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  Concept of Thevenin and Norton Theorems

Introduction:
   
Electric circuits can at times become extensive in number of elements and branches or loops which they comprise. This can make their analysis unwieldy on occasions, although there are systematic techniques that can be applied for the purpose. Dc circuits have only three primitive elements namely: constant voltage sources, constant current sources and the resistors. Thus any means of decreasing the complexity of the circuit to have fewer of such primitive elements will very much help the task of analysing a circuit. Two theorems that allow this reduction are Thevenin’s Theorem and Norton’s Theorem. Though, before examining such, there are some other formal ideas which require to be understood as they are employed in the application of such theorems.

Open-Circuit Load:

Consider the non-ideal current source and voltage source shown in figure below under open circuit load conditions. The subscript o/c is employed to designate this specific condition. In effect, the load has been disconnected and hence RL→∞ and the output are termed to as open-circuit.

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    Figure: Voltage Source and Current Source beneath Open-Circuit Load Conditions

In this case, no current flows into the load and hence IO/C = 0 for both the voltage and current source. The output voltage on other hand is not zero. In case of voltage source, the output voltage with no load joined will be equivalent to the cell voltage and hence VO/C = E. In case of current source, the open circuit voltage will be find out by the current I flowing via the internal resistance of the non-ideal source and hence VO/C = IRS. The utilization of an open circuit load permits the features of a circuit to be stipulated for infinite load resistance. It is thus particularly helpful for finding the value of ideal cell voltage related with a non-ideal voltage source.

Short-Circuit Load:

Consider the non-ideal current and voltage source shown in figure below beneath short-circuit load conditions. The subscript s/c is employed to designate this specific condition. In result, the load is set to its theoretical minimum with RL→0 and the output is termed to as short-circuit.

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Figure: Voltage and Current Source under Short-Circuit Load Conditions

In this case, the voltage across the load is forced to zero and hence VS/C = 0 for both voltage and current source. The output current on other hand is at maximum. In case of voltage source, the output current is recognized by the cell voltage and internal resistance and hence IS/C = E/RS. In case of current source, the short circuit gives no resistance to current flow and hence all the current given by the source flows into it and no current flows via the internal resistance, RS, and hence  the output current IS/C = I. The utilization of a short-circuit load permits the characteristics of a circuit to be stipulated for zero load resistance. It is thus particularly helpful for determining the ideal value of the current related with a non-ideal current source.

Thevenin’s Theorem:

This theorem was officially introduced by the French telegraph engineer Leon Charles Thevenin (1857-1926), in the year 1883, although similar discoveries had been made formerly. In modern terms precise to our analysis:

Thevenin’s Theorem states that: ‘As seen by a resistive load joined to it, any linear electric circuit comprising of a combination of current or voltage sources and resistors can be substituted by a single voltage source with the Thevenin voltage, VTH and a single internal resistance equivalent to the Thevenin resistance, RTH.’

Consider the scenario shown figure (a) and figure (b) below:

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Figure (a): An Electric Circuit comprising a Combination of Sources and Resistors

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 Figure (b): Thevenin Equivalent of the Circuit of figure (a)

The circuit of figure (a) arbitrarily includes two voltage sources, a current source and some resistors. A load resistance, RL is joined to the terminals of circuit on the right hand side, and a voltage, VL emerges across the load while a current, IL flows via it. Thevenin’s Theorem defines that this can be substituted by the circuit of figure (b) where there is a single non-ideal voltage source of voltage, VTH and internal resistance, RTH. In the second circuit similar voltage, VL is developed across the load when joined and the same current, IL, flows via it as in the circuit of figure (a).

The Thevenin voltage, VTH and Thevenin resistance, RTH are established from the circuit of figure (a) as shown:

The Thevenin voltage, VTH is set up as the output open-circuit voltage evaluated at the output terminals of the circuit with load disconnected.

The Thevenin resistance, RTH is set up as the resistance seen looking back into the output terminals of circuit with load disconnected.

For the aim of setting up the Thevenin resistance all active driving sources should be made inactive. In this case a non-ideal voltage source is made inactive by ‘shorting out’ the cell voltage (efficiently making it zero) and substituting the non-ideal source with its internal resistance and hence this is accounted for. The non-ideal current source is made inactive by ‘open-circuiting’ the current source (efficiently making it zero) and substituting the non-ideal source with its internal resistance.

Norton’s Theorem:

The second theorem surrounds an alternative form of equivalent circuit based on the non-ideal current source. This alternative was introduced by Edward Lawry Norton (1898-1983) while working as an electrical engineer in Bell Labs. In modern terms precise to our analysis:

Norton’s Theorem defines that: ‘as seen by a resistive load joined to it, any linear electric circuit comprising of a combination of current or voltage sources and resistors can be substituted by a single current source with the Norton Current, INR and a single internal resistance equivalent to the Norton resistance, RNR.’

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Figure: Norton Equivalent Circuit

It is thus also possible to find a Norton equivalent circuit for the circuit.

The Norton Current, INR, is set up as the output short-circuit current evaluated at the output terminals of the circuit with load disconnected and substituted by a short-circuit.

The Norton resistance, RNR is set up as the resistance seen looking back into the output terminals of the circuit with load disconnected. It is thus similar to the Thevenin resistance and is found in precisely the same manner.

This is a little harder to establish the Norton parameters for the circuit as it includes other circuit analysis methods. Though, a simpler approach can be taken on the basis that when both the Thevenin and the Norton equivalent circuits are valid, then they must both behave in precisely the same way for all load conditions. This must also comprise open-circuit and short-circuit load conditions. This signifies that the Thevenin equivalent model must produce similar current into a short-circuit load as the Norton equivalent model, namely the Norton current. It as well means that the open-circuit voltage generated by the Norton equivalent model must be similar to the Thevenin voltage. This is illustrated in figure shown below:  

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Figure: Thevenin Model under s/c and Norton Model under o/c conditions

Then for Thevenin model:

IS/C = VTH/RTH = INR => VTH/ INR = RTH

and for the Norton model:

VO/C = INR RNR = VTH => VTH/ INR = RNR

This is essentially an Ohm’s Law for equivalent circuits and confirms that as stated RNR = RTH that permits the value of the current source in Norton model to be drawn as shown in figure below:

INR = VTH/ RTH = 4.8 V/4.4 KΩ = 1.09 mA

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Figure: Norton Equivalent Circuit

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