Dr Tanmoy Debnathtanmoy.debnath@aiub.edu, Office: 6th floor

• PhD, Wireless Network Engineering,

Dublin Institute of Technology, Ireland

• MSc, Hardware for Wireless Communications,

Chalmers University of Technology, Sweden

• BSc, Electrical and Electronic Engineering,

Bangladesh University of Engineering and Technology (BUET), Bangladesh

Network Theorems (for linear circuits )

Theorems help us to solve so called complicated circuits -i.e. circuits that are not series / parallel / series parallel.

• Superposition Theorem

• Thévenin’s, Norton’s Theorems

• Maximum Power Transfer Theorem

• Recriprocity, Millman’s Theorems

Linear Systems• (Boylestad) The term linear indicates that the characteristics of the

network elements (such as the resistors) are independent of the voltage across or current through them.

• (Alexander, Sadiku) A linear circuit is one whose output is linearly related (or directly proportional) to its input. Linearity is a combination of both the homogeneity (scaling) and the additivity properties. A resistor is a linear element because the voltage-current relationship satisfies both properties.

• The homogeneity property requires that if the input is multiplied by a constant, then the output is multiplied by the same constant.

• For a resistor, If the input current is increased by a constant k, then the output voltage increases correspondingly by k, that is, k*V = k*iR

Linear Systems-contd.• The additivity property requires that the response to a sum of inputs is the sum of the responses to each input applied separately.

• Using the voltage-current relationship of a resistor, if v1 = i1R and v2 = i2R then applying (i1 + i2) gives v = (i1 + i2)R = i1R + i2R = v1 + v2

• Since P = i2R = v2/R (making it a quadratic function rather than a linear one), the relationship between power and voltage (or current) is nonlinear. Therefore, the theorems covered in this chapter are applicable to voltage and current and not to power.

Bilateral devices

• (Boylestad) The term, bilateral, refers to the fact that there is no change in the behavior or characteristics of an element if the current through or voltage across the element is reversed.

Source transformations

Superposition Theorem

• The superposition principle states that the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone. (Alexander, Sadiku)

1. Consider one source at a time

2. Kill all other sources - Open circuit the current source(s) Short circuit the voltage source(s)

3. Calculate the resulting current/ voltage

4. Add (or subtract) all the resulting currents/voltages keeping the polarity in mind (be careful!)

Thevenin’s and Norton’s Theorems

Léon Charles Thévenin(1857– 1926, France)

Edward Lawry Norton(1898 – 1983, USA)

Thevenin’s Theorem• It replaces a complex two-terminal linear circuit to a simpler one

which consists of an equivalent voltage source in series connection with an equivalent resistance.

VTH = Thévenin voltage RTH = Thévenin resistance

How to obtain the equivalent circuit?

1. Identify and isolate the circuit and terminals for which the Thévenin equivalent circuit is required

2. RTH: Kill the independent sources and determine the equivalent resistance of the circuit as seen by the load resistance.

- voltage sources should be short-circuited (just remove them and replace with plain wire)

- current sources should be open-circuited (just removed) VTH: Re-activate the sources and determine the open-circuit

voltage VTH across the circuit terminals (by using KCL, KVL, CDR,

VDR, Mesh analysis, Nodal analysis, superposition theorem…)

Practical things to remember

• RTH = Equivalent resistance seen by the load RL

• For open circuit: current through that branch is zero but there is voltage present

For short circuit: current through that is finite and but voltage across that element is zero

• Calculate VTH or ETH with caution

• If possible, convert (voltage/current) sources to make the circuit simpler

• If asked to calculate power, use the following equation:

Norton’s Theorem• It replaces a complex two-terminal linear circuit to a simpler one

which consists of an equivalent current source in parallel connection with an equivalent resistance.

IN = Norton short circuit currentRN = Equivalent Norton resistance

(Figure courtesy: Alexander, Sadiku)

How to obtain the equivalent circuit?

1. Identify and isolate the circuit and terminals for which the Norton equivalent circuit is required

2. RN:(RN = RTH) Kill the independent sources and determine the equivalent resistance of the circuit as seen by the load resistance.

- voltage sources should be short-circuited (just remove them and replace with plain wire)

- current sources should be open-circuited (just removed) IN: (Isc = IN) Re-activate all the sources and determine the short-

circuit current through the circuit terminals (by using KCL, KVL, CDR, VDR, Mesh analysis, Nodal analysis, superposition theorem…)

vTH+

-

RTH A

B

~Any circuitmade up of

resistors andsources

A

B

A

B

~ iN RN

Copyright: Dr Dave Shattuck, University of Houston, USA

If we have Thevenin equivalent circuit of a network it is possible to obtain the Norton equivalent by using source transformation.

• The Norton resistance RN is the equivalent resistance Req of the circuit. It is calculated after all sources are deactivated. This is same as RTH. i.e., RTH = RN = Req

• The current source is the current obtained by shorting the output of the network, i.e. it is equal to the short-circuit current for the two-terminal circuit after replacing the load resistance with a wire:

Isc = IN =

Practical things to remember

THTH R/V

Maximum power transfer to a load occurs when the load resistance (RL) matches the Thevenin’s resistance (RTH) of a given system, i.e., RL = RTH = RN

TH

TH

R

V

4

2

Maximum Power Transfer Theorem

Load voltage:

Delivered max power to the load:

L

LL R

VP

2

22

THL

L

L

TH

RR

R

R

V

THL

LTHL RR

RVV

DC operating efficiency under maximum power transfer condition is 50%

(Power delivered to the load / Power delivered by the source)

4

2NNRI

(Summary Courtesy: Alexander, Sadiku)