Download Series Circuits

Applied Circuit Analysis
Chapter 4 Series Circuits
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Multi-Element Circuits
• So far we have
considered circuits
limited to one resistor.
• From now on we will
consider circuits with
more than one resistor.
• We will begin by looking
at circuit topology.
2
Nodes Branches and Loops
• Circuit elements can be interconnected in
multiple ways.
• To understand this, we need to be familiar
with some network topology concepts.
• A branch represents a single element such
as a voltage source or a resistor.
• A node is the point of connection between
two or more branches.
• A loop is any closed path in a circuit.
3
Nodes
• A node is usually indicated by a dot in
a circuit, although we do not follow this
convention in this book.
• If a short circuit (wire) connects two
nodes, the nodes are considered as
one.
• The circuit shown has three nodes.
4
Recognizing Nodes
• It is important to keep track of the
topology of a circuit.
• Any single circuit can be drawn a
multitude of ways that are functionally
equivalent.
• Keeping track of nodes is an important
part of this.
5
Recognizing Nodes II
• Examine the two circuits shown here.
• They are equivalent circuits.
6
Network Topology
• A loop is independent if it contains at
least one branch not shared by any
other independent loops.
• Two or more elements are in series if
they share a single node and thus carry
the same current.
• Two or more elements are in parallel if
they are connected to the same two
nodes and thus have the same voltage.
7
Series Resistors
• Two resistors are considered in
series if the same current pass
through them
• Take the circuit shown:
• The total resistance is:
RT  R1  R2
• More generally, the total
resistance equals the sum of
the resistances.
RT  R1  R2  R3   RN
8
Series Resistors II
• Because the same current I passes
through each resistor, we can calculate
the voltage across each resistor:
V1  IR1
V2  IR2
VN  IRN
• This indicates the voltage drop across
each resistor depends on its resistance.
9
Series Resistors III
• We can examine the power dissipated
in series resistors as well.
• The power through the individual
resistors is:
P1  I 2 R1
P2  I 2 R2
PN  I 2 RN
10
Power in Series Resistors
• The total power delivered to the series
circuit is:
PT  P1  P2 
 PN
• Because the current through each
resistor is the same, the power can be
expressed as:
PT  I 2  R1  R2 
 RN 
• Or
PT  I 2 RT
11
Kirchoff’s Laws
• Ohm’s law is not sufficient for circuit
analysis.
• Kirchoff’s laws complete the needed
tools.
• There are two laws:
– Current law (KCL)
– Voltage law (KVL)
• KCL will be covered in the next chapter.
12
KVL
• Kirchoff’s voltage law is based on
conservation of energy.
• It states that the algebraic sum of
currents around a closed path (or loop)
is zero.
• It can be expressed as:
M
v
m 1
m
0
13
KVL II
• As an example, consider
the circuit shown.
• Starting at any branch
and go around the loop in
either direction.
• If we start at the voltage
source and go around
clockwise…
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KVL III
• The voltages we would see are –
V1,+V2,+V3,-V4, and +V5 in that order.
• For example, as we reach branch 3, the
positive terminal is met first, so the
voltage is written as positive.
• KVL will yield:
V1  V2  V3  V4  V5  0
15
Alternate KVL
• From the last example, one can see an
alternative way to express KVL.
• If we separate the negative and positive
voltages from the path we took, we
have:
V2  V3  V5  V1  V4
• Or
 voltage drops   voltage rises
16
Drops vs. Rises
• Voltage rises occur when we travel
across through an element going from
– to +.
• Voltage drops occur when we go from +
to -.
• A voltage rise is said to take place in an
active element.
• A voltage drop occurs in a passive one.
17
Voltage Sources in Series
• One application of KVL is dealing with
multiple voltage sources.
• A number of applications require
multiple voltages to be supplied to a
circuit.
• KVL helps us to understand how this
can be accomplished easily.
18
Voltage Sources in Series II
• Take the series connected
sources shown here.
• Applying KVL to the circuit:
• Or
Vab  V1  V2  V3  0
Vab  V1  V2  V3
19
Voltage Division
• Series resistors are often used to
provide voltage division.
• If we apply Ohm’s law to each resistor,
the voltage drops are:
V1  IR1 V2  IR2
Vn  IRn
• Because the resistors are in series, the
equivalent resistance is:
Req  R1  R2 
 Rn
20
Voltage Division II
• If a voltage V is applied across the
resistors, the current through them is:
I
V
Req
• We can thus express the voltage
across the resistors as
 R1
V1  
 Req


 R2 
 V V2  
 V

 Req 
 Rn
Vn  
 Req


 V

21
Voltage Division III
• The most common application is with
two resistors.
• Applying the formula that was just
presented, the voltages are:
R1
R2
V1 
V V2 
V
R1  R2
R1  R2
• One can see that two resistors may be
used to create any voltage between 0
and V.
22
Ground Connections
• Like measuring distance, voltage must
be measured from a reference point.
• The most common reference point
used is the earth.
• Or more specifically, the ground on
which the building you are in sits.
• This is why this reference point is
referred to as ground.
23
Grounding
• Electrical equipment that is connected
to ground is said to be grounded or
earthed.
• Part of the wiring of any building is a
wire that is connected to a large metal
rod driven deep into the ground.
• This ensures a good connection to
ground.
24
Grounding II
• Proper grounding is vital to making
electrical equipment safe.
• Imagine an electrical device sitting on a
wooden table.
• If the device is damaged, a charge
might accumulate on the frame of the
device, since the table will not conduct
electricity.
25
Grounding III
• Any person touching, or possibly even
just going near the device may get a
serious shock.
• In older homes, the incoming water
pipe was used a grounding as it was
galvanized steel.
• However, with the rise of plastic piping,
this is no longer the case.
26
Ground Symbols
• A ground is a point of reference.
• We attach the value of 0V to ground.
• The three symbols shown below all
represent ground.
• Earth ground is shown in a and b
• Chassis ground is shown in c.
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