RESISTIVE CIRCUITS

1. RESISTIVE CIRCUITS Here we introduce the basic concepts and laws that are fundamental to circuit analysis LEARNING GOALS • OHM’S LAW - DEFINES THE SIMPLEST PASSIVE ELEMENT: THE RESISTOR • KIRCHHOFF’S LAWS - THE FUNDAMENTAL CIRCUIT CONSERVATION LAWS- KIRCHHOFF CURRENT (KCL) AND KIRCHHOFF VOLTAGE (KVL) • LEARN TO ANALYZE THE SIMPLEST CIRCUITS • SINGLE LOOP - THE VOLTAGE DIVIDER • SINGLE NODE-PAIR - THE CURRENT DIVIDER • SERIES/PARALLEL RESISTOR COMBINATIONS - A TECHNIQUE TO REDUCE THE COMPLEXITY OF SOME CIRCUITS • WYE - DELTA TRANSFORMATION - A TECHNIQUE TO REDUCE COMMON RESISTOR CONNECTIONS THAT ARE NEITHER SERIES NOR PARALLEL • CIRCUITS WITH DEPENDENT SOURCES - (NOTHING VERY SPECIAL)

2. The unit of conductance is Siemens RESISTORS A resistor is a passive element characterized by an algebraic relation between the voltage across its terminals and the current through it Conductance If instead of expressing voltage as a function of current one expresses current in terms of voltage, OHM’s law can be written A linear resistor obeys OHM’s Law The constant, R, is called the resistance of the component and is measured in units of Ohm From a dimensional point of view Ohms is a derived unit of Volt/Amp Since the equation is algebraic the time dependence can be omitted

3. Some practical resistors Symbol

4. “A touch of reality” Linear approximation Linear range Ohm’s Law is an approximation valid while voltages and currents remain in the Linear Range Actual v-I relationship Notice passive sign convention Two special resistor values

5. OHM’S LAW OHM’S LAW THE EXAMPLE COULD BE GIVEN LIKE THIS GIVEN VOLTAGE AND CONDUCTANCE OHM’S LAW UNITS? REFERENCE DIRECTIONS SATISFY PASSIVE SIGN CONVENTION UNITS? CONDUCTANCE IN SIEMENS, VOLTAGE IN VOLTS. HENCE CURRENT IN AMPERES

6. DETERMINE CURRENT AND POWER ABSORBED BY RESISTOR

8. KIRCHHOFF CURRENT LAW ONE OF THE FUNDAMENTAL CONSERVATION PRINCIPLES IN ELECTRICAL ENGINEERING “CHARGE CANNOT BE CREATED NOR DESTROYED”

9. (A CONSERVATION OF CHARGE PRINCIPLE) NODE NODES, BRANCHES, LOOPS A NODE CONNECTS SEVERAL COMPONENTS. BUT IT DOES NOT HOLD ANY CHARGE. TOTAL CURRENT FLOWING INTO THE NODE MUST BE EQUAL TO TOTAL CURRENT OUT OF THE NODE NODE: point where two, or more, elements are joined (e.g., big node 1) LOOP: A closed path that never goes twice over a node (e.g., the blue line) The red path is NOT a loop BRANCH: Component connected between two nodes (e.g., component R4)

10. KIRCHHOFF CURRENT LAW (KCL) SUM OF CURRENTS FLOWING INTO A NODE IS EQUAL TO SUM OF CURRENTS FLOWING OUT OF THE NODE ALGEBRAIC SUM OF CURRENT (FLOWING) OUT OF A NODE IS ZERO ALGEBRAIC SUM OF CURRENTS FLOWING INTO A NODE IS ZERO

11. A node is a point of connection of two or more circuit elements. It may be stretched out or compressed for visual purposes… But it is still a node

12. A GENERALIZED NODE IS ANY PART OF A CIRCUIT WHERE THERE IS NO ACCUMULATION OF CHARGE ... OR WE CAN MAKE SUPERNODES BY AGGREGATING NODES INTERPRETATION: SUM OF CURRENTS LEAVING NODES 2&3 IS ZERO VISUALIZATION: WE CAN ENCLOSE NODES 2&3 INSIDE A SURFACE THAT IS VIEWED AS A GENERALIZED NODE (OR SUPERNODE)

13. WRITE ALL KCL EQUATIONS THE FIFTH EQUATION IS THE SUM OF THE FIRST FOUR... IT IS REDUNDANT!!!

14. FIND MISSING CURRENTS KCL DEPENDS ONLY ON THE INTERCONNECTION. THE TYPE OF COMPONENT IS IRRELEVANT KCL DEPENDS ONLY ON THE TOPOLOGY OF THE CIRCUIT

15. WRITE KCL EQUATIONS FOR THIS CIRCUIT • THE LAST EQUATION IS AGAIN LINEARLY • DEPENDENT OF THE PREVIOUS THREE • THE PRESENCE OF A DEPENDENT SOURCE • DOES NOT AFFECT APPLICATION OF KCL • KCL DEPENDS ONLY ON THE TOPOLOGY

16. Here we illustrate the use of a more general idea of node. The shaded surface encloses a section of the circuit and can be considered as a BIG node THE CURRENT I5 BECOMES INTERNAL TO THE NODE AND IT IS NOT NEEDED!!!

