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Transcript of KVL KCL NV
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Lecture 1
Circuit Elements, Ohms Law,Kirchhoffs Laws
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Overview
What a circuit element is?
Electrical Quantities Resistors and Ohms Law
Independent and dependent voltage
sources and current sources KVL & KCL
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Circuit Elements In circuits, we think about basic circuit
elements that are the building blocksof our circuits. This is similar to whatwe do in Chemistry with chemical
elements like oxygen or nitrogen. A circuit element cannot be broken down
or subdivided into other circuitelements.
A circuit element can be defined in
terms of the behavior of the voltage andcurrent at its terminals.
We define an electric circuit as aconnection of electrical devices thatform one or more closed paths.
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Basic Electrical Quantities
Basic Quantities: current, voltage andpower Current: Time rate of change of electric charge
I = dq/dt
1 Amp = 1 Coulomb/sec
Voltage: electromotive force or potential, V 1Volt = 1 Joule/Coulomb = 1 Nm/coulomb
Power: P = I V
1 Watt = 1 VoltAmp = 1 Joule/sec
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Current I
Normally we talk about the movement ofpositive charges although we know that, ingeneral, in metallic conductors current resultsfrom electron motion (conventionally positiveflow)
The sign of the current indicates thedirection of flow
Types of current:
Direct Current (DC): batteries and some specialgenerators Alternating Current (AC): household current which
varies with time
I(t)
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Voltage V
Voltage is the difference in energy levelof a unit charge located at each of two
points in a circuit, and therefore,represents the energy required to movethe unit charge from one point to theother
Circuit Element(s)
+ V(t)
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Default Sign Convention
Passive sign convention : current shouldenter the positive voltage terminal
Consequence for P = I V Positive (+) Power: element absorbs power
Negative (-) Power: element supplies power
Circuit Element+
I
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Electrical Analogies (Physical)
Electric Hydraulic
Base
quantity Charge (q) Mass (m)
Flow
variableCurrent (I) Fluid flow (G)
Potentialvariable
Voltage (V) Pressure (p)
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Active vs. Passive Elements
Active elements can generate energy Voltage and current sources
Batteries Passive elements cannot generate
energy Resistors
Capacitors and Inductors (but CAN storeenergy)
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The Basic Circuit Elements
There are 5 basic circuit elements:
1. Voltage sources
2. Current sources
3. Resistors
4. Inductors
5. Capacitors
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Voltage Sources
A voltage source is a two-terminalcircuit element that maintains avoltage across its terminals.
The value of the voltage is thedefining characteristic of a voltagesource.
Any value of the current can go
through the voltage source, in anydirection. The current can also bezero. The voltage source does notcare about current. It caresonly about voltage.
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Voltage Sources Ideal and
Practical A voltage source maintains a voltage across its
terminals no matter what you connect to thoseterminals.
We often think of a battery as being a voltagesource. For many situations, this is fine. Othertimes it is not a good model. A real battery willhave different voltages across its terminals insome cases, such as when it is supplying a largeamount of current. As we have said, a voltagesource should not change its voltage as the
current changes. We sometimes use the term ideal voltage source
for our circuit elements, and the term practicalvoltage source for things like batteries. We willfind that a more accurate model for a battery isan ideal voltage source in series with a resistor.
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Voltage Sources
There are 2 types of voltage sources:
1. Independent voltage sources
2. Dependent voltage sources, of whichthere are 2 forms:i. Voltage-dependent voltage sources
ii. Current-dependent voltage sources
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Voltage Sources Schematic Symbol for
Independent Sources
The schematicsymbol that we usefor independentvoltage sources isshown here.
Independent
voltage
source
+
-
vS=
#[V]
This is intended to indicate that the schematic symbol can belabeled either with a variable, like vS, or a value, with somenumber, and units. An example might be 1.5[V]. It could alsobe labeled with both.
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Voltage Sources SchematicSymbols for Dependent Voltage Sources
The schematic symbolsthat we use fordependent voltage
sources are shownhere, of which thereare 2 forms:
i. Voltage-dependent
voltage sourcesii. Current-dependent
voltage sources
Voltage-
dependent
voltage
source
+vS=
vX
-
Current-dependent
voltage
source
+vS=
iX
-
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Voltage-
dependent
voltage
source
+vS=
vX
-
Notes on Schematic
Symbols for Dependent Voltage Sources
The schematic symbols that we use fordependent voltage sources areshown here, of which there are 2forms:
i. Voltage-dependent voltage
sourcesii. Current-dependent voltagesources
The symbol m is the coefficient of thevoltage vX. It is dimensionless. Forexample, it might be 4.3 vX. The vX is avoltage somewhere in the circuit.
