Series and Parallel Resistors

Series and Parallel Resistors

Dr.-Eng. Hisham El-Sherif Electronics and Electrical Engineering Department

ELCT708: Electronics for Biotechnology

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Objectives

To calculate the equivalent resistance of series and parallel resistors.



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Examples for resistors in parallel and series

R4

R5

Series

R6 R4 // R5


R7 // R8 ELCT708: Electronics for Biotechnology

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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

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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)



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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 -


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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


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The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances.

For N resistors in series then,



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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



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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


i3

VR _

i2 = VR / R2 i3 = VR / R3


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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



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• 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


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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



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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


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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


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Example

Find Req of the circuit



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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


Ω

Ω

Ω

Ω

Ω


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Ω

Ω

Ω

Ω

10V 10V

Ω

Ω

Ω

Ω

Ω

Ω Ω

10V

10V Ω

Ω

Ω



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