circuit analysis bw
TRANSCRIPT
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ME2143/ME2143E Sensors and Actuators
Review of Electrical Circuits Theory
Chew Chee Meng
Department of Mechanical EngineeringNational University of Singapore
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Outline
IntroductionBasic Electrical ElementsKirchhoff’s LawsMethod of SuperpositionEquivalent CircuitsPractical Considerations
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IntroductionAll mechatronic and measurement systems contain electrical circuits and components
Typical elements of electrical circuits
IntroductionBasic mechanical quantities
DisplacementVelocityForce
What about electrical domain?
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Introduction
Basic electrical quantitiesChargeCurrentVoltage
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Introduction
Charge Fundamental electric quantity Unit: coulombs (C)
Atomic structure of matter:Consists of a nucleus (neutrons and protons) surrounded by electrons
Elementary chargesA proton has a charge of 1.6 × 10-19 CAn electron has a charge of -1.6 × 10-19 C
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IntroductionElectrical current (charge in motion)
time rate of flow of electrical charge through a conductor or circuit elementunit: amperes, A (or C /s)
q(t) : quantity of charge flowing through a cross-section of the circuit element.
Current flow direction
Electrons
( )( ) dq ti tdt
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IntroductionDirect Current vs Alternating Current
direct current (dc): constant with time.alternating current, (ac): varies with time, reversing direction periodically (typically sinusoidal).
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IntroductionCurrent Measurements
How to measure current in a circuit?
Refer to: http://www.youtube.com/watch?v=y_o34SY77yo
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IntroductionVoltage (potential difference, electromotive force (emf))
difference in electrical potential between 2 pointsSI unit: volt, V (or J/C)
Let Va be the electrical potential at point A and Vb at point B, then the voltage across A and B, Vab (A wrt B) is
Vab=Va-VbAlso, Vba=-Vab
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IntroductionNotations
When vab is positive (negative), electric potential at a is higher(lower) than that at b
When v is positive(negative), electric potential at arrow end is higher (lower) than that at the non-arrow end
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IntroductionHow to measure voltage?
http://www.youtube.com/watch?v=t0Zzoz4nM0I&
4.889V
A
B
Vab
Voltmeter
-ve+ve
Vab
+
-
IntroductionDigital Multimeter (DMM) can be used to measure:
VoltageCurrentResistance
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IntroductionGround
Typical reference for electric potentialSymbol
“Voltage at point A, Va = 3.8V” means potential at point A is 3.8V with respect to ground potential
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IntroductionWhen current flows through an element and voltage appears across the element, energy is transferred.When positive charge or current entersthrough positive (negative) polarity into an element, energy is absorbed (supplied) by the element
Energy
supplied
by the
element
Energy
absorbed
by the
element+
+
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IntroductionPower and Energy
Power absorbed by an element:
Energy absorbed from time t1 to t2:
( ) ( ) ( )p t v t i t
2
1
( )t
t
w p t dt t1 < t2
*Remark: This formula is based on convention that current reference i enters the positive polarity of the voltage . Positive => energy is absorbed by the element.
*
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Basic Electrical Elements
An electrical circuit is an interconnection of electrical elements and energy sources.Energy sources
Voltage source (Vs), current source (Is)Ideal energy sources: Contain no internal
resistance, inductance, or capacitance.Three basic passive* electrical elements
Resistors (R), capacitors (C) , inductors (L)
*Passive elements: Require no additional power supply (compared with integrated circuits (ICs))
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Basic Electrical Elements
Ideal voltage source (Vs)
Ideal independent voltage source
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Basic Electrical Elements
Ideal voltage source (Vs)
Ideal dependent voltage source (rhomboidal shape symbol)
• Depends on a current or voltage that appears elsewhere in the circuit
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Basic Electrical Elements
Ideal current source (Is)
Ideal independent current source
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Basic Electrical Elements
Ideal current source (Is)
Ideal dependent current sourceDepends on a current or voltage that appears
elsewhere in the circuit
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ResistorA dissipative element: Converts electrical energy into heat
Symbol
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ResistorIdeal resistor
Voltage-current characteristics defined by Ohm’s law:
where R is a constant called resistance (SI Unit: Ohm, )
v Ri
v
i
R = v / i
Resistor
Method of reading resistor’s value of wire-lead resistors
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(%)10 toleranceabR c
10 k5%
ResistorVariable resistors
Potentiometer (pot)Trim pot
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Schematic symbols
Three terminals
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ResistorResistance related to physical parameters
ALR
is called the resistivitywhich is a property of the material
http://www.youtube.com/watch?v=wUgJgK2aTG0&feature=related
How to measure resistance?
