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CIRCUIT THEOREMS

ELEC 308ELEC 308

Equivalent Circuits

In general, two circuits are said to be equivalent if th h th i h t i ti d they have the same v-i characteristics as measured at a given port using the ‘black box’ approach

The term black box implies an opaque container whose contents are hidden from view

Model the contents of the box by studying the l i hi b h l d i relationship between the voltage and current at its

terminals

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Source Transformation

Another network reduction technique in terms of i l t i it (i i h t i ti )equivalent circuits (i.e., same v-i characteristics)

A voltage source in series with a resistor is replaced with a current source in parallel with the same resistor, and vice versa.

Applicable to both dependent and independent sources.

The remaining part of the circuit is not affected by such substitution.

Source Transformation

The arrow of the current source is directed toward the positive terminal of the voltage source.

Not possible for R = 0 or R = ∞

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Thevenin’s Theorem

A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage by an equivalent circuit consisting of a voltage source VTh in series with a resistor RTh

Original CircuitThevenin

Equivalent circuit

VTh is the open-circuit voltage across terminals a-bof the original Circuit.

RTh is the equivalent resistance looking into terminals a-bwhen all independent sources are turned off (assuming the original circuit has no dependent sources.)

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Norton’s Theorem

A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current by an equivalent circuit consisting of a current source IN in parallel with a resistor RN

Original CircuitNorton

Equivalent circuit

IN is the short-circuit current through terminals a-bof the original Circuit.

RTh is the equivalent resistance looking into terminals a-bwhen all independent sources are turned off (assuming the original circuit has no dependent sources.)

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Thevenin-Norton Transformation

Thevenin and Norton equivalent circuits are l t d i th i th th d f related in the same way as in the method of source

transformation (i.e., one can be derived from the other) via the relationship

Calculate any two quantities using the method that takes the least effort and use Ohm’s law to find the third.

Example 3.28

Obtain the Thevenin equivalent circuit as seen by

th l d i t the load resistance.

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Example 3.29

Obtain the Norton equivalent circuit as seen by

th l d i t the load resistance.

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Example 3.35

Obtain the Thevenin equivalent circuit as seen by

th l d i t the load resistance.

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Example

Obtain the Thevenin and Norton equivalent circuit

t t i l b Th fi d th t th h Rat terminals a-b. Then, find the current through RL

when RL = 6, 16, and 36 Ω

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Maximum Power Transfer

Example

Find RL that will absorb maximum power from the circuit to the left of terminals a b What is the circuit to the left of terminals a-b. What is the maximum power?

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Practical Source Modeling

Model as an ideal source connected either in series or in parallel with an internal/source series or in parallel with an internal/source resistance

voltage source current source

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Example

The terminal voltage of a voltage source is 12 V when connected to a 2 W load When the load is when connected to a 2-W load. When the load is disconnected, the terminal voltage rises to 12.4 V.

(a) Calculate the source voltage and its internal resistance

(b) Determine the voltage when an 8-Ω load is connected to the sourceconnected to the source