Current-dependent
voltage
source
+vS=
iX
-The symbol r is the coefficient of the current iX.It has dimensions of [voltage/current]. Forexample, it might be 4.3[V/A] iX. The iX is a
current somewhere in the circuit.
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Current Sources
A current source is a two-terminal circuitelement that maintains a current through itsterminals.
The value of the current is the definingcharacteristic of the current source.
Any voltage can be across the currentsource, in either polarity. It can also be
zero. The current source does not careabout voltage. It cares only aboutcurrent.
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Current Sources - Ideal A current source maintains a current through its
terminals no matter what you connect to thoseterminals.
While there will be devices that reasonably model
current sources, these devices are not as familiaras batteries. We sometimes use the term ideal current source
for our circuit elements, and the term practicalcurrent source for actual devices. We will findthat a good model for these devices is an idealcurrent source in parallel with a resistor.
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Current Sources
There are 2 types of current sources:
1. Independent current sources
2. Dependent current sources, of which thereare 2 forms:
i. Voltage-dependent current sources
ii. Current-dependent current sources
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Independent
current
source
iS=#[A]
Current Sources Schematic Symbol for
Independent Sources
The schematic symbolsthat we use for
current sources areshown here.
This is intended to indicate that the schematic symbol can be labeled eitherwith a variable, like iS, or a value, with some number, and units. An examplemight be 0.2[A]. It could also be labeled with both.
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Current Sources Schematic
Symbols for Dependent Current SourcesThe schematic symbols
that we use fordependent currentsources are shownhere, of which thereare 2 forms:
i. Voltage-dependentcurrent sources
ii. Current-dependentcurrent sources
Voltage-
dependent
current
source
iS=
gvX
Current-
dependent
current
source
iS=
iX
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Current-
dependent
current
source
iS=
iX
Voltage-
dependent
current
source
iS=
gvX
Notes on Schematic
Symbols for Dependent Current Sources
The schematic symbols that we usefor dependent current sourcesare shown here, of which thereare 2 forms:
i. Voltage-dependent current
sourcesii. Current-dependent currentsources
The symbol g is the coefficient ofthe voltage vX. It has dimensionsof [current/voltage]. For example,it might be 16[A/V] vX. The vX is avoltage somewhere in the circuit.
The symbolb is the coefficient of thecurrent iX. It is dimensionless. Forexample, it might be 53.7 iX. The iX is a
current somewhere in the circuit.
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Voltage and Current Polarities
Previously, we have emphasized theimportant of reference polarities ofcurrents and voltages.
Notice that the schematic symbols forthe voltage sources and current sourcesindicate these polarities.
The voltage sources have a + and a toshow the voltage reference polarity. The
current sources have an arrow to show thecurrent reference polarity.
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Why do we have
these dependent sources? Students who are new to circuits often questionwhy dependent sources are included. Somestudents find these to be confusing, and they doadd to the complexity of our solution techniques.
However, there is no way around them. We needdependent sources to be able to model amplifiers,and amplifier-like devices. Amplifiers are crucialin electronics. Therefore, we simply need tounderstand and be able to work with dependent
sources.
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Resistors
A resistor is a two terminalcircuit element that has aconstant ratio of the voltageacross its terminals to thecurrent through its terminals.
The value of the ratio of voltageto current is the definingcharacteristic of the resistor.
Real-world devices that aremodeled by resistors:incandescent light bulbs, heatingelements (stoves, heaters, etc.),long wires
In many cases a light bulb
can be modeled with a
resistor.
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Resistors Definition and Units
A resistor obeys the expression
where R is the resistance. If something obeys this
expression, we can think of it,and model it, as a resistor.
This expression is called OhmsLaw. The unit ([Ohm] or [W]) is
named for Ohm, and is equal to a[Volt/Ampere]. IMPORTANT: use Ohms Law
only on resistors. It does nothold for sources.
To a first-order approximation,
the body can modeled as aresistor. Our goal will be toavoid applying large voltagesacross our bodies, because itresults in large currentsthrough our body. This is not
good.
R
R
vR
i
+
R
viR -
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RX=
#[]
vX
iX
-+
Schematic Symbol for Resistors
The schematic symbols that we use forresistors are shown here.
