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Chapter 2 – Resistive Circuits
Objectives:
• to learn about resistance and Ohm’s Law
• to learn how to apply Kirchhoff’s laws to resistive
circuits
• to learn how to analyze circuits with series and/or
parallel connections
• to learn how to analyze circuits that have wye or
delta connections
• Read pages 14 – 50
• Homework Problems - TBA
Fall 2001
ENGR201 Circuits I - Chapter 2
1
Resistance - Definition
• Resistance is an intrinsic property of matter and
is a measure of how much a device impedes the
flow of current.
• The greater the resistance of an object, the
smaller the amount of current that will flow for a
given applied voltage.
• The resistance of an object depends on the
material used to construct the object (copper has
less resistance than plastic), the geometry of the
object (size and shape), and the temperature of
the object. (R = L/A)
Fall 2001
ENGR201 Circuits I - Chapter 2
2
Resistance – Applications
• Sometimes we want to minimize the resistance
of an object (in a conductor, for instance).
• Sometimes we want to maximize the resistance
(in an insulator).
• Sometimes we to relate the resistance of the
object to some physical parameter (such as a
photoresistor or RTD).
• Sometimes we want to precisely control the
resistance of an element in order to influence the
behavior of a circuit such as an amplifier.
Fall 2001
ENGR201 Circuits I - Chapter 2
3
Resistance - Sizing
Resistors come in all shapes and sizes (see Figure
2.1 in your text). However, several common
parameters are used to characterize resistors:
 ohmic value (nominal) measured in Ohms (),
 maximum power rating measured in Watts
(W), and
 precision (or tolerance) measured as a
percentage of the ohmic value.
Fall 2001
ENGR201 Circuits I - Chapter 2
4
Ohm’s Law
Ohm’s Law - describes the relationship
between the current through and the
voltage across a resistor.
Different devices connected to a power source demand
different amounts of power from that source. That is,
different devices present differing amounts of loading.
I = 0.5A
12V
6W
I = 1A
12V
12W
The 6w bulb offers more resistance to the flow of current
than the 12w bulb.
Fall 2001
ENGR201 Circuits I - Chapter 2
5
Ohm’s Law –
Mathematical Definition
+
V
-
R
I
R = V/I
I = V/R
V = IR
• Rather than specify the load that a device represents in
terms of its voltage/power rating, we can specify that load
in terms of its resistance.
• The smaller the resistance the greater the load (the greater
the power demand).
I = 0.5A
I = 1A
12V
6W
12V
R = 12V/0.5A = 24
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12W
R = 12V1A = 12
ENGR201 Circuits I - Chapter 2
6
Example
12V
12W
12W
6V
How much current will a 12V/12W lamp demand if 6V is
applied to it? How much power is demanded?
A 12w/12v lamp will draw 1A of current:
• P = VI  12W = 12V  I  I = 1A
• V = IR (Ohm’s Law)  R = 12V/1A = 12
• Therefore, if V = 6V  I = 6V/1A = 12
• P = 6v  0.5A = 3W = 0.25  12W.
• Since both the voltage and current are halved, the
power is cut by a factor of four.
Fall 2001
ENGR201 Circuits I - Chapter 2
7
Short & Open Circuits
+
V
-
R = V/I
R is the resistance of the device, measured in ohms ().
The greater the value of R, the smaller the value of I.
I = 0A
(air, plastic, wood)
I=
(wire)
12V
12V
Open Circuit, R = 
I = 0 regardless of the
value of V (NO LOAD)
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V
Short Circuit, R =0
V = 0 regardless of the
value of I
ENGR201 Circuits I - Chapter 2
8
Ohm’s Law – Voltage Polarity & Current Direction
• Ohms’ law relates the magnitude of the voltage with
the magnitude of the current AND
• the polarity of the voltage to the direction of the
current.
R
+
V
-
I = V/R
Resistors always absorb power, so resistor current
always flows through a voltage drop.
Fall 2001
ENGR201 Circuits I - Chapter 2
9
Ohm’s Law - Graphically
+
V
-
R
I = V/R
Ohms’ Law can be represented graphically – called a
VI characteristic:
V
m = Slope = V/I = R
I
Ideal resistor, VI characteristic
Fall 2001
ENGR201 Circuits I - Chapter 2
10
V
Non-ideal Resistors
Open circuit, slope = 
(I = 0)
I
Short circuit, slope = 0
(V = 0)
V
Pmax
I
Pmax
Fall 2001
Practical resistor VI characteristic
ENGR201 Circuits I - Chapter 2
11
Conductance
• Resistance is a measure of how much a device impedes
the flow of current. Conductance is a measure of how little
a device impedes the flow of current.
• Resistance and conductance are simply two different ways
to describe the voltage-current characteristic of a device.
• At times, especially in electronic circuits, it is
advantageous to work in terms of conductance rather than
resistance
Fall 2001
ENGR201 Circuits I - Chapter 2
12
Conductance - Units
Resistance:
G = I/V, S
(seimens)
Old style symbol for conductance
Old style units = mho
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R = V/I,

(ohms)
ENGR201 Circuits I - Chapter 2
(G = 1/R = R-1)

