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Series and Parallel Resistors Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department ELCT708: Electronics for Biotechnology 1 Objectives To calculate the equivalent resistance of series and parallel resistors. Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department ELCT708: Electronics for Biotechnology 2 Examples for resistors in parallel and series R4 R5 Series R6 R4 // R5 Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department R7 // R8 ELCT708: Electronics for Biotechnology 3 Elements in Series • Suppose two elements are connected with nothing coming off in between • KCL says that the elements carry the same current. • We say these elements are in series. i1 – i2 = 0 Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department i1 = i2 ELCT708: Electronics for Biotechnology 4 Resistors in Series i + R1 + i R1 - R2 + i R2 - VTOTAL R3 + i R3 - - • Each resistor has the same current (labeled i). • Each resistor has voltage iR, given by Ohm’s law. • The total voltage drop across all 3 resistors is VTOTAL = i R1 + i R2 + i R3 = i (R1 + R2 + R3) Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department ELCT708: Electronics for Biotechnology 5 i R1 + R2 R3 v - • When we look at all three resistors together as one unit, we see that they have the same I-V relationship as one resistor, whose value is the sum of the resistances: • So we can treat these resistors as just one equivalent resistance, as long as we are not interested in the i individual voltages. Their effect on the rest of the circuit is the same, whether lumped together or not. Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department R1 + R2 + R3 + v - ELCT708: Electronics for Biotechnology 6 Example Find Req Solution v1 = iR1, v2 = iR2 If we apply KVL to the loop, we have −v + v1 + v2 = 0 Then v = v1 + v2 v = i(R1 + R2) , Req = R1 + R2 v = iReq Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department ELCT708: Electronics for Biotechnology 7 The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances. For N resistors in series then, Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department ELCT708: Electronics for Biotechnology 8 Parallel Elements • KVL tells us that any set of elements which are directly connected by wire at both ends carry the same voltage. • We say these elements are in parallel. KVL clockwise, start at top: Vb – Va = 0 Va = Vb Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department ELCT708: Electronics for Biotechnology 9 Parallel Resistors • Resistors in parallel carry the same voltage. All of the resistors below have voltage VR . • The current flowing through each resistor could definitely be different. Even though they have the same voltage, the resistances could be different. i1 = VR / R1 + R1 i1 R2 i2 R3 Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department i3 VR _ i2 = VR / R2 i3 = VR / R3 ELCT708: Electronics for Biotechnology 10 Example :- Find Requ From Ohm’s law, v = i1R1 = i2R2 Applying KCL at node a gives the total current i as i - i1 - i2 = 0 i = i1 + i2 where Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department ELCT708: Electronics for Biotechnology 11 • If we view the three resistors as one unit, with a current iTOTAL going in, and a voltage VR, this unit has the following I-V relationship: iTOTAL = i1 + i2 + i3 = VR(1/R1 + 1/R2 + 1/R3) in other words, VR = (1/R1 + 1/R2 + 1/R3)-1 iTOTAl So to the outside world, the parallel resistors look like one: iTOTAL iTOTAL + + VR R1 _ i1 R2 VR R3 i2 i3 REQ _ 1 1 1 1 = + + Req R1 R2 R3 Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department ELCT708: Electronics for Biotechnology 12 In case of two equal resistance in parallel The equivalent resistance of two parallel resistors is equal to the product of their resistances divided by their sum. R1 = R2, then Req = R1/ 2. R1 = R2 = · · · = RN = R, then Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department ELCT708: Electronics for Biotechnology 13 Example:- Resistance in series and Parallel Find total power expended in the circuit Ω Ω Ω Ω 10V Ω R = R1 + R2 + … + Rn Ω 10V Ω R = (5 + 35 + 25 + 10 + 5 + 50 + 15)kΩ V 2 10 2 10 2 100 = = = = 0.7mW p= R R (5 + 35 + 25 + 10 + 5 + 50 + 15)kΩ 145kΩ Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department ELCT708: Electronics for Biotechnology 14 Example Find RAB Combining resistors, The circuit is now reduced to: and further to: RAB = 8K Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department ELCT708: Electronics for Biotechnology 15 Example Find Req of the circuit Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department ELCT708: Electronics for Biotechnology 16 Example Find equivalent resistance Ω Ω Ω Ω Ω 10V Ω 10V Ω Ω Ω Ω Ω Ω Ω 10V 10V Ω Ω Ω Ω Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department Ω Ω Ω Ω Ω ELCT708: Electronics for Biotechnology 17 Ω Ω Ω Ω 10V 10V Ω Ω Ω Ω Ω Ω Ω 10V 10V Ω Ω Ω Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department ELCT708: Electronics for Biotechnology 18
