Download Series and Parallel Circuits Preamble Series Resistors Kirchoff

Series and Parallel Circuits
(R. Bolton - 2012)
Series and Parallel Circuits
(R. Bolton - 2012)
Preamble

Series and Parallel Circuits
Physics, 8th Edition Custom Edition
Cutnell & Johnson
Chapter 20.6-20.8,
20 6-20 8 20.10
20 10
Pages 610-618, 619-622
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Series and Parallel Circuits
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Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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Series Resistors
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Series and Parallel Circuits
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Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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Kirchoff’ Voltage Law (KVL)
The figure shows a series resistor circuit.


Our first job is to determine how the two
series resistors combine to form an
equivalent
i l t resistor.
i t

In this section of my lectures we will be
developing the two common types of
resistor arrangements; series and
parallel.
To do this we will introduce some very
important laws and rules that you must
become familiar with.
Around any closed-circuit loop, the
sum of the potential drops equals the
sum of the potential rises.
An alternative way of stating KVL is
V
l
loop
=0
Note: In this circuit the same current flows
g each of the resistors.
through
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Physics 155.3: Introduction to Electricity and Magnetism
Series and Parallel Circuits
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Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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Note that only voltage sources (i.e., a
battery) are considered as a potential rise.
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Series and Parallel Circuits
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This is due to the convention off considering how
positive charge is affected when going around a
circuit. Since positive charge gains potential when
going from a negative terminal on the voltage
source to the positive terminal on the voltage
source it experiences a potential rise.
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The figure is redrawn showing polarity
information
If this is the case, where are the potential
drops in the loop shown above?
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Series and Parallel Circuits
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Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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Series and Parallel Circuits
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Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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Series Resistance
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Using Ohm’s Law
VR1 = IR1
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loop
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VR2 = IR2
Using Kirchoff’s Voltage Law (KVL)
V
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
Rseries = R1 + R2 + R3 +  + Rn
= V − IR1 − IR2 = 0
Rearranging (and using Ohm’s Law)
V = IR1 + IR2
V = I (R1 + R2 ) V = IRseries
where

Rseries = R1 + R2
(R. Bolton - 2012)
Physics 155.3: Introduction to Electricity and Magnetism
Series and Parallel Circuits
Resistors in series (i.e., resistors with
the same current flowing
g through
g them))
add.
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Note: If the resistors do not have
the same current flowing through
them they are NOT in series!
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Physics 155.3: Introduction to Electricity and Magnetism
Series and Parallel Circuits
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Series and Parallel Circuits
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Example 1
Series Resistor Voltages
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Series and Parallel Circuits
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Example 1
Solution:
Consider the following circuit

(4.00V, 8.00V)
Using Ohm’s Law
V = IRs
+
I
+
I=
+
V
12 V
12 V
12 V
=
=
=
= 0 .4 A
Rs (R1 + R2 ) (10Ω + 20Ω ) 30Ω
V1 = IR1 = 0.4 A ⋅ 10Ω = 4.0V

V2 = IR2 = 0.4 A ⋅ 20Ω = 8.0V
What is the voltage
g across each resistor?
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Series and Parallel Circuits
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Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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Physics 155.3: Introduction to Electricity and Magnetism
Series and Parallel Circuits
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Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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V2 = IR2
Note that it is possible to determine the
voltage across each resistor in a series
combination by recalling that the same
current goes through each series resistor and
th t th
that
the sum off th
the resistor
i t voltages
lt
mustt
equal the battery voltage.
Take for example the resistor R2 from the
previous circuit:
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Series and Parallel Circuits
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But the current is equal to
I=

V
R1 + R2
Therefore
 V 
 R2
V2 = 
 R1 + R2 

Rearranging
 R2 

V2 = V 
 R1 + R2 
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Physics 155.3: Introduction to Electricity and Magnetism
Series and Parallel Circuits
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Series and Parallel Circuits
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Voltage Divider Rule (VDR)
(for series resistors)
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Series and Parallel Circuits
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Series and Parallel Circuits
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Consider the following circuit
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Physics 155.3: Introduction to Electricity and Magnetism
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Example 3
Very Important Circuit (1)!!!
(2.61V)
Series and Parallel Circuits
Series and Parallel Circuits
Physics 155.3: Introduction to Electricity and Magnetism
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What is the voltage across the 4.7kΩ
resistor?
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Example 2
Solution:
Consider the following circuit
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Physics 155.3: Introduction to Electricity and Magnetism
Series and Parallel Circuits
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Example 2
Voltage Divider Circuit
The voltage across any resistor in a
series combination of resistances
is equal to the voltage across the
series combination multiplied by
the value of the resistor in
question over the total series
resistance.
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Series and Parallel Circuits
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What is the value of E for 20mA to flow in
the circuit (in the direction shown)?
What is VA? VB? VAB?
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Physics 155.3: Introduction to Electricity and Magnetism
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Is the circuit a series circuit?
Is the circuit a parallel circuit?
What is the direction of the current?
What is the symbol on the bottom left?
What does it do?
What are the polarities of the voltage
drops across the resistors?
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Series and Parallel Circuits
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Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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Parallel Resistors
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(50V,-7V,57V)
Before going too far consider
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Example 3
Solution:
Procedure
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Series and Parallel Circuits
Physics 155.3: Introduction to Electricity and Magnetism
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Physics 155.3: Introduction to Electricity and Magnetism
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At any circuit node, the sum of the
currents into the node equals
q
the
sum of the currents out of the
node.