E.g. Resistance of a homogeneous material of length L and with uniform cross-sectional area, A:
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CapacitorA passive element that stores energy in the form of an electric field
Dielectricmaterial
Conducting plates
Dielectric material (an insulator): increases capacitance
i ++++++++
--------
Current flow* results in opposite charge built up on the conducting plates
Symbol:
*Strictly, current does not flow through a capacitor
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CapacitorCapacitor and its fluid-flow analogy
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Capacitor
where q (unit: coulombs, C): amount of accumulated chargeappearing on each capacitor plate C (unit: farads, F (coulombs/volts)): capacitancev: voltage across the capacitori : current flowing into the positive polarity of the capacitor
Cvq dtdvCi
v(t)+ -
i(t)
C
or
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Capacitor
Voltage across a capacitor cannot change instantaneously (why?)
Can be used for timing purposes in electrical circuits (e.g. RC circuit)
Used in low-pass filter
With DC sources, capacitor behaves like an open circuit during steady state condition
Vin
Vin
Vc
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Capacitor
Capacitance: A property of dielectric materialplate geometry and separation
Typical values: 1pF (picofarads, 10-12) to 1000F (microfarads, 10-
6)
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Capacitor
Primary types of commercial capacitors:Electrolytic (polarized, have a positive and negative ends)TantalumCeramic diskMylar
Capacitance codes: Three-digit code, e.g. 102, implies 10x102
pF = 1 nFTwo-digit code, e.g. 22, implies 22 pF
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Inductor
A passive energy storage element that stores energy in the form of a magnetic field.
Symbol:
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Inductor
Inductor’s characteristics governed by Faraday’s lawof induction:
where λ = total magnetic flux (webers, Wb) through the coil windings due to the current
dtdtV
)(
Fig 2.10, p15 of Alciatore and Histand, 2003
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InductorFor ideal coil, ,
hence,
where L is the inductance (henry, H (=Wb/A)) of the coil.
or
=> Current cannot change instantaneously
dtdiLtv
0
0
1 tidttvL
tit
t
Li
Note: With DC sources, Inductor behaves like a short circuit during steady state condition
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Inductor
Source: http://www.wikipedia.org/
Fuel injectorElectric motors
Typical inductor values: 1H to 100mHPresent in motors, relays, power supplies, oscillators circuits, etc
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Branches, Nodes and LoopsBranch: Any portion of a circuit with two terminalsconnected to it
Node: Junction of two or more branches
Loop: Any closed connectionof branches
Node
Circuitelements
Loop 1 Loop 2
Loop 3
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Kirchhoff’s Circuit LawsAnalysis of circuits: calculate voltagesand currents anywhere in a circuit*Kirchhoff’s circuit laws: essential for analysis of circuits which involve various electrical elements ranging from basic elements to semiconductor components like transistors, op amps, etc
Kirchhoff’s current law (KCL)Kirchhoff’s voltage law (KVL)
*Named after Gustav Kirchhoff (1824-1887)
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Kirchhoff’s Current LawSum of currents flowing into a node is zero:
I1+I2-I3 = 0
N
iiI
10
Eg:
(I3 has negative signbecause it is flowing away from the node)
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Kirchhoff’s Current LawAlternatively, the sum of the currents entering a node equals to the sum of the currents leavingthe node
I1+I2 = I3
Eg:
(Sum of currents entering node)
(Sum of currents leaving node)
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Kirchhoff’s Voltage LawSum of voltages around a closed loop is zero:
N
iiV
10
Start from a node (e.g. A) and end at the same node
Either clockwise or anti-clockwise is fine
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Kirchhoff’s Voltage Law
Loop 1: -va + vb + vc = 0Loop 2: -vc – vd + ve = 0Loop 3: va – vb + vd – ve = 0
Eg:
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Analysis of circuitsProcedure:
• First, assign current variable to each branch and assume its flow direction.
