Superposition, Thevenin / Norton Equivalents, Maximum Power Transfer

1
Superposition, Thevenin / Norton Equivalents,
Maximum Power Transfer

• Circuits 1
• Fall 2005
• Harding University
• Jonathan White

2
Outline Ch. 4

• Superposition
• Method of analyzing a circuit by turning off all
  sources but 1 and then finding their
  contributions individually. End by summing up
  all the contributions.
• Thevenin Equivalent Circuits
• A circuit at a given 2 terminals can be replaced
  by a voltage source with a resistor in series.
• Norton Equivalent Circuits
• A circuit at a given 2 terminals can also be
  replaced with a current source and a parallel
  resistor.
• Maximum Power Transfer
• When you have a load, when does it receive the
  maximum power? Weve already answered this in
  lab.

3
Superposition

• Resistors are linear elements, meaning that the
  output is linearly related to the input.
• Voltages around a loop can simply be added up
  no non linear math is required.
• Instead of analyzing circuits like we did in Ch.
  2 and Ch. 3, we can analyze them using
  Superposition.
• Definition The voltage across (or current
  through) a resistor is the algebraic sum of all
  the contributions due to each source acting
  alone.
• So, another way to analyze a circuit is to find
  the contribution of each source individually and
  them add them up at the end to get the total.

4
Superposition 2

• We only consider 1 independent source at a time
  when we use superposition. This means that we
• Replace voltage sources with a wire (0 V).
• Replace current sources with an open circuit (no
  current can flow).
• Dependent sources are left intact since they are
  controlled by circuit variables.

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Superposition 3

• To solve a circuit using superposition
• Turn off all independent sources but 1. Use the
  techniques of Ch. 2 and Ch. 3 to solve for the
  desired voltage or current.
• Repeat for each independent source.
• Find the total voltage or contribution by taking
  the algebraic sum.

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Superposition Exp. 1
Find the voltage over the 2 Ohm resistor using
superposition.
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Superposition Exp. 2
Find the voltage over the 5 ohm resistor using
superposition.
V -
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Equivalent Circuits

• A model of the real thing.
• Used to capture only the necessary details of a
  potentially complex circuit.
• Examples of various models
• Battery
• OSI network layer
• Function calls
• You (as a user), dont really care how the
  function operates, just that it does.

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Thevenin Equivalent Circuits

• Consists of a voltage source and a resistor in
  series.
• Used to provide a black box picture from the
  view of a load. The load, looking back in to the
  circuit, only wants to know the voltage and
  current that is provided to it.

10
Finding a TEC

• Steps
• Find the open circuit voltage disconnect the
  load from the circuit and calculate the voltage
  looking in to the circuit.
• Find the open circuit equivalent resistance
  looking back in to the circuit
• Remove all independent current sources
• Replace all independent voltage sources with
  wires.
• Rth is then that equivalent resistance and Vth is
  just the voltage that you found.

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TEC Example - 1
Find the Thevenin Equivalent Circuit
a
b
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TEC Example - 2
Find the Thevenin Equivalent Circuit
a
b
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Norton Equivalent Circuits

• Consists of a current source with a resistor in
  parallel.
• Electrically equivalent to the Thevenin model
• Rth is the same
• In is equal to Vth / Rth
• When finding Norton equivalents, I often
  recommend just finding the Thevenin equivalent
  and then just switch at the end.

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Norton Example
Find the Norton Equiv. Circuit
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Source Transformations

• Like the Wye-Delta transformation, we can
  transform a voltage 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.
• Like superposition, however, this is often more
  work than just using mesh currents to solve the
  problem.

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Source Transformation Exp.
Find i0 and the voltage over the 3 ohm resistor
using source transformations.
i0
V -
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Maximum Power Transfer

• When does the load receive maximum power? see
  notes
• When RL Rth

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