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LoopAnalysispp1

# LoopAnalysispp1 - LOOP ANALYSIS The second systematic...

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The second systematic technique to determine all currents and voltages in a circuit IT IS DUAL TO NODE ANALYSIS - IT FIRST DETERMINES ALL CURRENTS IN A CIRCUIT AND THEN IT USES OHM’S LAW TO COMPUTE NECESSARY VOLTAGES THERE ARE SITUATIONS WHERE NODE ANALYSIS IS NOT AN EFFICIENT TECHNIQUE AND WHERE THE NUMBER OF EQUATIONS REQUIRED BY THIS NEW METHOD IS SIGNIFICANTLY SMALLER LOOP ANALYSIS

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+ - + - 1 R 2 R 3 R V 18 V 12 + 2 R V + 1 R V + 3 R V Apply node analysis to this circuit There are 4 non-reference nodes There is one supernode There is one node connected to the reference through a voltage source We need three equations to compute all node voltages …BUT THERE IS ONLY ONE CURRENT FLOWING THROUGH ALL COMPONENTS AND IF THAT CURRENT IS DETERMINED ALL VOLTAGES CAN BE COMPUTED WITH OHM’S LAW I STRATEGY: 1. Apply KVL (sum of voltage drops =0) 0 ] [ 18 ] [ 12 3 2 1 = + + + R R R V V V V V 2. Use Ohm’s Law to express voltages in terms of the “loop current.” 0 ] [ 18 ] [ 12 3 2 1 = + + + + I R V I R I R V RESULT IS ONE EQUATION IN THE LOOP CURRENT!!! SHORTCUT Skip this equation Write this one directly 3 V 2 V 1 V 4 V
LOOPS, MESHES AND LOOP CURRENTS EACH COMPONENT IS CHARACTERIZED BY ITS VOLTAGE ACROSS AND ITS CURRENT THROUGH A LOOP IS A CLOSED PATH THAT DOES NOT GO TWICE OVER ANY NODE. THIS CIRCUIT HAS THREE LOOPS 1 2 3 4 5 6 7 A BASIC CIRCUIT a bc d e f fabcdef A MESH IS A LOOP THAT DOES NOT ENCLOSE ANY OTHER LOOP. fabef, ebcde ARE MESHES A LOOP CURRENT IS A (FICTICIOUS) CURRENT THAT IS ASSUMED TO FLOW AROUND A LOOP fabef ebcde 1 I 2 I CURRENTS LOOP ARE 3 2 1 , , I I I A MESH CURRENT IS A LOOP CURRENT ASSOCIATED TO A MESH. I1, I2 ARE MESH CURRENTS CLAIM: CLAIM: IN A CIRCUIT, THE CURRENT THROUGH ANY COMPONENT CAN BE EXPRESSED IN TERMS OF THE LOOP CURRENTS 1 2 3 4 5 6 7 A BASIC CIRCUIT a d e f FACT: FACT: NOT EVERY LOOP CURRENT IS REQUIRED TO COMPUTE ALL THE CURRENTS THROUGH COMPONENTS 1 I 3 I 3 1 3 1 = = = I I I I I I I c b e b f a CURRENTS LOOP TWO USING 3 2 2 1 3 1 I I I I I I I I I c b e b f a + = = = EXAMPLES THE DIRECTION OF THE LOOP CURRENTS IS SIGNIFICANT FOR EVERY CIRCUIT THERE IS A MINIMUM NUMBER OF LOOP CURRENTS THAT ARE NECESSARY TO COMPUTE EVERY CURRENT IN THE CIRCUIT. SUCH A COLLECTION IS CALLED A MINIMAL SET (OF LOOP CURRENTS).

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FOR A GIVEN CIRCUIT LET B = NUMBER OF BRANCHES N = NUMBER OF NODES THE MINIMUM REQUIRED NUMBER OF LOOP CURRENTS IS ) 1 ( = N B L MESH CURRENTS ARE ALWAYS INDEPENDENT AN EXAMPLE
FOR A GIVEN CIRCUIT LET B = NUMBER OF BRANCHES N = NUMBER OF NODES THE MINIMUM REQUIRED NUMBER OF LOOP CURRENTS IS ) 1 ( = N B L MESH CURRENTS ARE ALWAYS INDEPENDENT AN EXAMPLE 2 ) 1 6 ( 7 6 7 = = = = L N B TWO LOOP CURRENTS ARE REQUIRED. THE CURRENTS SHOWN ARE MESH CURRENTS. HENCE THEY ARE INDEPENDENT AND FORM A MINIMAL SET KVL ON LEFT MESH KVL ON RIGHT MESH USING OHM’S LAW REPLACING AND REARRANGING IN MATRIX FORM THESE ARE LOOP EQUATIONS FOR THE CIRCUIT DETERMINATION OF LOOP CURRENTS

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LEARNING BY DOING: WRITE THE MESH EQUATIONS
WRITE KVL ON EACH MESH

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LoopAnalysispp1 - LOOP ANALYSIS The second systematic...

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