This is intended to indicate that the schematic symbolcan be labeled either with a variable, like RX, or a
value, with some number, and units. An example
might be 390[]. It could also be labeled with both.
XX
X
vR
i
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Ohms Law
The Rest of
the Circuit
R
i(t)
+
v(t)
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Open Circuit
What if R = ?
i(t) = v(t)/R = 0
v(t)
The
Rest oftheCircuit
i(t)=0
+
i(t)=0
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Short Circuit
What if R = 0 ?
v(t) = R i(t) = 0
The
Rest oftheCircuit
v(t)=0
i(t)
+
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Series
Two elements are in series if the currentthat flows through one must also flowthrough the other.
R1 R2
Series
R1 R2
Not Series
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Parallel
Two elements are in parallel if they areconnected between (share) the same two(distinct) end nodes.
Parallel Not Parallel
R1
R2
R1
R2
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Resistor Polarities
Previously, we have emphasized the important ofreference polarities of current sources andvoltages sources. There is no correspondingpolarity to a resistor. You can flip it end-
for-end, and it will behave the same way.However, even in a resistor, direction matters inone sense; we need to have defined thevoltage and current in the passive signconvention to use the Ohms Law equationthe way we have it listed here.
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Getting the Sign Right with Ohms Law
If the reference current is in the
direction of the reference voltage
drop (Passive Sign Convention),
then
RX=
#[]
vX
iX
-+
XX
X
vR
i
If the reference current is in the
direction of the reference voltagerise (Active Sign Convention),then
RX=
#[]
vX
iX
-+
XX
X
vR
i
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Why do we have to worry
about the sign in Ohms Law? It is reasonable to ask why the sign in Ohms Law
matters. We may be used to thinking that resistanceis always positive.
Unfortunately, this is not true. The resistors we use,particularly the electronic components we callresistors, will always have positive resistances.However, we will have cases where a device will have aconstant ratio of voltage to current, but the value ofthe ratio is negative when the passive sign convention
is used. These devices have negative resistance.They provide positive power.This can be done usingdependent sources.
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Why do we have to worryabout the sign in Everything?
This is one of the central themes in circuit analysis.The polarity, and the sign that goes with thatpolarity, matters. The key is to find a way to getthe sign correct every time.
This is why we need to define reference polaritiesfor every voltage and current.
This is why we need to take care about whatrelationship we have used to assign reference
polarities (passive sign convention and active signconvention).
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In this part of the module, we willcover the following topics:
Some Basic Assumptions
Kirchhoffs Current Law (KCL)
Kirchhoffs Voltage Law (KVL)
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Some Fundamental Assumptions
Wires Although you may not have stated it,or thought about it, when you havedrawn circuit schematics, you haveconnected components or devices
with wires, and shown this with lines. Wires can be modeled pretty well as
resistors. However, their resistanceis usually negligibly small.
We will think of wires as connectionswith zero resistance. Note that thisis equivalent to having a zero-valuedvoltage source.
This picture shows wires
used to connect electrical
components. This particular
way of connectingcomponents is called
wirewrapping, since the
ends of the wires are
wrapped around posts.
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Some Fundamental Assumptions Nodes
A node is defined as a placewhere two or morecomponents are connected.
The key thing to rememberis that we connectcomponents with wires. Itdoesnt matter how many
wires are being used; it onlymatters how manycomponents are connectedtogether.
+
-
vA
C
RD
iB
RF
RE
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Basic Laws of Electric CircuitsNodes, Branches, and Loops:
A node: A node can be defined as a connection point betweentwo or more branches.
A branch: A branch is a single electrical element or device.
A circuit with 5 branches.
A circuit with 3 nodes.
2
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Basic Laws of Electric CircuitsNodes, Branches, and Loops:
If we start at any point in a circuit (node), proceed throughconnected electric devices back to the point (node) fromwhich we started, without crossing a node more than one time,
we form a closed-path.
A loop is a closed-path.
An independent loop is one that contains at least
one element not contained in another loop.
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Basic Laws of Electric CircuitsNodes, Branches, and Loops:
The relationship between nodes, branches and loopscan be expressed as follows:
# branches = # loops + # nodes - 1
or
B = L + N - 1
In using the above equation, the number of loops arerestricted to be those that are independent.
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How Many Nodes?
To test ourunderstanding ofnodes, lets look at
the example circuitschematic givenhere.
How many nodes are
there in thiscircuit?
