DC Network Theorem

Electric Circuit

DC Network TheoremSuperposition theoremCurrent through, or voltage across, an element in a linear bilateral network is equal to algebraic sum of the currents or voltages produced independently by each source• used to find the solution to networks with two or more sources not in series or parallel• does not require the use of a mathematical technique such as determinants • each source is treated independently, and the algebraic sum is found to determine unknown quantity

• No. of networks to be analyzed = No. of sources • requires that sources to be removed and replaced without affecting the final result • voltage source must be set in zero (short circuited)• removing a current source requires that its terminals be opened (open circuit)

DC Network Theorem: Superposition theorem

• Fig. 4.1.1 (c) represents when RHS battery acts alone • Fig. 4.1.1 (b) represents when LHS battery acts alone

• II = II' -II“ I2= I2" - I2‘ I = l' + 1"

DC Network Theorem: Superposition theorem Ex: Using superposition theorem find current through resistor R3 (4 Ω) shown

Considering the effect of E1 = 54 V

DC Network Theorem: Superposition theorem Ex: Considering the effect of E1 = 48 V

DC Network Theorem: Superposition theorem Ex: Using superposition theorem find current through resistor R2 (6 Ω) shown in Fig. 4.5 (a). Also find total power dissipated in the resistor R2 (6 Ω).

Considering the effect of the E (36 V) Considering the effect of the I (9 A) source

Total current through the resistor R2 (6 Ω)

Power dissipated in the resistor R2 (6 Ω)

DC Network Theorem: Thevenin theorem Thevenin’s theorem states that: Any two-terminal linear bilateral dc network can be replaced by an equivalent circuit consisting of a voltage source and a series resistor as in Fig. 4.6 (a).

Fig. 4.6(a) Fig. 4.6(b) Fig. 4.6(c) • a mathematical technique for replacing a given network, as viewed from two output terminals, • by a single voltage source with a series resistance• It makes the solution of complicated networks (particularly, electronic networks) quite quick and easy.• Application of this extremely useful

DC Network Theorem: Thevenin theorem Let, it is required to find current flowing through load resistance RL as in Fig.

Fig. 4.7(a) Fig. 4.7(b) Fig. 4.7(c) We will proceed as under:1. Remove RLfrom the circuit terminals A and B and redraw the circuit as shown in Fig. 4.7. Obviously, the terminals have become open-circuited. ·2. Calculate the open-circuit voltage Voc which appears across terminals A and B when they are open i.e. when RL is removed.

Voc = drop across R2 = lR2 , where I is the circuit current when A and B are open

DC Network Theorem: Thevenin theorem

Fig. 4.7(a) Fig. 4.7(b) Fig. 4.7(c)

• Now, imagine the battery to be removed from the circuit, leaving its internal resistance r behind and redraw the circuit, as shown in Fig. 4.7• When viewed inwards from terminals A and B, the circuit consists of two parallel paths: one containing R2 and the other containing (Rt + r)• Equivalent resistance of-the network, as viewed from these terminals

This resistance is also called, *Thevenin resistance RTh.

DC Network Theorem: Thevenin theorem

Steps to follow

1. Remove that portion of the network across which the Thevenin’s equivalent circuit is to be found.

2. Mark the terminals of the remaining two terminal network.3. calculate RTH by first setting all sources to zero (Voltage sources are

replaced by SC and current sources by OC) and then finding the resultant resistance between the two marked terminals (internal resistances of the sources must remain when the sources are set to zero).

4. Calculate VTH by first returning all sources to their original position and finding the open circuit voltage (VOC) between the terminals. It is the same voltage that would be measured by a voltmeter placed between the marked terminals.

5. Draw the Thevenin equivalent circuit with the portion of the circuit previously removed between the terminals of the equivalent circuit.

DC Network Theorem: Thevenin theorem Ex: Find the Thevenin equivalent circuit forthe network in Fig. 4.8 (a). Then find the current through R L for value of 2 Ω and 100 Ω.

Solution:• Steps 1 and 2 produce the network of Fig. • RL has been removed and the two “holding” terminals have been defined as a and b.• Step 3: Replacing the voltage source E1 with SC yields network, where

• For determining RTh

DC Network Theorem: Thevenin theorem Ex: Find the Thevenin equivalent circuit forthe network in Fig. 4.8 (a). Then find the current through R L for value of 2 Ω and 100 Ω.

Solution:• Steps 3 • OC voltage ETh = voltage drop across 6 Ω resistor.• Applying VDR,

• For determining RTh

DC Network Theorem: Thevenin theorem Ex: Find the Thevenin equivalent circuit forthe network in Fig. 4.8 (a). Then find the current through R L for value of 2 Ω and 100 Ω.

Solution:•

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DC Network Theorem: Norton theorem Norton’s theorem states that Any two terminal linear bilateral dc network can be replaced by an equivalent circuit consisting of a current source and a parallel resistor

• Every voltage source with a series internal resistance has a current source equvalent. • Current source equivalent of Thevenin’s network can be determined by Norton’s theorem.

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DC Network Theorem: Norton theorem • Steps leading to proper values of IN and RN are:

• Remove that portion of the network across which the Thevenin’s equivalent circuit is to be found.

• Mark the terminals of the remaining two terminal network.• calculate RN by first setting all sources to zero (Voltage sources are replaced

by SC and current sources by OC) and then finding the resultant resistance RN between the two marked terminals (internal resistances of the sources must remain when the sources are set to zero). Since RN=RTH, the procedure and value obtained using the approach described for Thevenin’s theorem will determine the proper value of RN.

• Calculate IN by first returning all sources to their original position and finding the short circuit current (ISC) between the terminals. It is the same current that would be measured by an ammeter placed between the marked terminals.

• Draw the Norton equivalent circuit with the portion of the circuit previously removed between the terminals of the equivalent circuit.

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DC Network Theorem: Norton theorem Ex: Find the Norton equivalent circuit for the network in the shaded area. • Identifying the terminals of interest

• Determining RN for the network

• Determining IN,

• Norton equivalent circuit

Network Theorem: Maximum power transfer theorem The maximum power transfer theorem states that

• A load will receive maximum power from a linear bilateral dc network when its total resistive value is exactly equal to the Thevenin resistance of the network as seen by the load.

• For the network of Fig. the maximum power will be delivered to the load when RL = RTH

• Determining RN for the network

• Determining IN,

• Norton equivalent circuit

Network Theorem: Maximum power transfer theorem

Proof:

• Circuit current I = E/( RL + RTH)

• Power consumed by the load PL = I2RL = E2RL/( RL + RTH)2

• For PL to be maximum, dPL/dRL = 0

• Differentiating and equating to zero, we have RL = RTH.

• So, Max. power Pmax = E2RL/4R2L = E2/4RL = E2/4RTH

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Network Theorem: Maximum power transfer theorem Ex: For the network shown in Fig. determine the value of R for maximum

power to R, and calculate the power delivered under these conditions.

• Determining RTH or RN

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• Determining ETH

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