• Then assign appropriate polarity to the voltage across each passive element (current entering into +ve polarity).
• Apply KVL for loops or apply KCL for nodes to generate sufficient equations together with constitutive equations of the elements (eg. Ohm’s law) to solve the unknown current and voltage variables
i1 i2 i3
-
vA
+
+ vB - - vD +
-
vE
+
Passive
element
+
vC
-
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Systematic Circuits Analysis Methods
For resistive circuits Node-voltage methodMesh-current method
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Series Resistance Circuit
Apply KVL to the closed loop starting from node A (clockwise):
-Vs+VR1+VR2 = 0Constitutive equations, Ohm’s law:
VR1 = IR1
VR2 = IR2
=> -Vs+IR1+IR2 = 0
Fig 2.13, p18, Alciatore and Histand, 2003
?
??
Circuit: R1 and R2 connected in series with a voltage source Vs
To find: I, VR1 and VR2 (need three equations to solve)
Hence, I = Vs/(R1+R2) , VR1= IR1 , VR2 = IR2
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Series Resistance CircuitSince Vs = I(R1+R2)=IReq where Req=R1+R2
Vs
I
Req
+
-
i.e. the two resistors can be replaced by a single resistor Reqof value R1+R2.
Fig 2.13, p18, Alciatore and Histand, 2003
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Series Resistance Circuit
In general, N resistors connected in series is equivalent to a resistor with resistance:
N
iieq RR
1
where Ri is the resistance of ithresistor connected in series
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Series Resistance CircuitVoltage divider
1
11
1 2R s
RV IR VR R
2
22
1 2R s
RV IR VR R
iR RViand,
In general, voltage across the resistor Ri of N series connected resistors branch is given by:
sN
jj
iR V
R
RVi
1
Fig 2.13, p18, Alciatore and Histand, 2003
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Series Resistance CircuitVoltage divider : Create different reference voltages by selecting appropriate resistors.Question:
Given a 12 V battery, is it appropriate to use the voltage divider to directly create a voltage source or supply of say, 5 V, for a device directly?
Vout
Vin=12VR1
R2
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Parallel Resistance Circuit
Constitutive equation, Ohm’s law:I1 = Vs/R1
I2 = Vs/R2
Applying KCL at node A:I - I1 - I2 = 0
=> 1 2
1 2
s sV VI I IR R
Fig 2.14, p20, Alciatore and Histand, 2003
?
? ?
Circuit: R1 and R2 connected in parallel with a voltage source Vs
To find: I, I1 and I2 (need three equations to solve)
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Parallel Resistance Circuit
eq
ss
ss
RV
RRV
RV
RVI
2121
11
i.e. the two resistors can be replaced by a single resistor Req of value =
1 2
1 2
1 2
11 1
R RR R
R R
Vs
I
Req
+
-
Fig 2.14, p20, Alciatore and Histand, 2003
Since where1 2
1 1 1
eqR R R
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Parallel Resistance Circuit
In general, N resistors connected in parallel is equivalent to a resistor of resistance, Req , given by:
1
1 1N
ieq iR R
where Ri is the resistance of ith resistor
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Parallel Resistance CircuitCurrent divider
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1 1 2
SV RI IR R R
12
2 1 2
SV RI IR R R
Fig 2.14, p20, Alciatore and Histand, 2003
1 2
1 2S eq
R RV IR IR R
That is, and1 2I R 2 1I R
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Series Capacitors/Inductors Circuit
By applying KVL, it can be shown that:
1
1 1N
ieq iC C
In general
1
N
eq ii
L L
In general
L1 L2
21 LLLeq
C1 C2
21
21
CCCCCeq
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Parallel Capacitors/Inductors Circuit
By applying KCL, it can be shown that:
1
N
eq ii
C C
In general
1
1 1N
ieq iL L
In general
C2
C1
21 CCCeq 21
21
LLLLLeq
L2
L1
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Principle of Superposition
Apply to linear circuits (for example, those which consist of multiple ideal sources and passive elements)For a linear system:
SystemInput, u1 Output, y1
Input, u2 Output, y2
Input
au1+ bu2
Output
ay1+by2
where a and b are some constantsSystem
System
Given:
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Principle of Superposition
If more than one independent voltage or current source is present in any given circuit, each branch voltage and current is the sum of the independent voltages or currentswhich would arise from each voltage or current source acting individually when all the other independent sources are zero*.
