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Transcript 
ECE 3301
General Electrical Engineering
Section 03
Simple Resistive Circuits
1
Resistivity
Length l (m)
Cross-sectional
Area A (m2 )
I
𝑙
𝑅=𝜌
𝐴
Resistivity ρ (Ω-m)
Assuming the Current is uniformly distributed across the
Cross-Sectional Area.
2
Simple Resistive Circuits
The simplest circuit is a voltage source
connected across a resistance as
demonstrated below.
IS
VS
R
VR= VS
3
Simple Resistive Circuits
The voltage source VS is placed across the
resistance R and the current IS flows in the
closed loop.
IS
VS
R
VR= VS
4
Simple Resistive Circuits
Assuming VS and R are known, the current
is given by Ohm’s Law.
+𝑉𝑆 βˆ’ 𝑉𝑅 = 0
IS
𝑉𝑆 βˆ’ 𝐼𝑆 𝑅 = 0
VS
R
VR= VS
𝐼𝑆 𝑅 = 𝑉𝑆
𝑉𝑆
𝐼𝑆 =
𝑅
5
Resistances in Series
A slightly more
complicated resistive
circuit has two
resistors in series.
IS
I1
V1
VS
R1
I2
V2
R2
6
Resistances in Series
IS
Note that resistors R1
and R2 are connected
end-to-end.
In this configuration
they share the same
current
I1
V1
VS
R1
I2
V2
R2
IS = I1 = I2
7
Series Connection
When the same continuous current
flows through circuit elements,
those circuit elements are connected
in series.
8
Resistances in Series
IS
The voltage source VS is
placed across the series
combination of R1 and R2.
The current IS is established
in the closed loop.
V1
R1
V2
R2
VS
9
Resistances in Series
IS
There will be a voltage
V1 β€œdropped” across
resistor R1
There will be a voltage
V2 β€œdropped” across
resistor R2
V1
R1
V2
R2
VS
10
Resistances in Series
Applying Kirchhoff’s
Voltage Law to this
closed loop gives the
equation
+𝑉𝑆 βˆ’ 𝑉1 βˆ’ 𝑉2 = 0
IS
V1
R1
V2
R2
VS
𝑉𝑆 = 𝑉1 + 𝑉2
11
Resistances in Series
IS
Using Ohm’s Law, we may
express the voltage across each
resistance by the product of the
current and resistance, IR.
This gives the equation
V1
R1
V2
R2
VS
𝑉𝑆 = 𝐼𝑆 𝑅1 + 𝐼𝑆 𝑅2
12
Resistances in Series
IS
Factoring IS reveals
𝑉𝑆 = 𝐼𝑆 𝑅1 + 𝑅2
The quantity in parentheses
represents the equivalent
resistance of the series
connection of R1 and R2.
V1
R1
V2
R2
VS
13
Resistances in Series
IS
𝑅eq = 𝑅1 + 𝑅2
The current in the loop
is given by
𝑉𝑆
𝐼𝑆 =
𝑅eq
VS
Req
14
I
This same procedure may
be applied to any number
of resistors connected in
series.
The same current, I, flows
through the resistors.
a
Vab
The total voltage from
point a to point b is Vab.
V1
R1
V2
R2
Vk
Rk
b
15
I
Applying Kirchhoff’s
Voltage Law to this
combination of resistors
leads to the equation:
𝑉ab = 𝑉1 + 𝑉2 + βˆ™βˆ™βˆ™ +𝑉k
a
Vab
V1
R1
V2
R2
Vk
Rk
b
16
I
Using Ohm’s Law, the same
voltage may be expressed
a
𝑉ab = 𝐼𝑅1 + 𝐼𝑅2 + βˆ™βˆ™βˆ™ +𝐼𝑅k
Vab
V1
R1
V2
R2
Vk
Rk
b
17
I
a
Factoring I from each term in
the summation reveals
𝑉ab = 𝐼 𝑅1 + 𝑅2 + βˆ™βˆ™βˆ™ +𝑅k
Vab
The term in parentheses
represents the equivalent
resistance of the series
combination of resistances.
V1
R1
V2
R2
Vk
Rk
b
18
I
a
𝑅eq = 𝑅1 + 𝑅2 + βˆ™βˆ™βˆ™ +𝑅k
π‘˜
𝑅eq =
𝑅𝑛
Vab
Req
𝑛=1
b
19
Resistances in Series
For resistors in series, the equivalent
resistance is equal to the sum of the
resistances of the individual resistors.
