LOOP ANALYSS The second systematic technique to determine all currents and voltages in a circuit T S DUAL TO NODE ANALYSS - T FRST DETERMNES ALL CURRENTS N A CRCUT AND THEN T USES OHM S LAW TO COMPUTE NECESSARY VOLTAGES THERE ARE STUATON WHERE NODE ANALYSS S NOT AN EFFCENT TECHNQUE AND WHERE THE NUMBER OF EQUATONS REQURED BY THS NEW METHOD S SGNFCANTLY SMALLER
Apply node analysis to this circuit V V - V R R R R 3 V R - 8V There are non reference nodes There is one super node There is one node connected to the reference through a voltage source We need three equations to compute all node voltages V R 3 BUT THERE S ONLY ONE CURRENT FLOWNG THROUGH ALL COMPONENTS AND F THAT CURRENT S DETERMNED ALL VOLTAGES CAN BE COMPUTED WTH OHM S LAW STRATEGY:. Apply KVL (sum of voltage drops ). Use Ohm s Law to express voltages in terms of the loop current. [ V ] V V 8[ V ] V R R R3 [ V ] R R 8[ V ] R3 RESULT S ONE EQUATON N THE LOOP CURRENT!!! SHORTCUT V V 3 V Skip this equation Write this one directly
LOOPS, MESHES AND LOOP CURRENTS a b 3 7 c f 6 e 5 d A BASC CRCUT EACH COMPONENT S CHARACTERZED BY TS VOLTAGE ACROSS AND TS CURRENT THROUGH A LOOP S A CLOSED PATH THAT DOES NOT GO TWCE OVER ANY NODE. THS CRCUT HAS THREE LOOPS fabef ebcde fabcdef A MESH S A LOOP THAT DOES NOT ENCLOSE ANY OTHER LOOP. fabef, ebcde ARE MESHES A LOOP CURRENT S A (FCTCOUS) CURRENT THAT S ASSUMED TO FLOW AROUND A LOOP,, 3 ARE LOOP CURRENTS A MESH CURRENT S A LOOP CURRENT ASSOCATED TO A MESH., ARE MESH CURRENTS CLAM: N A CRCUT, THE CURRENT THROUGH ANY COMPONENT CAN BE EXPRESSED N TERMS OF THE LOOP CURRENTS EXAMPLES FACT: NOT EVERY LOOP CURRENT S REQURED TO COMPUTE ALL THE CURRENTS THROUGH COMPONENTS a a f b e b c b 3 7 3 3 c f 6 e 5 d A BASC CRCUT 3 THE DRECTON OF THE LOOP CURRENTS S SGNFCANT USNG TWO LOOP CURRENTS a f be bc 3 FOR EVERY CRCUT THERE S A MNMUM NUMBER OF LOOP CURRENTS THAT ARE NECESSARY TO COMPUTE EVERY CURRENT N THE CRCUT. SUCH A COLLECTON S CALLED A MNMAL SET (OF LOOP CURRENTS). 3
FOR A GVEN CRCUT LET B NUMBER OF BRANCHES N NUMBER OF NODES THE MNMUM REQURED NUMBER OF LOOP CURRENTS S L B ( N ) MESH CURRENTS ARE ALWAYS NDEPENDENT AN EXAMPLE DETERMNATON OF LOOP CURRENTS KVL ON LEFT MESH KVL ON RGHT MESH v v v v S 5 3 USNG OHM S LAW v i R, v i R, v ( i i ) R v i R, v i R 3 3 5 5 REPLACNG AND REARRANGNG B 7 N 6 L 7 (6 ) TWO LOOP CURRENTS ARE REQURED. THE CURRENTS SHOWN ARE MESH CURRENTS. HENCE THEY ARE NDEPENDENT AND FORM A MNMAL SET
