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Bridging Theory in
PracticeTransferring Technical Knowledgeto Practical Applications
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Characteristics andModeling
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Intended Audience: Engineers with a basic knowledge of resistive circuits
Engineers desiring a more intuitive understanding of capacitiveand inductive circuits
Topics Covered:
Introduction to Load Modeling
Introduction to Capacitors and RC networks
Introduction to Inductors and RL networks
Example Load Models
Expected Time:
Approximately 120 minutes
RLC LoadCharacteristics and Modeling
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Introduction to Load Modeling
Introduction to Capacitors and RC Networks
Introduction to Inductors and RL Networks
Example Load Models:
Turning on an Incandescent Lamp
Switching a Relay
RLC LoadCharacteristics and Modeling
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RLC LoadCharacteristics and Modeling
Introduction to Load Modeling
Introduction to Capacitors and RC
Networks Introduction to Inductors and RL
Networks
Example Load Models:Turning on an Incandescent Lamp
Switching a Relay
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ElectromechanicalPower Conversion
Electrical power can be converted tomechanical power
Electrical power can turn-on a motor
Electrical power can drive a Solenoid
Electrical power can be converted to heat
Electrical power can a light a LED
(=
)
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Load Modeling
Power converters (the loads) can be modeled byequivalent circuits composed of simple RLC passivecomponents
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RLC LoadCharacteristics and Modeling
Introduction to Load Modeling
Introduction to Capacitors and RC
Networks Introduction to Inductors and RL
Networks
Example Load Models:Turning on an Incandescent Lamp
Switching a Relay
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Capacitors
Physical object with the ability to store electriccharge(i.e. electric voltage)
Consists of two electrically isolated metalelectrodes, typically two conductive parallelplates
Is mostly used to store energy or for filtering
purposes The isolating material the dielectric defines
the type of capacitor: e.g. tantalum or ceramiccapacitor
Circuit s mbol:
C
C i
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The capacitance of a parallel plate capacitor isproportional to:
C = Capacitance;a = Area of each parallel plate;d = Distance between parallel plates;
Larger value capacitors have larger plate areas and lessspacing between plates
They can store more energy (and are more expensive)
Capacitors:Physical Properties
C ~a
d
d
a
C i
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The capacitance of a parallel plate capacitor is given by:
C = Capacitance
Units of: F = A s / V
= Permittivity =0
r
Units of: A s / V m = F / m
0 = Permittivity of vacuum = 8.854x10-12
Units of: A s / V m = F / m
r = Relative permittivity = 1 (free air)
Units of: (dimensionless)
Permittivity1) : the ability of a dielectric to store electrical potentialenergy under the influence of an
electric field1) Websters 9th edition
C = a
d
Capacitors:Physical Properties
d
a
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Relative Size ofCapacitance Capacitance of a free air (r = 1) parallel plate
capacitor with the dimensions of A=1m2 andd=1mm is:
Typically, capacitance values in the 1F range areuncommon
Capacitances typically range from microFarads topicoFarads
1 microFarad = 1F = 10-6F
1 nanoFarad = 1nF = 10-9F
1 icoFarad = 1 F = 10-12F
( ) ( )
= = =
11 1
1r 1
1
. x F/m ( m )1 111111 1AC . x F111111
d x m111
C it El t i l
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Capacitors ElectricalProperties
The stored electrical charge Q in a capacitor isproportional to the voltage V across the capacitor:Q ~ V
The proportional factor between stored electrical
charge and voltage difference is the capacitancevalue of the capacitor:Q = C V
Q = 8 As = 8 CoulombsV = 16V
C = Q/V = 8 A s / 16V = 0.5 Farad (F)
Unit [C] = A s / V = F
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ara e an er aCapacitance
Parallel capacitors Serial capacitors
C1
C2
C
C = C1 + C2 1C
1+
1C
1=
C
1
C1 C2
C
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Capacitor Experiment #1
An ideal current source is connected to acapacitor
V, I
t
IC
VC
tON
IIDEAL
CIIDEAL
tON IC
VC+
- The constant current
causes the voltage
to linearly rise across
the capacitor.
Constant current source supplies
the current regardless of the
voltage drop across the load.
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Capacitor Experiment #2
An ideal current source is disconnected from acapacitor
V, I
t
tON
IIDEALVC
IC
tOFF
CIIDEAL
tOFF IC
VC+
-
If the constant current
source is removed,
the voltage across the
capacitor remains
constant.
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An ideal current source is connected to acapacitor
Capacitor Experiment #3
The rate ofvoltage
change is proportional
to the current.
CIIDEAL
tON IC
VC+
-
IC1
VC1
tON
V, I
t
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A variable ideal current source is connected toa capacitor
Capacitor Experiment #3
The rate ofvoltage
change is proportional
to the current.
CIIDEAL
tON IC
VC+
-
IC1
VC1
tON
V, I
t
IC2 VC2
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A voltage source is connected to a capacitorthrough a resistor
Capacitor Experiment #4
CVIDEAL
tON RIC
VC+
-+-
The peak current
in the capacitor is
limited by theresistor.