Note: In this circuit the same voltage is across
each of the resistors.
Series and Parallel Circuits
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Series and Parallel Circuits
Our first job is to determine how the two
parallel resistors combine to form an
equivalent
i l t resistor.
i t
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Kirchoff’s Current Law (KCL)
The figure shows a parallel resistor circuit.
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Series and Parallel Circuits
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Note: Currents out of the node are
considered positive and currents into the
node are considered as negative
negative.
An alternative way of stating KCL is
I
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node
=0
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Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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The figure is redrawn showing current
information
Series and Parallel Circuits
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Using Ohm’s Law
VR1 = I1 R1

VR2 = I 2 R2
But
VR1 = VR2 = V

Therefore
I1 =
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Physics 155.3: Introduction to Electricity and Magnetism
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Using Kirchoff’s Current Law (KCL)
I

V V
+
R1 R2
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Series and Parallel Circuits
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In order for this equation to be true
(i.e., V
V=V)
V)
1
R parallel
ll l
1
1 
I = V  + 
 R1 R2 
V
R2
Physics 155.3: Introduction to Electricity and Magnetism
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V V
= −I + +
=0
R1 R2
Rearranging
I=

node
d
I2 =
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Series and Parallel Circuits
V
R1
=
1
1
+
R1 R2
Using Ohm’s Law (again)
 1
1 
V = V  +  R parallel
  R1 R2 
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Physics 155.3: Introduction to Electricity and Magnetism
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Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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Parallel Resistance
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R parallel

=

1
1
1
1
+
+
++
R1 R2 R3
Rn

Note: If the resistors do not have
the same voltage across them they
are NOT in parallel!
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Series and Parallel Circuits
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Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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(1.2A, 0.6A)
Using Ohm’s Law
V = IR parallel
 R + R2 
V
V

I=
=
= V  1
R
R
R parallel
1 2
 R1R2 
R1 + R2
 30Ω 
I = 12V 
 = 1 .8 A
 10Ω ⋅ 20Ω 
What is the total current and the current
through each of the resistors?
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Series and Parallel Circuits
Example 4
Solution:
Consider the following circuit

R1 R2
R1 + R2
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Example 4
Parallel Resistor Currents

It turns out that we have two resistors
in parallel a lot of the time in the
circuits we use.
The previous formula can be simplified
to
R parallel =
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Physics 155.3: Introduction to Electricity and Magnetism
Series and Parallel Circuits
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Two resistors in parallel
Resistors in parallel (i.e., resistors with
the same voltage
g across them)) do NOT
add.
1
Series and Parallel Circuits
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and
I1 =
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V 12V
=
= 1 .2 A
R1 10Ω
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Physics 155.3: Introduction to Electricity and Magnetism
I2 =
V
12V
=
= 0 .6 A
R2 20Ω
Series and Parallel Circuits
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Series and Parallel Circuits
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(R. Bolton - 2012)
Note that it is possible to determine the
current through each resistor in a
parallel combination by recalling that
the same voltage is across each parallel
resistor and that the sum of the resistor
currents must equal the total current.
Take for example the resistor R2 from
the previous example:
(R. Bolton - 2012)
Series and Parallel Circuits
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Physics 155.3: Introduction to Electricity and Magnetism
Series and Parallel Circuits
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Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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Two Resistors in Parallel
The current through any resistor in
a parallel combination of
resistances is equal to the total
current through the parallel
combination multiplied by the
value of the equivalent resistance
of the parallel resistors over the
value of the resistor in question.
Series and Parallel Circuits
Rearranging and substituting for V
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V
V
=
 R1 R2  (R parallel )


 R1 + R2 
 RR 
I  1 2 
R + R2  I (R parallel )
I2 =  1
=
R2
R2
Current Divider rule (CDR)
(for parallel resistors)

But the total current is equal
q
to
I=
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Series and Parallel Circuits
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V = I 2 R2
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Physics 155.3: Introduction to Electricity and Magnetism
Series and Parallel Circuits


It turns out that we have two resistors
in parallel a lot of the time in the
circuits we use.
The previous formula can be simplified
 R1 

I 2 = I 
 R1 + R2 

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Does this make sense?
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Physics 155.3: Introduction to Electricity and Magnetism
Series and Parallel Circuits
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Series and Parallel Circuits
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Example 5
Parallel Resistor Current

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Example
p 5
Solution:
(2.13mA)
Consider the following circuit

What is the current through the 4.7kΩ
resistor?

Before going too far, think about it…
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Series and Parallel Circuits
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Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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Example 6
Very Important Circuit (2)!!!

Series and Parallel Circuits
Physics 155.3: Introduction to Electricity and Magnetism
(R. Bolton - 2012)

Before going too far consider




What is Vab?
Physics 155.3: Introduction to Electricity and Magnetism
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Series and Parallel Circuits
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Series and Parallel Circuits
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Procedure
Consider the following circuit

Series and Parallel Circuits
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Is the circuit a series circuit?
Is the circuit a parallel circuit?
What is the value of Is?
What is the symbol on the bottom left?
What does it do? Can there be more than
one in a circuit?
What is the polarity of the voltage,
voltage Vab?
(R. Bolton - 2012)
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Physics 155.3: Introduction to Electricity and Magnetism
Series and Parallel Circuits
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Series and Parallel Circuits
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Example 6
Solution:
(R. Bolton - 2012)
Physics 155.3: Introduction to Electricity and Magnetism
(0V)
Series and Parallel Circuits
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