Slide 1

Unit 4Network TheoremsSuperposition TheoremSource TransformationThevenins and Nortons theoremMaximum Power transfer theorem

Principle of Superposition-states that the response (a desired current or voltage) in a linear circuit having more than one independent source can be obtained by adding the responses caused by the separate independent sources acting alone.Superposition Theorem

In any linear resistive network, the voltage across or the current through any resistor or source may be calculated by adding algebraically all the individual voltages or currents caused by the separate independent sources acting alone with all other independent voltage sources replaced by short circuits and all other independent current sources replaced by open circuits.Superposition TheoremIf there are N independent sources, then there will be N experiments to be performed each having only one of the independent sources active and the others inactive/turned off/zeroed out.

Use superposition to determine the unknown branch current ix.Problem 1

vs=3Vis=2A69ix

Use superposition to determine the current i and voltage v

Problem 2

Problem 3Use superposition principle to determine the value of ix.

10V3A212ix+v-ix

Use superposition to obtain the voltage across each current source.

2A71553V4iiv2v1

Practice Problem 1

Practice Problem 2 Use superposition to determine the current ix.

2A3.5V15735ix

Equivalent SourcesAn ideal current source has the voltage necessary to provide its rated current.An ideal voltage source supplies the current necessary to provide its rated voltage.A real voltage source cannot supply arbitrarily large amounts of current.A real current source cannot have an arbitrarily large terminal voltage.

Source Transformation

Source Transformation

Vs

Rs

Is

Rs

+

Open Circuit VoltageIf the current flowing from a source is zero, then the source is connected to an open circuit.The voltage at the source terminals with i(t) equal to zero is called the open circuit voltage: voc(t)Short Circuit CurrentIf the voltage across the source terminals is zero, then the source is connected to a short circuit.The current that flows when v(t) equals zero is called the short circuit current: isc(t)

Source TransformationEquivalent sources can be used to simplify the analysis of some circuits.Averaging CircuitHow can source transformation make analysis of this circuit easier?

+Vout

1kW

1kW

1kW

V1

V2

++A voltage source in series with a resistor is transformed into a current source in parallel with a resistor.A current source in parallel with a resistor is transformed into a voltage source in series with a resistor.

When performing source transformation, the head of the current source arrow corresponds to the +ve terminal of the voltage source.

Problem 4Compute the current through the 4.7-k resistor after transforming the 9-mA source into an equivalent voltage source.

9mA3V5k4.7k3kI

Calculate the value of v0 by source transformation

Problem 5

Compute the voltage V across the 1-M resistor using repeated source transformations.

75A4 M6 M200 k1M3V40A+ V -Problem 6

Calculate the current through the 2- resistor by making use of source transformations.

5A31747921A3VX+VX -IPractice Problem 1

Practice Problem 2Compute the current IX through the 47-k resistor after performing a source transformation on the voltage source.

1mA5V5k47kIX

Thevenins TheoremAny circuit with sources (dependent and/or independent) and resistors can be replaced by an equivalent circuit containing a single voltage source and a single resistor in series.

Nortons TheoremAny circuit with sources (dependent and/or independent) and resistors can be replaced by an equivalent circuit containing a single current source and a single resistor in parallel.

Thevenins/Nortons theorem implies that we can replace arbitrarily complicated networks with simple networks for purposes of analysis.

Independent Sources (Thevenin)

Circuit with independent sources

RTh

Voc

Thevenin equivalent circuit+No Independent Sources

Circuit without independent sources

RTh

Thevenin equivalent circuit

Thevenin/Norton AnalysisPick a good breaking point in the circuit (cannot split a dependent source and its control variable).

2. Thevenin: Compute the open circuit voltage, VOC. Norton: Compute the short circuit current, ISC. 3. Compute the Thevenin equivalent resistance, RTh.If there are only independent sources, then short circuit all the voltage sources and open circuit the current sources (just like superposition).

4. Thevenin: Replace circuit with VOC in series with RTh, . Norton: Replace circuit with ISC in parallel with RTh, .

Only steps 2 & 4 differ from Thevenin & Norton!

Determine the Thevenin and Norton equivalents for the part of the circuit to the left of RL.

3712V6RLProblem 7

Use Thevenins theorem to find the current through the 2- resistor in the circuit.

454269VI2Problem 8

Determine the Thevenin and Norton equivalent circuits for the network faced by the 1-k resistor.

2k3k4V1k2mAProblem 9

Determine the Thevenin and Norton equivalent circuits for the network faced by the 1 resistor and the current i. Problem 10

Practice Problem 1Determine the Thevenin and Norton equivalents of the circuit below.

2k1k5k7mA3V

Maximum Power TransferAn independent voltage source in series with a resistance RS, or an independent current source in parallel with a resistance Rs, delivers a maximum power to that load resistance RL for which RL=RS.

RSRL+

vL

_iLvS