Basic Theory of Circuits, SJTU Chapter 4 Circuit Theorems Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Linearity Substitution Maximum Power Transfer Superposition Tellegen Theorems Source transformation Reciprocity Thevenin’s Source Transfer Norton’s Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU 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. Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Example 4.2 Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU 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. Basic Theory of Circuits, SJTU

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

Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Fig. 4.6 For Example 4.3 Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Substitution Theorem  Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Substitution Theorem  Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Substitution Theorem  Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Substitution Theorem    Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Substitution Theorem If the voltage across and current through a 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. Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Substitution Theorem OR . Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Thevenin’s Theorem A linear two-terminal circuit can be replaced by an equivalent circuit consisting of a voltage source Vth in series with a resistor Rth (accompanied voltage source), where Vth is the open-circuit voltage at the terminals and Rth is the input or equivalent resistance at the terminals when the independent source are turned off. Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU (a) original circuit, (b) the Thevenin equivalent circuit c d Proof by figures. Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Simple Proof by figures Applying superposition theorem + V=Voc-RoI Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU 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. Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Case 2 If the network has dependent sources, there are two methods to get Rth: Turn off all the independent sources, apply a voltage source v0 (or current source i0) at terminals a and b and determine the resulting current i0 (or resulting voltage v0), then Rth= v0/ i0 Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Case 2 If the network has dependent sources, there are two methods to get Rth: 2. Calculate the open-circuit voltage Voc and short-circuit current Isc at the terminal of the original circuit, then Rth=Voc/Isc Rth=Voc/Isc Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Examples Basic Theory of Circuits, SJTU

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

Basic Theory of Circuits, SJTU (a) Original circuit, (b) Norton equivalent circuit d (c) N Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Examples Basic Theory of Circuits, SJTU

Maximum Power Transfer RL a b Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU

Maximum Power Transfer (several questions) If the load RL is invariable, and RTh is variable, then what should RTh be to make RL get maximum power? 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? Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Examples Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Tellegen Theorem If there are b branches in a lumped circuit, and the voltage uk, current ik of each branch apply passive sign convention, then we have Basic Theory of Circuits, SJTU

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 Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Example Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Reciprocity Theorem 2  6  3  3  6  2  Basic Theory of Circuits, SJTU

Reciprocity Theorem (only applicable to reciprocity networks) 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. Basic Theory of Circuits, SJTU

Reciprocity Theorem (only applicable to reciprocity networks) Case 2 Basic Theory of Circuits, SJTU

Reciprocity Theorem (only applicable to reciprocity networks) Case 3 Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Example All resistors are 1 , find out i. i + E -- + E -- i Basic Theory of Circuits, SJTU

Source Transfer Property Voltage source transfer Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Source Transfer Current source transfer Basic Theory of Circuits, SJTU

Basic Theory of Circuits, SJTU Summary Maximum Power Transfer Tellegen Theorem Inference of Tellegen Theorem Reciprocity Theorem Source Transfer Linearity Property Superposition Source Transformation Substitution Theorem Thevenin’s Theorem Norton’s Theorem Basic Theory of Circuits, SJTU