+
-vA
C
RD
iB
RF
RE
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How Many Nodes Correct Answer
In this schematic,there are threenodes. These nodesare shown in dark blue
here. Some students countmore than threenodes in a circuit likethis. When they do, itis usually because
they have consideredtwo points connectedby a wire to be twonodes.
+
-
vA
RC
RD
iB
RF
RE
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How Many Nodes Wrong Answer
In the example circuitschematic given here,the two red nodes arereally the same node.There are not fournodes.
Remember, two nodesconnected by a wirewere really only onenode in the first place.
+
-
vA
RC
RD
iB
RF
RE
Wire connecting two nodesmeans that these are really asingle node.
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Some Fundamental Assumptions
Closed Loops A closed loop can be
defined in this way: Startat any node and go in anydirection and end up where
you start. This is a closedloop.
Note that this loop doesnot have to followcomponents. It can jumpacross open space. Most
of the time we will followcomponents, but we willalso have situations wherewe need to jump betweennodes that have noconnections.
+
-
vA
RC
RD
iB
RF
RE
vX
+
-
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How Many Closed Loops To test our
understanding of
closed loops, letslook at theexample circuitschematic given
here. How many closedloops are therein this circuit?
+
-
vA
RC
RD
iB
RF
RE
vX
+
-
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How Many Closed Loops
An Answer There are several closed
loops that are possiblehere. We will show a few
of them, and allow you tofind the others.
The total number ofsimple closed loops in thiscircuit is 13.
Finding the number will not
turn out to be important.What is important is torecognize closed loopswhen you see them.
+
-
vA
RC
RD
iB
RF
RE
vX
+
-
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+
-
vA
RC
RD
iB
RF
RE
vX
+
-
Closed Loops Loop #1 Here is a loop we
will call Loop #1.
The path isshown in red.
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+
-
vA
RC
RD
iB
RF
RE
vX
+
-
Closed Loops Loop #2 Here is Loop #2.
The path is shown inred.
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+
-
vA
RC
RD
iB
RF
RE
vX
+
-
Closed Loops Loop #3 Here is Loop #3.
The path is shown inred.
Note that this pathis a closed loop that
jumps across the
voltage labeled vX.This is still a closedloop.
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+
-
vA
RC
RD
iB
RF
RE
vX
+
-
Closed Loops Loop #4 Here is Loop #4. The
path is shown in red.
Note that this path isa closed loop that
jumps across thevoltage labeled vX.This is still a closed
loop. The loop alsocrossed the currentsource. Rememberthat a current sourcecan have a voltage
across it.
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+
-
vA
RC
RD
iB
RF
RE
vX
+
-
A Not-Closed Loop The path is shown in
red here is notclosed.
Note that this pathdoes not end whereit started.
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Kirchhoffs Current Law (KCL)
With these definitions, we are
prepared to state KirchhoffsCurrent Law:
The algebraic (orsigned) summation of
currents through aclosed surface mustequal zero.
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Kirchhoffs Current Law(KCL) Some notes.
The algebraic (or signed)summation of currentsthrough any closed surface
must equal zero.
This definition is often stated as applying to nodes. It applies to any closedsurface. For any closed surface, the charge that enters must leavesomewhere else. A node is just a small closed surface. A node is theclosed surface that we use most often. But, we can use any closedsurface, and sometimes it is really necessary to use closed surfaces that
are not nodes.
This definition essentially means that charge does not build up at aconnection point, and that charge is conserved.
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KCL (Kirchhoffs Current Law)
The sum of currents entering the node is zero:
Analogy: mass flow at pipe junction
i1(t)
i2(t) i4(t)
i5(t)
i3(t)
n
j
j ti1
0)(
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Current Polarities
Again, the issue of thesign, or polarity, or direction,of the current arises. Whenwe write a Kirchhoff CurrentLaw equation, we attach asign to each referencecurrent polarity, dependingon whether the referencecurrent is entering or leavingthe closed surface. This canbe done in different ways.
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Kirchhoffs Current Law (KCL) a Systematic Approach
The algebraic (or signed) summation ofcurrents through any closed surface mustequal zero.
For this set of material, we will always assign a positive sign to aterm that refers to a reference current that leaves a closedsurface, and a negative sign to a term that refers to a referencecurrent that enters a closed surface.
For most students, it is a good idea to choose one way to writeKCL equations, and just do it that way every time. The idea isthis: If you always do it the same way, you are less likely to getconfused about which way you were doing it in a certain equation.
