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SJTU1 Chapter 4 Circuit Theorems

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SJTU2 Linearity Property Linearity is the property of an element describing a linear relationship between cause and effect. A linear circuit is one whose output is linearly ( or directly proportional) to its input.

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SJTU3 Fig. 4.4 For Example 4.2

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SJTU4 Superposition(1) The superposition principle states that 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.

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SJTU5 Steps to Apply Superposition Principle: 1.Turn off all independent source except one source. Find the output(voltage or current) due to that active source using nodal or mesh analysis. 2.Repeat step 1 for each of the other independent sources. 3.Find the total contribution by adding algebraically all the contributions due to the independent sources. Superposition(2)

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SJTU6

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7 Fig. 4.6 For Example 4.3

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SJTU8 Source Transformation(1) A source transformation is the process of replacing a voltage source V s in series with a resistor R by a current source i s in parallel with a resistor R, or vice versa. V s =i s R or i s =V s /R

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SJTU9 Source Transformation(2) It also applies to dependent sources:

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SJTU10 Fig. 4.17 for Example, find out Vo

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SJTU11 So, we get v o =3.2V

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SJTU12 7 2A 6V 2A I Example: find out I (use source transformation )

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SJTU13 Substitution Theorem I1=2A, I2=1A, I3=1A, V3=8V

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SJTU14 Substitution Theorem If the voltage across and current through any branch of a dc bilateral network are known, this branch can be replaced by any combination of elements that will maintain the same voltage across and current through the chosen branch.

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SJTU15 Substitution Theorem OR

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SJTU16 Thevenin’s Theorem A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source V th in series with a resistor R th, where V th is the open-circuit voltage at the terminals and R th is the input or equivalent resistance at the terminals when the independent source are turned off.

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SJTU17 (a) original circuit, (b) the Thevenin equivalent circuit d c

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SJTU18 + V=Voc-RoI Simple Proof by figures

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SJTU19 Thevenin’s Theorem Consider 2 cases in finding Rth: Case 1 If the network has no dependent sources, just turn off all independent sources, calculate the equivalent resistance of those resistors left. Case 2 If the network has dependent sources, there are two methods to get Rth: 1.

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SJTU20 Thevenin’s Theorem Case 2 If the network has dependent sources, there are two methods to get Rth: 1.Turn off all the independent sources, apply a voltage source v 0 (or current source i 0 ) at terminals a and b and determine the resulting current i 0 (or resulting voltage v 0 ), then R th = v 0 / i 0

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SJTU21 Case 2 If the network has dependent sources, there are two methods to get Rth: 2. Calculate the open-circuit voltage V oc and short-circuit current I sc at the terminal of the original circuit, then Rth=V oc /I sc Thevenin’s Theorem Rth=V oc /I sc

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SJTU22 Examples

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SJTU23 Norton’s Theorem A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a current source I N in parallel with a resistor R N, where I N is the short-circuit current through the terminals and R N is the input or equivalent resistance at the terminals when the independent sources are turned off.

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SJTU24 (a) Original circuit, (b) Norton equivalent circuit d (c) N

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SJTU25 Examples

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SJTU26 Maximum Power Transfer RLRL a b Replacing the original network by its Thevenin equivalent, then the power delivered to the load is

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SJTU27 Power delivered to the load as a function of R L We can confirm that is the maximum power by showing that

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SJTU28 If the load R L is invariable, and R Th is variable, then what should R Th be to make R L get maximum power? Maximum Power Transfer (several questions) If using Norton equivalent to replace the original circuit, under what condition does the maximum transfer occur? Is it true that the efficiency of the power transfer is always 50% when the maximum power transfer occurs?

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SJTU29 Examples

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SJTU30 Tellegen Theorem If there are b branches in a lumped circuit, and the voltage u k, current i k of each branch apply passive sign convention, then we have

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SJTU31 Inference of Tellegen Theorem If two lumped circuits and have the same topological graph with b branches, and the voltage, current of each branch apply passive sign convention, then we have not only

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SJTU32 Example

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SJTU33 Reciprocity Theorem 2 3 6 3 6 2

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SJTU34 Case 1 The current in any branch of a network, due to a single voltage source E anywhere else in the network, will equal the current through the branch in which the source was originally located if the source is placed in the branch in which the current I was originally measured. Reciprocity Theorem (only applicable to reciprocity networks)

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SJTU35 Reciprocity Theorem (only applicable to reciprocity networks) Case 2

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SJTU36 Reciprocity Theorem (only applicable to reciprocity networks) Case 3

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SJTU37 example

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SJTU38 Source Transfer Voltage source transfer An isolate voltage source can then be transferred to a voltage source in series with a resistor.

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SJTU39 Source Transfer Current source transfer Examples

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SJTU40 Summary Linearity Property Superposition Source Transformation Substitution Theorem Thevenin’s Theorem Norton’s Theorem Maximum Power Transfer Tellegen Theorem Inference of Tellegen Theorem Reciprocity Theorem Source Transfer

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