Series and Parallel Resistors

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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
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