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Kirchhoffs Current Law (KCL) For this set of material, we
will always assign a positive
sign to a term that refers toa current that leaves aclosed surface, and anegative sign to a term thatrefers to a current thatenters a closed surface.
In this example, we havealready assigned referencepolarities for all of thecurrents for the nodesindicated in darker blue.
For this circuit, and using my
rule, we have the followingequation:
+
-
vA
RC
RD
iB
RF
RE
iA
iB
iC
iE
iD
0A C D E Bi i i i i
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Kirchhoffs Current Law (KCL) Example Done Another Way
Some prefer to write thissame equation in a differentway; they say that thecurrent entering the closedsurface must equal the
current leaving the closedsurface. Thus, they write :
+
-
vA
RC
RD
iB
RF
RE
iA
iB
iC
iE
iD
0A C D E Bi i i i i
A D B C Ei i i i i
Compare this to theequation that we wrote inthe last slide:
These are the same
equation. Use either
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Kirchhoffs Voltage Law (KVL)
Now, we are prepared to state KirchhoffsVoltage Law:
The algebraic (or signed)summation of voltages arounda closed loop must equal zero.
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Kirchhoffs Voltage Law
(KVL) Some notes.
The algebraic (or signed)summation of voltages around aclosed loop must equal zero.
This applies to all closed loops. While we usually write equations forclosed loops that follow components, we do not need to. The onlything that we need to do is end up where we started.
This definition essentially means that energy is conserved. If wemove around, wherever we move, if we end up in the place westarted, we cannot have changed the potential at that point.
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Voltage Polarities
Again, the issue of thesign, or polarity, or direction, of
the voltage arises. When wewrite a Kirchhoff Voltage Lawequation, we attach a sign toeach reference voltage polarity,depending on whether the
reference voltage is a rise or adrop. This can be done indifferent ways.
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Kirchhoffs Voltage Law
(KVL) a Systematic Approach
The algebraic (or signed) summation of voltagesaround a closed loop must equal zero.
For this set of material, we will always go around loops clockwise. We willassign a positive sign to a term that refers to a reference voltage drop,and a negative sign to a term that refers to a reference voltage rise.
For most students, it is a good idea to choose one way to write KVLequations, and just do it that way every time. The idea is this: If youalways do it the same way, you are less likely to get confused aboutwhich way you were doing it in a certain equation.
(At least we will do this for planar circuits. For nonplanar circuits,clockwise does not mean anything. If this is confusing, ignore it for now.)
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+
-
vA
RC
RD
iB
RF
RE
vX
+
-
vF
+
-
vE
- +
Kirchhoffs Voltage Law
(KVL) an Example For this set of material, we will
always go around loops clockwise.We will assign a positive sign to aterm that refers to a voltage drop,
and a negative sign to a term thatrefers to a voltage rise.
In this example, we have alreadyassigned reference polarities forall of the voltages for the loopindicated in red.
For this circuit, and using our rule,starting at the bottom, we havethe following equation:
0A X E Fv v v v
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+
-
vA
RC
RD
iB
RF
RE
vX
+
-
vF
+
-
vE
- +
Kirchhoffs Voltage Law
(KVL) Notes For this set of material, we will
always go around loopsclockwise. We will assign apositive sign to a term that refers
to a voltage drop, and a negativesign to a term that refers to avoltage rise.
Some students like to use thefollowing handy mnemonicdevice: Use the sign of the
voltage that is on the side of thevoltage that you enter. Thisamounts to the same thing.
0A X E Fv v v v
As we go up through thevoltage source, we enter thenegative sign first. Thus, vAhas a negative sign in the
equation.
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+
-
vA
RC
RD
iB
RF
RE
vX
+
-
vF
+
-
vE
- +
Kirchhoffs Voltage Law
(KVL) Example Done Another Way
Some textbooks, and somestudents, prefer to write this same
equation in a different way; they
say that the voltage drops mustequal the voltage rises. Thus, they
write the following equation:
0A X E Fv v v v
X F A Ev v v v
Compare this to the equation thatwe wrote in the last slide:
These are the same equation.Use either method.
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How many of these equations
do I need to write?
This is a very important question. In general, it boils down to theold rule that you need the same number of equations as you haveunknowns.
Speaking more carefully, we would say that to have a singlesolution, we need to have the same number of independentequations as we have variables.
At this point, we are not going to introduce you to the way toknow how many equations you will need,or which ones to write. It is assumed thatyou will be able to judge whether you havewhat you need because the circuits will befairly simple. Later we will developmethods to answer this question specifically and efficiently.
