply a method such as mesh analysis,nodal analysis, superposition and so on.

1. Convert all voltage sources to current sources2. Combine parallel current sources3. Convert the resulting current source to a voltage source, and the

desired single-source network is obtained

9.7 - Substitution Theorem The substitution theorem states:

If the voltage across and the current through any branchof a dc bilateral network is known, this branch can bereplaced by any combination of elements that will maintainthe same voltage across and current through the chosenbranch

Simply, for a branch equivalence, the terminal voltageand current must be the same

9.8 - Reciprocity Theorem The reciprocity theorem is applicable only to single -

source networks and states the following: The current I in any branch of a network, due to a single

voltage source E anywhere in the network, will equal thecurrent through the branch in which the source wasoriginally located if the source is placed in the branch inwhich the current I was originally measured

The location of the voltage source and the resulting currentmay be interchanged without a change in current

9.9 - Application Speaker system

Insert Fig 9.111Insert Fig 9.111