ELECTRICAL STUDY MATERIAL | NORTON’S THEOREM

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NORTON’S THEOREM

Norton’s theorem states that any two terminals   A & B of a network composed of linear resistances (see fig.(a)) and independent sources (voltage or current, combination of voltage and current sources) may be replaced by an equivalent current source and a parallel resistance. The magnitude of current source is the current measured in the short circuit placed across the terminal pair  A & B . The parallel resistance is the equivalent resistance looking into the terminal pair  A & B  with all independent sources has been replaced by their internal resistances.

Any linear dc circuit, no matter how complicated, can also be replaced by an equivalent circuit consisting of one dc current source in parallel with one resistance.  Precisely, Norton’s theorem is a dual of Thevenin’s theorem. To find a current  ” IL ” through the load resistance ” RL ” (as shown in fig.(a)) using Norton’s theorem, the following steps are  followed:

norton theorem study material 1

fig. (a)

Step-1: Short the output terminal after disconnecting the load resistance ( RL ) from the terminals  A & B  and then calculate the short circuit current IN (as shown in fig.(b)). In general, one can apply any of the techniques (mesh-current, node-voltage and superposition method) learnt in earlier lessons to compute  IN (experimentally just measure the short-circuit current using an ammeter).

NORTON THEOREM STUDY MATERIAL 2

Step-2: Redraw the circuit with each practical sources replaced by its internal resistance while the short–circuit across the output terminals removed (note: voltage sources should be short-circuited (just replace with plain wire) and current sources should be open-circuited (just removed)). Look backward  into the resulting circuit from the load terminals ( A & B ), as suggested by the eye in fig.(c).

NORTON THEOREM STUDY MATERIAL 3

Step-3: Calculate the resistance that would exist between the load terminals   A & B  ( or equivalently one can think as if a voltage source is applied across the load terminals and then trace the current distribution through the circuit (fig.(c)) in order to calculate the resistance across the load terminals). This resistance is denoted as RN , is shown in fig.(d).Once again, calculating this resistance may be a difficult task but one can try to use the standard circuit reduction technique or  Y-Δ or Δ-Y transformation techniques. It may be noted that the value of Norton’s resistance RN is truly same as that of Thevenin’s resistance RTh in  a circuit.

Step-4: Place RN in parallel with current  IN to form the Norton’s equivalent circuit (replacing the imaginary fencing portion or fixed part of the circuit with an equivalent practical current source) as shown in fig. (d).

Step-5: Reconnect the original load to the Norton current circuit; the load’s voltage, current and power may be calculated by a simple arithmetic operation only.

NORTON THEOREM STUDY MATERIAL 4

Load current    IL = (RL/ RN+RL) × IN

Voltage across the load    VL = IL × RL

Power absorbed by the load  PL=IL2 × RL

Remarks:

(i)  Similar to the Thevenin’s theorem, Norton’s theorem has also a similar advantage over the normal circuit reduction technique or any other  technique when it is used to calculate load current (  IL ), load voltage ( VL ) and load power (PL ) for different loads.

(ii)Fortunately, with help of either Norton’s theorem or Thevenin’s theorem one can find the choice of load resistance   RL that results in the maximum power transfer to the load.

(iii) Norton’s current source may be replaced by an equivalent Thevenin’s voltage source . The magnitude of voltage source ( VTh ) and its internal resistances (RTh ) are expressed by the following relations

VTh = IN × RN RTh = RN ( with proper polarities at the terminals)

In other words, a source transformation converts a Thevenin’s equivalent circuit into a Norton equivalent circuit or vice-verse.

THEVENIN’S THEOREM STUDY MATERIAL

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