*To zero a source, current source replaced by open circuit and voltage source by short circuit.
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Example: Superposition
Ans: I=II1+II2+IV
=I1-I2
RI2 I1
V
I
RI2
II2
RV
IV
RI1
II1
To find I
(a)(b)
(c)
?
II1: Portion of Iarising from I1II2: Portion of Iarising from I2IV: Portion of Iarising from V
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Equivalent Circuits
Equivalent circuits
Portion of circuit to be replaced with an equivalent circuit
Equivalent circuit
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Equivalent Circuits
Equivalent circuit - one that has identical V-I relationship as viewed from a given pair of terminals
Equivalent circuit
V V
Portion of circuit to be replaced with an equivalent circuit
II
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Thévenin EquivalentThévenin’s theorem: Given a pair of terminals in a linear resistive network, the network may be replaced by an independent voltage source VOC in series with a resistance RTH.:
I
VVOC
RTHI
V
VOC - Thévenin voltage RTH - Thévenin resistance
Vin
R1
R2
Linear resistivenetwork
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Thévenin Equivalent (procedures)
Thévenin voltage - open circuit voltageacross the terminals.Thévenin resistance – equivalent resistance across the terminals when independent voltage sources are shorted and independent current sources are replaced by open circuit.(Applicable only if there is no dependent sources in the circuit)
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Example: Thévenin Equivalent
Vin
R1
R2
21
2
RRRVV inoc
Find VOC by voltage dividerformula,
A
B
Find the Thevenin equivalent circuit as seen from terminals A and B
Solution:
VOC
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Example: Thévenin Equivalent
R1
R21 2
1 21 2
||THR RR R R
R R
Find RTH across the terminals A & B after replacing the voltage source with a short circuit:
A
B
Solution (cont):
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Example: Thévenin Equivalent
Thévenin Equivalent:
VOCRTH
+Vin
R1
R2
A
B
A
B
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Norton EquivalentNorton equivalent: Linear resistive network can be replaced by an independent current source ISC and Thevenin resistance RTH in parallel with the source.
ISCRTH V
ILinear
resistive network
I
V
ISC - Norton current RTH - Thevenin resistance
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Norton Equivalent (procedures)ISC - current that would flow through the terminals if they were shorted together.To convert to Thevenin equivalent circuit, we can compute Thevenin voltage VOC as follows:
ISCRTH V
I
VOC
RTHI
V
Thevenin equivalentNorton equivalent
THSCOC RIV
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Example: Find Norton Equivalent circuit across A and B
First, find the short circuit current (ISC) across AB:
Vo Io
R1
R2
A
B
Applying KCL at node X:
1
0o Xo SC
V V I IR
Vo Io
R1
R2
A
BISC
oo
SC IRVI
1
X
(since VX = 0)
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Example - Norton Equivalent (cont.)Next, find the Thevenin resistance:
Replace voltage source with short circuit and current source with open circuit and inspect the equivalent resistance across the terminals. R1
R2
A
B
RTH= RAB= R1
Thus the Norton equivalent circuit would be:
R1
A
Bo
o IRV
1
Practical ConsiderationsBreadboard
For prototyping circuits
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Points are internally connected as shown
Instruments for powering and making measurements in circuits
Practical Considerations
Impedance (AC concept of resistance) matchingMaximum power transmission
In order to transmit maximum power to a load from a source, the load’s impedance should match the source’s impedance (see textbook for proof).
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For example, when you select speakers, the audio amplifier output impedance should be considered for maximum power transmission to a load (speaker).
Practical Considerations
GroundingVery important to provide a common ground defining a common voltage reference among all instruments and voltage sources used in a circuit or system.
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Not to confuse the signal ground with the chassis ground. The chassis ground is internally connected to the ground wire on the power cord and may not be connected to the signal ground (COM).
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Review of Electrical Circuits Theory
IntroductionBasic Electrical ElementsKirchhoff’s LawsPrinciple of Superposition Equivalent CircuitsPractical Considerations