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How many more laws
are we going to learn?
This is another very important question. Until, we get toinductors and capacitors, the answer is, none.
Speaking more carefully, we would say that most of the
rules that follow until we introduce the other basicelements, can be derived from these laws.
At this point, you have the tools to solve many, manycircuits problems. Specifically, you have Ohms Law, andKirchhoffs Laws. However, we need to be able to use these
laws efficiently and accurately.
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Example #1
Lets do an exampleto test out our new
found skills.
In the circuit shownhere, find the
voltage vXand thecurrent iX.
R4=
20[]
R3=
100[]
vS1
=
3[V]
+
-
vX
+
-
iX
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Example #1 Step 1
The first step insolving is to define
variables we need.
In the circuit shownhere, we will define
v4 and i3.
R4=
20[]
R3
=
100[]
vS1
=
3[V]
+
-
vX
+
-
iX
v4+ -
i3
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Example #1 Step 2
The second step insolving is to write some
equations. Lets startwith KVL.
1 4
4
0, or
3[V] 0.
S X
X
v v v
v v
R4=
20[]
R3
=
100[]
vS1
=
3[V]
+
-
vX
+
-
iX
v4+ -
i3
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Example #1 Step 3
Now lets write OhmsLaw for the resistors.
4 4
3 3
, and
.
X
X
v i R
v i R
Notice that there is a sign in OhmsLaw.
R4=
20[]
R3
=
100[]
vS1
=
3[V]
+
-
vX
+
-
iX
v4+ -
i3
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Example #1 Step 4
Next, lets write KCL forthe node marked in
violet.
3
3
0, or
.
X
X
i i
i i
Notice that we can write KCL for a node,or any other closed surface.
R4=
20[]
R3
=
100[]
vS1
=
3[V]
+
-
vX
+
-
iX
v4+ -
i3
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Example #1 Step 5
We are ready to solve.
We have substituted into our KVLequation from other equations.
3[V] 20[ ] 100[ ] 0, or
3[V]25[mA].
120[ ]
X X
X
i i
i
R4=
20[
]
R3=
100[]
vS1
=
3[V]
+
-
vX
+
-
iX
v4
+ -
i3
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Example #1 Step 6
Next, for the otherrequested solution.
We have substituted into Ohms Law,using our solution for iX.
3 3 3, or
25[mA] 100[ ] 2.5[V].
X X
X
v i R i R
v
R4=
20[
]
R3=
100[]
vS1
=
3[V]
+
-
vX
+
-
iX
v4
+ -
i3
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Example #2 Lets do
anotherexample. Findthe voltage vX,the currents iXand iQ, and thepowerabsorbed byeach of the
dependentsources.
Problem 2.28 is on page 61 of the text. The
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Example #3 Problem 2.28
Problem 2.28 is on page 61 of the text. Thedependent source coefficient has units of [A/V].
Problem 2.20 is on page 59 of the text.
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Example #4 Problem 2.20
p g
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Kirchhoffs Voltage Law:
Basic Laws of Circuits
Consideration 1: Sum of the voltage drops around a circuitequal zero. We first define a drop.
We assume a circuit of the following configuration. Notice that
no current has been assumed for this case, at this point.
+
+
+
+
_
_
_
_
v1
v2
v4
v3
Figure 18/11/2012
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Consideration 1.We define a voltage drop as positive if we enter the positive terminaland leave the negative terminal.
+ _v1
The drop moving from left to right above is + v1.
+_ v1
The drop moving from left to right above is v1.
Figure 2
Figure 3
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Basic Laws of Circuits
Kirchhoffs Voltage Law:
+
+
+
+
_
_
_
_
v1
v2
v4
v3Figure 4
Consider the circuit of Figure 4 onceagain. If we sum the voltage drops in the clockwise direction around thcircuit starting at point a we write:
- v1 v2 + v4 + v3 = 0
- v3 v4 + v2 + v1 = 0
a
drops in CW direction starting at a
drops in CCW direction starting at a
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Basic Laws of Circuits
Kirchhoffs Voltage Law:Consideration 2: Sum of the voltage rises around a circuit
equal zero. We first define a drop.
We define a voltage rise in the following diagrams:
+_ v1 Figure 5
+ _v1 Figure 6
The voltage rise in moving from left to right above is + v1.