18. KIRCHHOFF VOLTAGE LAW ONE OF THE FUNDAMENTAL CONSERVATION LAWS IN ELECTRICAL ENGINERING THIS IS A CONSERVATION OF ENERGY PRINCIPLE “ENERGY CANNOT BE CREATE NOR DESTROYED”

20. KIRCHHOFF VOLTAGE LAW (KVL) KVL IS A CONSERVATION OF ENERGY PRINCIPLE KVL: THE ALGEBRAIC SUM OF VOLTAGE DROPS AROUND ANY LOOP MUST BE ZERO

21. PROBLEM SOLVING TIP: KVL IS USEFUL TO DETERMINE A VOLTAGE - FIND A LOOP INCLUDING THE UNKNOWN VOLTAGE THE LOOP DOES NOT HAVE TO BE PHYSICAL LOOP abcdefa

22. BACKGROUND: WHEN DISCUSSING KCL WE SAW THAT NOT ALL POSSIBLE KCL EQUATIONS ARE INDEPENDENT. WE SHALL SEE THAT THE SAME SITUATION ARISES WHEN USING KVL A SNEAK PREVIEW ON THE NUMBER OF LINEARLY INDEPENDENT EQUATIONS EXAMPLE: FOR THE CIRCUIT SHOWN WE HAVE N = 6, B = 7. HENCE THERE ARE ONLY TWO INDEPENDENT KVL EQUATIONS THE THIRD EQUATION IS THE SUM OF THE OTHER TWO!!

23. DEPENDENT SOURCES ARE HANDLED WITH THE SAME EASE GIVEN THE CHOICE USE THE SIMPLEST LOOP

24. There are no loops with only one unknown!!! - Vx/2 + The current through the 5k and 10k resistors is the same. Hence the voltage drop across the 5k is one half of the drop across the 10k!!!

25. KVL ON THIS LOOP VOLTAGE DIVISION: THE SIMPLEST CASE SINGLE LOOP CIRCUITS BACKGROUND: USING KVL AND KCL WE CAN WRITE ENOUGH EQUATIONS TO ANALYZE ANY LINEAR CIRCUIT. WE NOW START THE STUDY OF SYSTEMATIC, AND EFFICIENT, WAYS OF USING THE FUNDAMENTAL CIRCUIT LAWS WRITE 5 KCL EQS OR DETERMINE THE ONLY CURRENT FLOWING • THE PLAN • BEGIN WITH THE SIMPLEST ONE LOOP CIRCUIT • EXTEND RESULTS TO MULTIPLE SOURCE • AND MULTIPLE RESISTORS CIRCUITS IMPORTANT VOLTAGE DIVIDER EQUATIONS

26. SUMMARY OF BASIC VOLTAGE DIVIDER VOLUME CONTROL?

27. A “PRACTICAL” POWER APPLICATION HOW CAN ONE REDUCE THE LOSSES?

28. THE CONCEPT OF EQUIVALENT CIRCUIT THIS CONCEPT WILL OFTEN BE USED TO SIMPLFY THE ANALYSIS OF CIRCUITS. WE INTRODUCE IT HERE WITH A VERY SIMPLE VOLTAGE DIVIDER AS FAR AS THE CURRENT IS CONCERNED BOTH CIRCUITS ARE EQUIVALENT. THE ONE ON THE RIGHT HAS ONLY ONE RESISTOR IN ALL CASES THE RESISTORS ARE CONNECTED IN SERIES

29. FIRST GENERALIZATION: MULTIPLE SOURCES Voltage sources in series can be algebraically added to form an equivalent source. We select the reference direction to move along the path. Voltage drops are subtracted from rises i(t) KVL Collect all sources on one side

30. APPLY KVL TO THIS LOOP APPLY KVL TO THIS LOOP SECOND GENERALIZATION: MULTIPLE RESISTORS VOLTAGE DIVISION FOR MULTIPLE RESISTORS

31. THIS ELEMENT IS INACTVE (SHORT-CIRCUITED) SINGLE NODE-PAIR CIRCUITS THESE CIRCUITS ARE CHARACTERIZED BY ALL THE ELMENTS HAVING THE SAME VOLTAGE ACROSS THEM - THEY ARE IN PARALLEL

32. THE CURRENT DIVISION APPLY KCL USE OHM’S LAW TO REPLACE CURRENTS DEFINE “PARALLEL RESISTANCE COMBINATION” BASIC CURRENT DIVIDER THE CURRENT i(t) ENTERS THE NODE AND SPLITS - IT IS DIVIDED BETWEEN THE CURRENTS i1(t) AND i2(t)