For resistances in series, the equivalent
resistance is always larger than the
largest resistance in the group.
20
Resistances in Parallel
Consider the resistive circuit below:
Node
IS
I2
I1
VS
V1
V2
R1
R2
21
Resistances in Parallel
Note that resistances R1 and R2 are connected
side-by side.
The voltage source VS is placed across both R1
and R2.
Node
IS
I2
I1
VS
V1
V2
R1
R2
22
Resistances in Parallel
+𝑉𝑆 βˆ’ 𝑉1 = 0
𝑉𝑆 = 𝑉1
IS
+𝑉𝑆 βˆ’ 𝑉2 = 0
𝑉𝑆 = 𝑉2
Node
I2
I1
VS
+𝑉1 βˆ’ 𝑉2 = 0
𝑉1 = 𝑉2
V1
V2
R1
R2
𝑉𝑆 = 𝑉1 = 𝑉2
23
Parallel Connection
When the same voltage appears across
circuit elements, those circuit elements
are connected in parallel.
24
Resistances in Parallel
Applying Kirchhoff’s Current Law to the node
results in the equation:
𝐼𝑆 βˆ’ 𝐼1 βˆ’ 𝐼2 = 0
𝐼𝑆 = 𝐼1 + 𝐼2
IS
I2
I1
VS
VS
VS
R1
R2
25
Resistances in Parallel
By applying Ohm’s Law, we can replace
the current in each resistance by the
quotient of the voltage divided by the
resistance, V/R.
IS
I2
I1
VS
VS
VS
R1
R2
26
Resistances in Parallel
𝑉S 𝑉S
𝐼S =
+
𝑅1 𝑅2
IS
I2
I1
VS
VS
VS
R1
R2
27
Resistances in Parallel
The term VS can be factored out of both terms.
1
1
𝐼S = 𝑉S
+
𝑅1 𝑅2
IS
I2
I1
VS
VS
VS
R1
R2
28
Resistances in Parallel
The term in brackets has the units of
conductance, the reciprocal of resistance.
This suggests an equivalent conductance:
𝐺eq
1
1
1
=
=
+
𝑅eq 𝑅1 𝑅2
𝐺eq = 𝐺1 + 𝐺2
29
Resistances in Parallel
Putting the term in brackets over a common
denominator leads to the equation:
1
𝑅2 + 𝑅1
=
𝑅eq
𝑅1 𝑅2
𝑅eq
𝑅1 𝑅2
=
𝑅1 + 𝑅2
30
Resistances in Parallel
This connection occurs frequently, so it is worthwhile to
remember that for two resistors in parallel :
π‘ƒπ‘Ÿπ‘œπ‘‘π‘’π‘π‘‘ π‘œπ‘“ π‘‘β„Žπ‘’ π‘…π‘’π‘ π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’π‘ 
πΈπ‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ π‘…π‘’π‘ π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’ =
π‘†π‘’π‘š π‘œπ‘“ π‘‘β„Žπ‘’ π‘…π‘’π‘ π‘–π‘ π‘‘π‘Žπ‘›π‘π‘’π‘ 
31
Resistances in Parallel
An arbitrary number of resistances
connected in parallel.
Node
I
βˆ™βˆ™βˆ™
I1
I2
Ik
Vab
R1
R2
βˆ™βˆ™βˆ™
Rk
32
Resistances in Parallel
The same voltage Vab appears across each
of the resistances.
Node
I
βˆ™βˆ™βˆ™
I1
I2
Ik
Vab
R1
R2
βˆ™βˆ™βˆ™
Rk
33
Resistances in Parallel
Applying Kirchhoff’s current Law at the
node results in the equation:
𝐼 = 𝐼1 + 𝐼2 + βˆ™βˆ™βˆ™ +πΌπ‘˜
Node
I
βˆ™βˆ™βˆ™
I1
I2
Ik
Vab
R1
R2
βˆ™βˆ™βˆ™
Rk
34
Resistances in Parallel
Applying Ohm’s Law, we can replace the
current through each resistance with the
quotient of the voltage divided by the
resistance, V/R.