WRTE THE MESH EQUATONS v R BOOKKEEPNG BRANCHES 8 NODES 7 LOOP CURRENTS NEEDED i R AND WE ARE TOLD TO USE MESH CURRENTS! THS DEFNES THE LOOP CURRENTS TO BE USED DENTFY ALL VOLTAGE DROPS v R3 i R 3 v R ( i i R v R ) v R5 i R i R 5 WRTE KVL ON EACH MESH TOP MESH: v BOTTOM: v USE OHM S LAW S vr vs vr R vr5 vr vs3 vr3
DEVELOPNG A SHORTCUT WRTE THE MESH EQUATONS V - R V - R R 3 WHENEVER AN ELEMENT HAS MORE THAN ONE LOOP CURRENT FLOWNG THROUGH T WE COMPUTE NET CURRENT N THE DRECTON OF TRAVEL R 5 R DRAW THE MESH CURRENTS. ORENTATON CAN BE ARBTRARY. BUT BY CONVENTON THEY ARE DEFNED CLOCKWSE NOW WRTE KVL FOR EACH MESH AND APPLY OHM S LAW TO EVERY RESSTOR. AT EACH LOOP FOLLOW THE PASSVE SGN CONVENTON USNG LOOP CURRENT REFERENCE DRECTON V R ( ) R R5 V R3 R ( ) R
EXAMPLE: FND o USNG LOOP ANALYS AN ALTERNATVE SELECTON OF LOOP CURRENTS SHORTCUT: POLARTES ARE NOT NEEDED. APPLY OHM S LAW TO EACH ELEMENT AS KVL S BENG WRTTEN KVL @ KVL @ REARRANGE k 6k 6k 9k 3* / and add k 6. 5mA 5 k 6k ma EXPRESS VARABLE OF NTEREST AS FUNCTON OF LOOP CURRENTS O KVL @ KVL @ NOW REARRANGE O THS SELECTON S MORE EFFCENT k 6k * /3 6k 9k 9 * / and substract 3 k 8 ma
F THE CRCUT CONTANS ONLY NDEPENDENT SOURCE THE MESH EQUATONS CAN BE WRTTEN BY NSPECTON A PRACTCE EXAMPLE MUST HAVE ALL MESH CURRENTS WTH THE SAME ORENTATON N LOOP K THE COEFFCENT OF k S THE SUM OF RESSTANCES AROUND THE LOOP. LOOP coefficient of coefficient of coefficient of k 6k 6k RHS 6[ V ] 3 THE RGHT HAND SDE S THE ALGEBRAC SUM OF VOLTAGE SOURCES AROUND THE LOOP (VOLTAGE RSES - VOLTAGE DROPS) THE COEFFCENT OF j S THE SUM OF RESSTANCES COMMON TO BOTH k AND j AND WTH A NEGATVE SGN. LOOP k 6k k 9k 6 LOOP 3 Loop 3 LOOP coefficient of coefficient of coefficient of 9k 3k 3k RHS 6[ V ] 3 (6 k ) (3 k ) (3k 6k k ) 3
LEARNNG EXTENSON. DRAW THE MESH CURRENTS. WRTE MESH EQUATONS MESH k k k) k 3[ ] ( V k (k 6k) (6V 3V MESH ) DVDE BY k. GET NUMBERS FOR COEFFCENTS ON THE LEFT AND ma ON THE RHS 3. SOLVE EQUATONS 8 3[ ma] 8 9[ ma] * / and add 3 33[ ma] 33 V O 6k [ V ] 5
WRTE THE MESH EQUATONS k V k k k. DRAW MESH CURRENTS 6k 3 9V BOOKKEEPNG: B 7, N. WRTE MESH EQUATONS. USE KVL MESH : k V 6k( 3) MESH : V k( ) k( 3) MESH 3: 9V 6k( ) k( 3 ) MESH : V k( ) k 3 9 CHOOSE YOUR FAVORTE TECHNQUE TO SOLVE THE SYSTEM OF EQUATONS EQUATONS BY NSPECTON 8k 6k3 V 8k k3 k V 6k k k3 9V k 6k 9V