Conductance:
+
V
-
+
V
-
13
Resistance – Power Equations
Resistance:
+
V
-
I
R
P = VI
P = V2/R
P = I2R
V = IR = I/G
P = VI (any device)
for a resistor:
P = V(V/R) = V2/R
or
P = (IR)I = I2R
Fall 2001
P = VI (any device)
In terms of conductance:
P = V(VG) = V2G
or
P = (I/G)I = I2/G
ENGR201 Circuits I - Chapter 2
14
Kirchhoff’s Laws
• Kirchhoff’s Current Law (KCL)
• Kirchhoff’s Voltage Law (KVL)
Kirchhoff’s Laws
A node is a “point” in a circuit where two or elements are
connected.
Node-A
Node-A
R
R
R
R
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+
-
R
ENGR201 Circuits I - Chapter 2
+
-
R
15
KCL
Kirchhoff’s Current Law
• The algebraic sum of all currents at any node in a circuit is
exactly zero.
• The sum of all currents entering = sum of all currents
leaving
• We neither gain nor lose current at a node.
Node-A
I2
I1
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I1-I2+I3-I4 = 0
I3
R
R
+
-
R
I4
ENGR201 Circuits I - Chapter 2
I1+I3 = I2+I4
16
KVL
Kirchhoff’s Voltage Law (KVL)
A loop is a closed path about a circuit that begins and ends
at the same node. However, no element may be traversed
more than once.
A
Five loops in the circuit shown are:
A-C-B-A
B-C-E-B
A-D-C-A
C-D-E-C
C
B
D
E
A-D-E-B-A
Are there more loops ?
Fall 2001
ENGR201 Circuits I - Chapter 2
17
KVL
V2
+
B
+
V6
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• The algebraic sum of all voltages about any
loop in a circuit is exactly zero.
• The sum of all increases (rises) = sum of all
voltage decreases (drops)
• We do not gain or lose voltage if we start and
end at the same node.
A
+
V3
+ V1 - C+
+ Vx Vy
E
+
V4
D
+
V5
-
By KVL:
• V2 + V3 - V1 = 0
• -V3 + V4 - Vx = 0
• V1 + Vy - V6 = 0
• Vx + V5 -Vy = 0
• V2 + V4 + V5 - V6 = 0
ENGR201 Circuits I - Chapter 2
18
Equivalent
Circuits
I1
Vs
Two circuits are equivalent if, for any
source connected to the circuits, they
demand the same amount power. The
two circuits “look” the same to the source
I2
+
Device
-
#1
Vs
+
Device
-
#2
P1 = VSI1
P2 = VSI2
P1 = P2  VSI1 = VSI2  I1 = I2
If the applied voltage is the same, two equivalent circuits will
demand the same amount of current from the source.
Fall 2001
ENGR201 Circuits I - Chapter 2
19
Series connection (all elements
have the same current)
Series Resistance
a
R1
R2
Rab 
b
a
Vab
b
a
R3
Rab
R4
R6
b
R5
R1
R2
I
R6
I
R3
R4
Vab
Rab
R5
Vab = I  Rab
Fall 2001
ENGR201 Circuits I - Chapter 2
20
Series Resistance
a
Vab
b
R1
R2
R
4
I
R6
I
R3
R5
Vab
Ra
b
Vab = I  Rab
By KCL: IR1 = IR2 = … = IR6 = I
By KVL: Vab = IR1+ IR2+ IR3+ IR4+ IR5+ IR6
Vab/I = (R1 + R2 + R3 + R4 + R5 + R6) = Rab
The equivalent resistance of two or more series-connected
resistors is the sum of the individual resistors.
Fall 2001
ENGR201 Circuits I - Chapter 2
21
Parallel Resistance Parallel connection (all the
elements have the same voltage)
I
Vab/I = Rab
A
I
Vab
R1 R2 R3 R4
R5
Vab
Rab
B
by KCL:
I = I1 + I2 + I3 + I4 + I5
I = Vab/R1 + Vab/R2 + Vab/R3 + Vab/R4 + Vab/R5
I = Vab[R1-1 + R2-1 + R3-1 + R4-1 + R5-1 ]
Vab/I = [R1-1 + R2-1 + R3-1 + R4-1 + R5-1 ]-1
Fall 2001
ENGR201 Circuits I - Chapter 2
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Parallel Resistance
Vab/I = Rab
I
Vab
I
R1 R2 R3 R4
R5
Vab
Rab
Rab = [R1-1 + R2-1 + R3-1 + R4-1 + R4-1 ]-1
Since G = 1/R = R-1
Rab = [G1 + G2 + G3 + G4 + G5]-1
Gab = G1 + G2 + G3 + G4 + G5
Fall 2001
ENGR201 Circuits I - Chapter 2
23
The Voltage Divider Rule (VDR)
The total voltage applied to a group of series-connected
resistors will be divided among the resistors. The fraction
of the total voltage across any single resistor depends on
what fraction that resistor is of the total resistance.
a
Vab
b
R1
R2
R3
I
R6
R4
+
V4
-
R5
RTOTAL = Rab = R1 + R2 + R3 + R4 + R5 + R6
R4


V4  Vab 
 R1  R 2  R3  R 4  R5  R6 
Fall 2001
ENGR201 Circuits I - Chapter 2
24
The Current Divider Rule (CDR)
The total current applied to a group of resistors connected in
parallel will be divided among the resistors. The fraction of
the total current through any single resistor depends on what
fraction that resistor is of the total conductance.
ITotal
Vab
R1 R2 R3 R4 R5
I5
GTOTAL = R1-1 + R2 -1 + R3 -1 + R4 -1 + R5 -1
 G5 


R51
I 5  ITotal 
 ITotal  1

1
1
1
1 
G
R
1

R
2

R
3

R
4

R
5


 Total 
Fall 2001
ENGR201 Circuits I - Chapter 2
25
CDR – Two Resistors
 R2 
I1  ITotal 
 R1  R 2 
For two resistors:
Itotal = I1 + I2
R1
I1
I2
R2
 R1 
I 2  ITotal 
 R1  R 2 
Observations:
• The smaller resistor will have the larger current.
• If R1= R2, then I1 = I2
• If R1 = nR2, then I2 = nI1
Fall 2001
ENGR201 Circuits I - Chapter 2
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