The voltage rise in moving from left to right above is - v1.8/11/2012
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Consider the circuit of Figure 7 onceagain. If we sum the voltage rises in the clockwise direction around thecircuit starting at point a we write:
+
+
+
+
_ _
_
v1
v2
v4
v3Figure 7
a
+ v1 + v2 - v4 v3 = 0
+ v3 + v4 v2 v1 = 0
rises in the CW direction starting at a
rises in the CCW direction starting at a
_
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Basic Laws of Circuits
Kirchhoffs Voltage Law:Consideration 3: Sum of the voltage rises around a circuit
equal the sum of the voltage drops.
Again consider the circuit of Figure 1 in which we start atpoint a and move in the CW direction. As we cross elements1 & 2 we use voltage rise: as we cross elements 4 & 3 we usevoltage drops. This gives the equation,
+
+
+
+
_
_
_
_
v1
v2
v4
v3
v1 + v2 = v4 + v3
1
2
3
4
p
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Comments.
We note that a positive voltage drop = a negative voltage rise.
We note that a positive voltage rise = a negative voltage drop.
We do not need to dwell on the above tongue twisting statements.
There are similarities in the way we state Kirchhoffs voltageand Kirchhoffs current laws: algebraic sums
However, one would never say that the sum of the voltagesentering a junction point in a circuit equal to zero.
Likewise, one would never say that the sum of the currentsaround a closed path in an electric circuit equal zero.
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Further details.For the circuit of Figure 8 there are a number of closed paths.Three have been selected for discussion.
+
+
+
+ +
+
+
+
+
+
+
-
- -
-
-
-
--
-
-
-v1
v2
v4
v3
v12
v11 v9
v8
v6
v5
v7
v10
+
-
Figure 8Multi-pathCircuit.
Path 1
Path 2
Path 3
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Further details.
For any given circuit, there are a fixed number of closed pathsthat can be taken in writing Kirchhoffs voltage law and stillhave linearly independent equations. We discuss this more, later.
Both the starting point and the direction in which we go around aclosed path in a circuit to write Kirchhoffs voltage law arearbitrary. However,one must end the path at the same point fromwhich one started.
Conventionally, in most text, the sum of the voltage dropsequal to zero is normally used in applying Kirchhoffsvoltage law.
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Illustration from Figure 8.
+
+
+
+ +
+
+
+
+
+
+
-
- --
-
-
--
-
-
-v1
v2
v4
v3
v12
v11 v9
v8
v6
v5
v7
v10
+
-
a
Blue path, starting at a
- v7 + v10 v9 + v8 = 0
b
Red path, starting at b
+v2 v5 v6 v8 + v9 v11
v12 + v1 = 0
Yellow path, starting at b
+ v2 v5 v6 v7 + v10 v11- v12 + v1 = 0
Using sum of the drops = 0
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Double subscript notation.Voltages in circuits are often described using double subscript notation.
a b
Consider the following:
Figure 9: Illustrating double subscript notation.
Vab means the potential of point a with respect to point b withpoint a assumed to be at the highest (+) potential and point bat the lower (-) potential.
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Double subscript notation.Task: Write Kirchhoffs voltage law going in the clockwisedirection for the diagram in Figure 10.
b a
xy
Going in the clockwise direction, starting at b, using rises;
vab + vxa + vyx + vby = 0
Figure 10: Circuit for illustrating double subscript notation.
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Equivalences in voltage notations
The following are equivalent in denoting polarity.
+
-
v1 v1
a
b
+ -v2
= =
v2 = - 9 volts means the right hand side
of the element is actually positive.
vab = v1
Assumes the upper terminal is positive in all 3 cases
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Application.Given the circuit of Figure 11. Find Vad and
Vfc.
5 V
8 V
15 V
12 V
20 V 10 V
30 V
a b c
d
ef
+ _
+
+
_
_
++
+
+
_
_
_
_
Using drops = 0; Vad + 30 15 5 = 0 Vab = - 10 V
Vfc 12 + 30 15 = 0 Vfc = - 3 V
Figure 11: Circuit for illustrating KVL.
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Single-loop circuits.We are now in a position to combine Kirchhoffs voltage and current
Laws to the solution of single loop circuits. We start by developing the
Voltage Divider Rule. Consider the circuit of Figure 12.
R2
R1
v2
v1
+ +
+
_
_
_
v i1
v = v1 + v2
v1 = i1R1, v2 = i1R2
Figure 12: Circuit for developing
voltage divider rule.
then,
v = i1(R1 + R2) , and i1 =
v
(R1 + R2)so,
v1 =vR1
(R1 + R2)
* You will be surprised by how much you use this in circuits.