CRCUTS WTH NDEPENDENT CURRENT SOURCES KVL THERE S NO RELATONSHP BETWEEN V AND THE SOURCE CURRENT! HOWEVER... MESH CURRENT S CONSTRANED MESH EQUATON MESH ma BY NSPECTON k 8k V k (ma) V 3 ma VO 6k 8k 9 [ V ] CURRENT SOURCES THAT ARE NOT SHARED BY OTHER MESHES (OR LOOPS) SERVE TO DEFNE A MESH (LOOP) CURRENT AND REDUCE THE NUMBER OF REQURED EQUATONS TO OBTAN V APPLY KVL TO ANY CLOSED PATH THAT NCLUDES V
EXAMPLE COMPUTE V O USNG MESH ANALYSS KVL FOR Vo TWO MESH CURRENTS ARE DEFNED BY CURRENT SOURCES ma ma MESH 3 USE KVL TO COMPUTE Vo BY NSPECTON 3 3V k(ma) k( ma) k k k k 3V 3 ma
CURRENT SOURCES SHARED BY LOOPS - THE SUPERMESH APPROACH. WRTE CONSTRANT EQUATON DUE TO MESH CURRENTS SHARNG CURRENT SOURCES 3 ma 3. WRTE EQUATONS FOR THE OTHER MESHES ma. DEFNE A SUPERMESH BY (MENTALLY) REMOVNG THE SHARED CURRENT SOURCE. SELECT MESH CURRENTS 5. WRTE KVL FOR THE SUPERMESH 6 k3 k k( ) k ( 3 ) SUPERMESH NOW WE HAVE THREE EQUATONS N THREE UNKNOWNS. THE MODEL S COMPLETE
CURRENT SOURCES SHARED BY MESHES - THE GENERAL LOOP APPROACH THE STRATEGY S TO DEFNE LOOP CURRENTS THAT DO NOT SHARE CURRENT SOURCES - EVEN F T MEANS ABANDONNG MESHES FOR CONVENENCE START USNG MESH CURRENTS UNTL REACHNG A SHARED SOURCE. AT THAT PONT DEFNE A NEW LOOP. N ORDER TO GUARANTEE THAT F GVES AN NDEPENDENT EQUATON ONE MUST MAKE SURE THAT THE LOOP NCLUDES COMPONENTS THAT ARE NOT PART OF PREVOUSLY DEFNED LOOPS A POSSBLE STRATEGY S TO CREATE A LOOP BY OPENNG THE CURRENT SOURCE THE LOOP EQUATONS FOR THE LOOPS WTH CURRENT SOURCES ARE ma ma THE LOOP EQUATON FOR THE THRD LOOP S 6[ V ] k3 k( 3 ) k( 3 ) k ( 3 ) THE MESH CURRENTS OBTANED WTH THS METHOD ARE DFFERENT FROM THE ONES OBTANED WTH A SUPERMESH. EVEN FOR THOSE DEFNED USNG MESHES.
FND VOLTAGES R V R S V S V R V 3 ACROSS S3 R 3 RESSTORS For loop analysis we notice... - V S Three independent current sources. Four meshes. One current source shared by two meshes. Careful choice of loop currents should make only one loop equation necessary. Three loop currents can be chosen using meshes and not sharing any source. 3 Now we need a loop current that does not go over any current source and passes through all unused components. HNT: F ALL CURRENT SOURCES ARE REMOVED THERE S ONLY ONE LOOP LEFT V S SOLVE FOR THE CURRENT. USE OHM S LAW TO CMPUTE REQURED VOLTAGES V V MESH EQUATONS FOR LOOPS WTH CURRENT SOURCES s S 3 S 3 KVL OF REMANNG LOOP R3 ( ) R ( 3 ) R ( 3) V R ( 3 ) R ( ) 3 R3 ( )