Superposition, Thevenin / Superposition, Thevenin / Norton Equivalents, Norton Equivalents,

Maximum Power TransferMaximum Power Transfer

Circuits 1Circuits 1Fall 2005Fall 2005

Harding UniversityHarding University

Jonathan WhiteJonathan White

Outline – Ch. 4Outline – Ch. 4

SuperpositionSuperposition Method of analyzing a circuit by turning off all sources but 1 and Method of analyzing a circuit by turning off all sources but 1 and

then finding their contributions individually. End by summing up then finding their contributions individually. End by summing up all the contributions.all the contributions.

Thevenin Equivalent CircuitsThevenin Equivalent Circuits A circuit at a given 2 terminals can be replaced by a voltage A circuit at a given 2 terminals can be replaced by a voltage

source with a resistor in series.source with a resistor in series.

Norton Equivalent CircuitsNorton Equivalent Circuits A circuit at a given 2 terminals can also be replaced with a A circuit at a given 2 terminals can also be replaced with a

current source and a parallel resistor.current source and a parallel resistor.

Maximum Power TransferMaximum Power Transfer When you have a load, when does it receive the maximum When you have a load, when does it receive the maximum

power? We’ve already answered this in lab.power? We’ve already answered this in lab.

SuperpositionSuperposition

Resistors are linear elements, meaning that the Resistors are linear elements, meaning that the output is linearly related to the input.output is linearly related to the input. Voltages around a loop can simply be added up – no Voltages around a loop can simply be added up – no

non linear math is required.non linear math is required. Instead of analyzing circuits like we did in Ch. 2 Instead of analyzing circuits like we did in Ch. 2

and Ch. 3, we can analyze them using and Ch. 3, we can analyze them using Superposition.Superposition. Definition: The voltage across (or current through) a Definition: The voltage across (or current through) a

resistor is the algebraic sum of all the contributions resistor is the algebraic sum of all the contributions due to each source acting alone.due to each source acting alone.

So, another way to analyze a circuit is to find the So, another way to analyze a circuit is to find the contribution of each source individually and them add contribution of each source individually and them add them up at the end to get the total.them up at the end to get the total.

Superposition 2Superposition 2

We only consider 1 independent source at We only consider 1 independent source at a time when we use superposition. This a time when we use superposition. This means that we:means that we: Replace voltage sources with a wire (0 V).Replace voltage sources with a wire (0 V). Replace current sources with an open circuit Replace current sources with an open circuit

(no current can flow).(no current can flow).

Dependent sources are left intact since Dependent sources are left intact since they are controlled by circuit variables.they are controlled by circuit variables.

Superposition 3Superposition 3

To solve a circuit using superposition:To solve a circuit using superposition: Turn off all independent sources but 1. Use Turn off all independent sources but 1. Use

the techniques of Ch. 2 and Ch. 3 to solve for the techniques of Ch. 2 and Ch. 3 to solve for the desired voltage or current.the desired voltage or current.

Repeat for each independent source.Repeat for each independent source. Find the total voltage or contribution by taking Find the total voltage or contribution by taking

the algebraic sum.the algebraic sum.

Superposition – Exp. 1Superposition – Exp. 1Find the voltage over the 2 Ohm resistor using superposition.

Superposition Exp. 2Superposition Exp. 2

+ V -

Find the voltage over the 5 ohm resistor using superposition.

Equivalent CircuitsEquivalent Circuits

A model of the real thing. A model of the real thing. Used to capture only the necessary details Used to capture only the necessary details

of a potentially complex circuit.of a potentially complex circuit. Examples of various models:Examples of various models:

BatteryBattery OSI network layerOSI network layer Function callsFunction calls

You (as a user), don’t really care how the function You (as a user), don’t really care how the function operates, just that it does.operates, just that it does.

Thevenin Equivalent CircuitsThevenin Equivalent Circuits

Consists of a voltage source and a resistor Consists of a voltage source and a resistor in series.in series. Used to provide a “black box” picture from the Used to provide a “black box” picture from the

view of a load. The load, looking back in to view of a load. The load, looking back in to the circuit, only wants to know the voltage and the circuit, only wants to know the voltage and current that is provided to it.current that is provided to it.

Finding a TECFinding a TEC

Steps:Steps: Find the open circuit voltage – disconnect the load Find the open circuit voltage – disconnect the load

from the circuit and calculate the voltage looking in to from the circuit and calculate the voltage looking in to the circuit.the circuit.

Find the open circuit equivalent resistance looking Find the open circuit equivalent resistance looking back in to the circuitback in to the circuit Remove all independent current sourcesRemove all independent current sources Replace all independent voltage sources with wires.Replace all independent voltage sources with wires.

RRthth is then that equivalent resistance and V is then that equivalent resistance and Vth th is just the is just the

voltage that you found.voltage that you found.

TEC Example - 1TEC Example - 1Find the Thevenin Equivalent Circuit:

a

b

TEC Example - 2TEC Example - 2Find the Thevenin Equivalent Circuit:

a

b

Norton Equivalent CircuitsNorton Equivalent Circuits

Consists of a current source with a resistor Consists of a current source with a resistor in parallel.in parallel.

Electrically equivalent to the Thevenin Electrically equivalent to the Thevenin modelmodel

RRthth is the same is the same IIn n is equal to Vis equal to Vth th / R/ Rthth

When finding Norton equivalents, I often When finding Norton equivalents, I often recommend just finding the Thevenin recommend just finding the Thevenin equivalent and then just switch at the end.equivalent and then just switch at the end.

Norton ExampleNorton ExampleFind the Norton Equiv. Circuit

Source TransformationsSource Transformations

Like the Wye-Delta transformation, we can transform a voltage Like the Wye-Delta transformation, we can transform a voltage source with a resistor in series into a current source with a resistor source with a resistor in series into a current source with a resistor in parallel without changing the rest of the circuit and vice versa.in parallel without changing the rest of the circuit and vice versa. Like superposition, however, this is often more work than just Like superposition, however, this is often more work than just

using mesh currents to solve the problem.using mesh currents to solve the problem.

Source Transformation Exp.Source Transformation Exp.

i0

+

V

-

Find i0 and the voltage over the 3 ohm resistor using source transformations.

Maximum Power TransferMaximum Power Transfer

When does the load receive maximum When does the load receive maximum power? – see notespower? – see notes When RWhen RLL = R = Rthth

Maximum Power ExampleMaximum Power ExampleFind the RL that achieves maximum power transfer. Find the power it absorbs. Note: You must find Vth to calculate the power.