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F.Y.PHY.P. / II / II / I DC NETWORK ANALYSIS  N. KAPOORWhat is network analysis? Generally speaking, network analysis is any structured technique used to mathematically analyze acircuit (a “network” of interconnected components). Quite often the technician or engineer willencounter circuits containing multiple sources of power or component configurations which defysimplification by series/parallel analysis techniques. In those cases, he or she will be forced to useother means. This chapter presents a few techniques useful in analyzing such complex circuits.

Superposition Theorem →

A linear network which contains two or more independent sources canbe analysed to obtain the various voltage and branch currents by allowing the source to act one at atime, then superposing the results. This principle applies because of the linear relationship betweencurrent and voltage. The theorem can be stated as follows.:--“The current through or voltage across, any element in a linear bilateral network is equal to thealgebraic sum of the current or voltages produced independently by each source.”Voltage sources to be suppressed while a single source acts are replaced by short circuits, currentsources are replaced by open circuits.Superposition cannot be directly applied to the computation of power, because power in an elementis proportional to the square of the current or the square of the voltage, which is non-linear.

Thevenin’s Theorem →

“ Any two-terminal linear bilateral dc network can be replaced by equivalentcircuit consisting of a voltage source (Vth) and a series resistor (Rth).

Where Vth → is open circuited voltage between the two terminals.

Rth → the resistance between two terminals of circuit obtained by looking “in” at theterminals with sources replaced by their internal resistance, if any.

Rth

Network RL  ≅ Vth RL 

Thus Thevenin's Theorem is a way to reduce a network to an equivalent circuit composed of asingle voltage source, series resistance, and series load.Steps to follow for Thevenin's Theorem:(1) Find the Thevenin source voltage by removing the load resistor from the original circuit andcalculating voltage across the open connection points where the load resistor used to be.(2) Find the Thevenin resistance by removing all power sources in the original circuit (voltagesources shorted and current sources open) and calculating total resistance between the openconnection points.(3) Draw the Thevenin equivalent circuit, with the Thevenin voltage source in series with the

Thevenin resistance. The load resistor re-attaches between the two open points of the equivalentcircuit.(4) Analyze voltage and current for the load resistor following the rules for series circuits.

Norton’s Theorem →“Any two-terminal linear bilateral dc network can be replaced by an equivalentcircuit consisting of a current source (IN) & a parallel resistor (RN).

IN  →Short circuit current between marked terminals.

RN→ the resistance between marked terminals of circuit obtained by looking “in” at the

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terminal with sources replaced by their internal resistance, if any. 

Network RL  ≅ IN RN RL 

Thus Norton's Theorem is a way to reduce a network to an equivalent circuit composed of a singlecurrent source, parallel resistance, and parallel load.Steps to follow for Norton's Theorem:(1) Find the Norton source current by removing the load resistor from the original circuit andcalculating current through a short (wire) jumping across the open connection points where the loadresistor used to be.(2) Find the Norton resistance by removing all power sources in the original circuit (voltage sourcesshorted and current sources open) and calculating total resistance between the open connection

points.(3) Draw the Norton equivalent circuit, with the Norton current source in parallel with the Nortonresistance. The load resistor re-attaches between the two open points of the equivalent circuit.(4) Analyze voltage and current for the load resistor following the rules for parallel circuits.

Maximum Power Transfer Theorem → “A resistive load will abstract maximum power from anetwork when the load resistance is equal to the resistance of the network as viewed from theoutput terminals, with all sources replaced by their internal resistances.”In fig. a load resistance of RL is connected across the terminals A & B of a network which consists ofa Vth and equivalent thevenin resistance Rth. According to this theorem, RL will abstract maximumpower from the network when RL = Rth .

PROOF →

  Rth

Vth RL

I =v

R R

th

L th+

power consumed by the load is PL = I2 RL =( )

Vth R

R R

L

L ih

2

2+

-------(1)

For PL to be maximumdP

dR

L

L

= 0

Differentiating eqn. (1) we get --- ( ) ( )( )

dP

dR

V

R R

V R

R R

L

L

th

L th

th L

L th

=+

++

−2 2

32

2

=( )

V

R R

R

R R

th

L th

L

L th

2

21

20

+−

+

=

  ∴ 1 -

20

R

R R

L

L th+

=   ∴  2 RL = RL + Rth 

.∴ RL = Rth

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  ∴ PL max ( )

V R

R

V

Rth th

th

th

th

2

2

2

24

=  

( )Maximum efficiency

power delivered to load

Power delivered by circuit

I RL

I R R

R

Ri e

th L

L

L

= =

+

= =

2

2 20 5 50. . . %

If you were designing a circuit for maximum power dissipation at the load resistance, this theorem

would be very useful. Having reduced a network down to a Thevenin voltage and resistance (or Norton current and resistance), you simply set the load resistance equal to that Thevenin or Nortonequivalent (or vice versa) to ensure maximum power dissipation at the load. Practical applications ofthis might include stereo amplifier design (seeking to maximize power delivered to speakers) or electric vehicle design (seeking to maximize power delivered to drive motor).