Chapter 9

	Network Theorems

Superposition Theorem

	Total current through or voltage across a resistor or branch
		Determine by adding effects due to each source acting independently
	Replace a voltage source with a short

Superposition Theorem

	Replace a current source with an open
	Find results of branches using each source independently
		Algebraically combine results

Superposition Theorem

	Power
		Not a linear quantity
		Found by squaring voltage or current
	Theorem does not apply to power
		To find power using superposition
		Determine voltage or current
		Calculate power

Thévenin’s Theorem

	Lumped linear bilateral network
		May be reduced to a simplified two-terminal circuit
		Consists of a single voltage source and series resistance

Thévenin’s Theorem

	Voltage source
		Thévenin equivalent voltage, ETh.
	Series resistance is Thévenin equivalent resistance, RTh

Thévenin’s Theorem

Thévenin’s Theorem

	To convert to a Thévenin circuit
		First identify and remove load from circuit
	Label resulting open terminals

Thévenin’s Theorem

	Set all sources to zero
	Replace voltage sources with shorts, current sources with opens
	Determine Thévenin equivalent resistance as seen by open circuit

Thévenin’s Theorem

	Replace sources and calculate voltage across open
	If there is more than one source
		Superposition theorem could be used

Thévenin’s Theorem

	Resulting open-circuit voltage is Thévenin equivalent voltage
	Draw Thévenin equivalent circuit, including load

Norton’s Theorem

	Similar to Thévenin circuit
	Any lumped linear bilateral network
		May be reduced to a two-terminal circuit
		Single current source and single shunt resistor

Norton’s Theorem

	RN = RTh
	IN is Norton equivalent current

Norton’s Theorem

Norton’s Theorem

	To convert to a Norton circuit
		Identify and remove load from circuit
	Label resulting two open terminals
	Set all sources to zero

Norton’s Theorem

	Determine open circuit resistance
		This is Norton equivalent resistance
	Note
		This is accomplished in the same manner as Thévenin equivalent resistance

Norton’s Theorem

	Replace sources and determine current that would flow through a short place between two terminals
	This current is the Norton equivalent current

Norton’s Theorem

	For multiple sources
		Superposition theorem could be used
	Draw the Norton equivalent circuit
		Including the load

Norton’s Theorem

	Norton equivalent circuit
		May be determined directly from a Thévenin circuit (or vice-versa) by using source transformation theorem

Norton’s Theorem

Maximum Power Transfer

	Load should receive maximum amount of power from source
	Maximum power transfer theorem states
		Load will receive maximum power from a circuit when resistance of the load is exactly the same as Thévenin (or Norton) equivalent resistance of the circuit

Maximum Power Transfer

	To calculate maximum power delivered by source to load
		Use P = V2/R
	Voltage across load is one half of Thévenin equivalent voltage

Maximum Power Transfer

	Current through load is one half of Norton equivalent current

Maximum Power Transfer

	Power across a load changes as load changes by using a variable resistance as the load

Maximum Power Transfer

Maximum Power Transfer

Efficiency

	To calculate efficiency:

Substitution Theorem

	Any branch within a circuit may be replaced by an equivalent branch
		Provided the replacement branch has same current  voltage
	Theorem can replace any branch with an equivalent branch
	Simplify analysis of remaining circuit

Substitution Theorem

	Part of the circuit shown is to be replaced with a current source and a 240 W shunt resistor
		Determine value of the current source

Substitution Theorem

Millman’s Theorem

	Used to simplify circuits that have
		Several parallel-connected branches containing a voltage source and series resistance
		Current source and parallel resistance
		Combination of both

Millman’s Theorem

	Other theorems may work, but Millman’s theorem provides a much simpler and more direct equivalent

Millman’s Theorem

	Voltage sources
		May be converted into an equivalent current source and parallel resistance using source transformation theorem
	Parallel resistances may now be converted into a single equivalent resistance

Millman’s Theorem

	First, convert voltage sources into current sources
	Equivalent current, Ieq, is just the algebraic sum of all the parallel currents

Millman’s Theorem

	Next, determine equivalent resistance, Req, the parallel resistance of all the resistors
	Voltage across entire circuit may now be calculated by:
	Eeq = IeqReq

Millman’s Theorem

	We can simplify a circuit as shown:

Reciprocity Theorem

	A voltage source causing a current I in any branch
		May be removed from original location and placed into that branch

Reciprocity Theorem

	Voltage source in new location will produce a current in original source location
		Equal to the original I

Reciprocity Theorem

	Voltage source is replaced by a short circuit in original location
	Direction of current must not change

Reciprocity Theorem

	A current source causing a voltage V at any node
		May be removed from original location and connected to that node
	Current source in the new location
		Will produce a voltage in original location equal to V

Reciprocity Theorem

	Current source is replaced by an open circuit in original location
	Voltage polarity cannot change