33. WHEN IN DOUBT… REDRAW THE CIRCUIT TO HIGHLIGHT ELECTRICAL CONNECTIONS!! IS EASIER TO SEE THE DIVIDER

34. APPLY KCL TO THIS NODE DEFINE “PARALLEL RESISTANCE COMBINATION” FIRST GENERALIZATION: MULTIPLE SOURCES

36. APPLY KCL TO THIS NODE Ohm’s Law at every resistor General current divider SECOND GENERALIZATION: MULTIPLE RESISTORS

37. SERIES COMBINATIONS PARALLEL COMBINATION SERIES PARALLEL RESISTOR COMBINATIONS UP TO NOW WE HAVE STUDIED CIRCUITS THAT CAN BE ANALYZED WITH ONE APPLICATION OF KVL(SINGLE LOOP) OR KCL(SINGLE NODE-PAIR) WE HAVE ALSO SEEN THAT IN SOME SITUATIONS IT IS ADVANTAGEOUS TO COMBINE RESISTORS TO SIMPLIFY THE ANALYSIS OF A CIRCUIT NOW WE EXAMINE SOME MORE COMPLEX CIRCUITS WHERE WE CAN SIMPLIFY THE ANALYSIS USING THE TECHNIQUE OF COMBINING RESISTORS… … PLUS THE USE OF OHM’S LAW

38. FIRST WE PRACTICE COMBINING RESISTORS SERIES 6k||3k (10K,2K)SERIES

39. …OTHER OPTIONS... FIRST REDUCE IT TO A SINGLE LOOP CIRCUIT SECOND: “BACKTRACK” USING KVL, KCL OHM’S

40. LEARNING BY DOING

41. AN EXAMPLE OF “BACKTRACKING” A STRATEGY. ALWAYS ASK: “WHAT ELSE CAN I COMPUTE?”

42. SERIES PARALLEL THIS IS AN INVERSE PROBLEM WHAT CAN BE COMPUTED?

43. http://www.wiley.com/college/irwin/0470128690/animations/swf/D2Y.swfhttp://www.wiley.com/college/irwin/0470128690/animations/swf/D2Y.swf THEN THE CIRCUIT WOULD BECOME LIKE THIS AND BE AMENABLE TO SERIES PARALLEL TRANSFORMATIONS IF INSTEAD OF THIS WE COULD HAVE THIS THIS CIRCUIT HAS NO RESISTOR IN SERIES OR PARALLEL

44. REPLACE IN THE THIRD AND SOLVE FOR R1 SUBTRACT THE FIRST TWO THEN ADD TO THE THIRD TO GET Ra

45. DELTA CONNECTION LEARNING EXAMPLE: APPLICATION OF WYE-DELTA TRANSFORMATION ONE COULD ALSO USE A WYE - DELTA TRANSFORMATION ...

46. ONE EQUATION, TWO UNKNOWNS. CONTROLLING VARIABLE PROVIDES EXTRA EQUATION KVL REPLACE AND SOLVE FOR THE CURRENT UNITS ARE EXPLICIT USE OHM’S LAW CIRCUITS WITH DEPENDENT SOURCES GENERAL STRATEGY TREAT DEPENDENT SOURCES AS REGULAR SOURCES AND ADD ONE MORE EQUATION FOR THE CONTROLLING VARIABLE A CONVENTION ABOUT DEPENDENT SOURCES. UNLESS OTHERWISE SPECIFIED THE CURRENT AND VOLTAGE VARIABLES ARE ASSUMED IN SI UNITS OF Amps AND Volts FOR THIS EXAMPLE THE MULTIPLIER MUST HAVE UNITS OF OHM A PLAN: SINGLE LOOP CIRCUIT. USE KVL TO DETERMINE CURRENT

47. KCL TO THIS NODE. THE DEPENDENT SOURCE IS JUST ANOTHER SOURCE THE EQUATION FOR THE CONTROLLING VARIABLE PROVIDES THE ADDITIONAL EQUATION ALGEBRAICALLY, THERE ARE TWO UNKNOWNS AND JUST ONE EQUATION SUBSTITUTION OF I_0 YIELDS VOLTAGE DIVIDER NOTICE THE CLEVER WAY OF WRITING mA TO HAVE VOLTS IN ALL NUMERATORS AND THE SAME UNITS IN DENOMINATOR A PLAN: IF V_s IS KNOWN V_0 CAN BE DETERMINED USING VOLTAGE DIVIDER. TO FIND V_s WE HAVE A SINGLE NODE-PAIR CIRCUIT

48. THE DEPENDENT SOURCE IS ONE MORE VOLTAGE SOURCE KVL TO THIS LOOP THE EQUATION FOR THE CONTROLLING VARIABLE PROVIDES THE ADDITIONAL EQUATION REPLACE AND SOLVE FOR CURRENT I … AND FINALLY A PLAN: ONE LOOP PROBLEM. FIND THE CURRENT THEN USE OHM’S LAW.

49. KCL KVL KVL KCL A PLAN: ONE LOOP ON THE LEFT - KVL ONE NODE-PAIR ON RIGHT - KCL ALSO A VOLTAGE DIVIDER