*
16
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Single-loop circuits.Find V1 in the circuit shown in Figure 13.
V
R3 R2
R1V1
I
+_
1 2 3
1 1
( )
, ,
VI
R R R
V IR so wehave
11
1 2 3( )
VRVR R R
Figure 3.13
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V
R3 R2
R1I
+_
Basic Laws of Circuits
Kirchhoffs Voltage Law: Single-loop circuits.Example 1: For the circuit of Figure 14, the following is known:
R1 = 4 ohms, R2 = 11 ohms, V = 50 volts, P1 = 16 watts
Find R3.
Figure 14: Circuit for example 1.
Solution:
P1 = 16 watts = I2R1
I = 2 amps
V = I(R1
+ R2
+ R3
), giving,
R1 + R2 + R3 = 25, then solve for R3,
R3 = 25 15 = 10 ohms
, thus,
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Single-loop circuits.Example 2: For the circuit in Figure 15 find I, V1, V2, V3, V4 and the
power supplied by the 10 volt source.
+ +
+
+
+
+
+_ _
_
_
_
_
_V1
V4
V3 V2
30 V 10 V
15 40
5
20
20 V
I
"a"
Figure 15: Circuit for example 2.
For convenience, we start at point a and sum voltage drops =0 in thedirection of the current I.
+10 V1 30 V3 + V4 20 + V2 = 0 Eq. 3.1
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Single-loop circuits. Ex.2 cont.We note that: V1 = - 20I, V2 = 40I, V3 = - 15I, V4 = 5I Eq. 3.2
We substitute the above into Eq. 3.1 to obtain Eq. 3.3 below.
10 + 20I 30 + 15I + 5I 20 + 40I = 0 Eq. 3.3
Solving this equation gives, I = 0.5 A.
Using this value of I in Eq. 3.2 gives;
V1 = - 10 V
V2 = 20 V
V3 = - 7.5 V
V4 = 2.5 V
P10(supplied) = -10I = - 5 W
(We use the minus sign in 10I because the current is entering the + terminal)
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Basic Laws of Circuits
Kirchhoffs Voltage Law:Single-loop circuits, Equivalent Resistance.
+ +
+
+
_
_
+_ _
_
+
+
_
_V1
V4
V3 V2
VS1 VS3
R2 R4
R3
R1
VS2
I
"a"
Given the circuit of Figure 3.16. We desire to develop an equivalent circuit
as shown in Figure 17. Find Vs and Req.
Figure 16: Initial circuit fordevelopment.
VSReq
+
_I Figure 17: Equivalent circuit
for Figure 16
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Basic Laws of Circuits
Kirchhoffs Voltage Law:Single-loop circuits, Equivalent Resistance.
+ +
+
+
_
_
+_ _
_
+
+
_
_V1
V4
V3 V2
VS1 VS3
R2 R4
R3
R1
VS2
I
"a"
Figure 16: Initial circuit.
Starting at point a, apply KVL going clockwise, using drops = 0, we have
VS1 + V1 VS3 + V2 + VS2 + V4 + V3 = 0
or
- VS1 - VS2 + VS3 = I(R1 + R2 + R3 + R4) Eq. 3.4
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Basic Laws of Circuits
Kirchhoffs Voltage Law:Single-loop circuits, Equivalent Resistance.
Consider again, the circuit of Figure 17.
VS
Req+
_ I
Figure 17: Equivalent circuit
of Figure 16.
Writing KVL for this circuit gives;
VS = IReq compared to - VS1 - VS2 + VS3 = I(R1 + R2 + R3 + R4)
Therefore;
VS = - VS1 - VS2 + VS3 ; Req = R1 + R2 + R3 + R4 Eq. 3.5
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Single-loop circuits, Equivalent Resistance.
We make the following important observations from Eq. 3.5:
The equivalent source of a single loop circuit can beobtained by summing the rises around the loop of
the individual sources.
The equivalent resistance of resistors in series is equalto the sum of the individual resistors.
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Basic Laws of Circuits
Kirchhoffs Voltage Law: Single-loop circuits.
Example 3: Find the current I in the circuit of Figure 18.
+ +
_
_ _
+
10 V 40 V
15 10
5
20
20 V
I
Figure 18: Circuit for
example 3.
From the previous discussion we have the following circuit.
50 V 50 +_ I
Therefore, I = 1 A