I
βˆ™βˆ™βˆ™
I1
I2
Ik
Vab
R1
R2
βˆ™βˆ™βˆ™
Rk
35
Resistances in Parallel
𝑉ab 𝑉ab
𝑉ab
𝐼=
+
+ βˆ™βˆ™βˆ™ +
𝑅1
𝑅2
π‘…π‘˜
Factoring Vab from each term leads to:
𝐼 = π‘‰π‘Žπ‘
1
1
1
+
+ βˆ™βˆ™βˆ™ +
𝑅1 𝑅2
π‘…π‘˜
36
Resistances in Parallel
𝐼 = π‘‰π‘Žπ‘
1
1
1
+
+ βˆ™βˆ™βˆ™ +
𝑅1 𝑅2
π‘…π‘˜
The term in brackets has the units of conductance and
can be expressed
𝐺eq
1
1
1
1
=
=
+
+ βˆ™βˆ™βˆ™ +
𝑅eq 𝑅1 𝑅2
π‘…π‘˜
37
Resistances in Parallel
1
1
1
1
=
+
+ βˆ™βˆ™βˆ™ +
𝑅eq 𝑅1 𝑅2
π‘…π‘˜
I
βˆ™βˆ™βˆ™
I1
I2
Ik
Vab
R1
R2
βˆ™βˆ™βˆ™
Rk
38
Resistances in Parallel
1
1
1
1
=
+
+ βˆ™βˆ™βˆ™ +
𝑅eq 𝑅1 𝑅2
π‘…π‘˜
I
Vab
Req
1
=
𝑅eq
π‘˜
𝑛=1
π‘˜
𝐺eq =
1
𝑅𝑛
𝐺𝑛
𝑛=1
39
Resistances in Parallel
For resistors in parallel, the equivalent
conductance is equal to the sum of the
conductances of the individual resistors.
For resistances in parallel, the equivalent
resistance is always smaller than the
smallest resistance in the group.
40
Resistances in Parallel
Warning: Do not make these common mistakes!
𝑅eq
1
1
1
β‰ 
+
+ βˆ™βˆ™βˆ™ +
𝑅1 𝑅2
π‘…π‘˜
𝑅eq
𝑅1 𝑅2 𝑅3
β‰ 
𝑅1 + 𝑅2 + 𝑅3
These are wrong!
41
Simplifying Resistive Circuits
R1
VS
R3
R2
R4
Are R1 and R3 connected in series?
No! They do not share the same current!
42
Simplifying Resistive Circuits
R1
VS
R3
R2
R4
Are R2 and R4 connected in parallel?
No! They do not share the same voltage!
43
Simplifying Resistive Circuits
R1
VS
R3
R2
R4
Are R1 and R2 connected in series?
No! They do not share the same current!
Are R1 and R2 connected in parallel?
No! They do not share the same voltage!
44
Simplifying Resistive Circuits
R1
VS
R3
R2
R4
Are R3 and R4 connected in series?
Yes! They share the same current!
45
Simplifying Resistive Circuits
R1
VS
R2
R3 + R4
Are R2 and (R3 + R4) connected in parallel?
Yes! They share the same voltage!
46
Simplifying Resistive Circuits
R1
VS
R2 || (R3 + R4)
Are these resistances connected in series?
Yes! They share the same current!
47
Simplifying Resistive Circuits
VS
𝑅eq
𝑅eq
R1 + R2 || (R3 + R4)
𝑅2 𝑅3 + 𝑅4
= 𝑅1 +
𝑅2 + 𝑅3 + 𝑅4
𝑅1 𝑅2 + 𝑅3 + 𝑅4 + 𝑅2 𝑅3 + 𝑅4
=
𝑅2 + 𝑅3 + 𝑅4
48
Simplifying Resistive Circuits
R2
VS
R5
R1
R3
R4
49
Simplifying Resistive Circuits
VS
R5
R1
R2 + R3
R4
50
Simplifying Resistive Circuits
VS
R5 || (R2 + R3)
R1
R4
51
Simplifying Resistive Circuits
VS
R1
R4 + R5 || (R2 + R3)
52
Simplifying Resistive Circuits
VS
𝑅eq
R1||{R4 + R5 || (R2 + R3)}
𝑅5 𝑅2 + 𝑅3
𝑅1 𝑅4 +
𝑅5 + 𝑅2 + 𝑅3
=
𝑅5 𝑅2 + 𝑅3
𝑅1 + 𝑅4 +
𝑅5 + 𝑅2 + 𝑅3
53
Series or Parallel?
R1
R2
R3
R7
R5
R4
VS
R8
R6
54
Series or Parallel?
R3
R2
R1
R4
R7
R8
R5
VS
R6
55
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