The voltage across
the capacitor willreach VIDEAL
Ideal voltage source
supplies the voltage
regardless of the current
load.tON
V, I
t
VIDEAL/R ICVC
VIDEAL
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Capacitor Experiment #5
A voltage source is connected through avariable resistor
CVIDEAL
tON R IC
VC
+
-VC
+
-
+-
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Capacitor Experiment #5
A voltage source is connected through avariable resistor
tON t
R = R1
CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-
IC1
VC1
V, I
VIDEAL
R1
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Capacitor Experiment #5
A voltage source is connected through avariable resistor
tON
V, I
t
IC1
VC1
R1
CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-
VIDEAL
R1 > R2
VIDEAL
R1
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Capacitor Experiment #5
A voltage source is connected through avariable resistor
tON
V, I
t
IC1
VC1
R1
CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-
IC2
VIDEAL
R1 > R2
VIDEAL
R1
VIDEAL
R2
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Capacitor Experiment #5
A voltage source is connected through avariable resistor
tON
V, I
t
IC1
VC1
CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-
VC2
IC2
VIDEAL
R1 > R2
VIDEAL
R1
VIDEAL
R2
Capacitors are
charged faster
through smaller
resistors
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Capacitor Experiment #5
A voltage source is connected through avariable resistor
tON t
R1 < R3
CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-
IC1V
C1
V, I
VIDEAL
R1
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Capacitor Experiment #5
A voltage source is connected through avariable resistor
tON t
R1 < R3
CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-
IC1VC1
V, I
VIDEAL
R1
IC3
VIDEAL
R3
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Capacitor Experiment #5
A voltage source is connected through avariable resistor
tON t
R1 < R3
CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-
IC1VC1
V, I
VIDEAL
R1
IC3
VIDEAL
R3
VC3
Capacitors are
charged faster
through smaller
resistors
Capacitor Experiment
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Capacitor Experiment#6
0 t
VC
VIDEAL
tC
R1 < R3
CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-
tC= RC
The rise time of the capacitor's voltage ismonitored:
0.63VIDEAL
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The rise time of the capacitor's voltage ismonitored:
Capacitor Experiment #6
0 t
VC
0.63VIDEAL
3tC
R1 < R3
CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-
tC= RC
tC
0.95VIDEAL0.87VIDEAL
2tC
De elopment of Mathematical
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Current is defined as the amount of chargewhich is transferred in a certain period of time:I = Q / t
Development of MathematicalCapacitor Model: IC vs. VC
dqi or dq i dt
dt= = (1)
The relations above are derivatives for very small changesdifferentials can be used for quasi linear changes:
i=q/t or q=i.t (1a)
Development of Mathematical
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Current is defined as the amount of chargewhich is transferred in a certain period of time:I = Q / t
Capacitance is defined as the stored charge ona capacitor vs. the voltage across the capacitor,
C = Q / V
Development of MathematicalCapacitor Model: IC vs. VC
dqi or dq i dt
dt= =
dqC or dq C dv
dv= =
(1)
(2)
In differential form:
C=q/t or q=C.v (2a)
Development of Mathematical
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Current is defined as the amount of chargewhich is transferred in a certain period of time:I = Q / t
Capacitance is defined as the stored charge ona capacitor vs. the voltage across the capacitor,
C = Q / V
Setting (2) equal to (1) results in:
Development of MathematicalCapacitor Model: IC vs. VC
dqi or dq i dt
dt= =
dqC or dq C dv
dv= =
(1)
(2)
dvi dt C dv or i C
dt
= =
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rs
CVINVC across
plates
time
Voltage
across
Capacitor
Current
through
CapacitorVC
IC
R
VIN
Capacitor & Resistor
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Capacitor & ResistorNetworks
In general, there are two basic options forcapacitor placement:
C in Series with Signal Path C from Signal Path to Ground
VINC
R VOUTC
RVIN VOUT
Capacitor & Resistor
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Capacitor & ResistorNetworks
Initially a DC voltage is applied at thesignal input IN.
Current passes through the capacitorand the voltage across the capacitorincreases
C in Series with Signal Path C from Signal Path to Ground
VIN
C
R
VOUT+ VC -
I
VINC
RVOUT
+VC
-
I
a ac o e o
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Initially a DC voltage is applied at the signalinput IN.
Current passes through the capacitor and thevoltage across the capacitor increases
When the voltage across the capacitor is equalto the input voltage the current stops
C in Series with Signal Path C from Signal Path to Ground
VIN
C
R
VOUT+ VIN -
I=0AVIN
C
RVOUT
+VIN
-
I=0A
a ac o e oNetworks
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Initially a DC voltage is applied at the signalinput IN.
Current passes through the capacitor and the
voltage across the capacitor increases
When the voltage across the capacitor is equalto the input voltage the current stops
Depending on the capacitors placement, the
C in Series with Signal Path C from Signal Path to Ground
VIN
C
R
0V
+ VIN -
I=0A VINC
RVIN
+VIN-
I=0A
Networks
Capacitance in Series with
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Capacitance in Series withSignal Path
VX
I
VOUT
t1 t2
VIN
C
R
VOUT
+ VC -
I
t1
t2
VX
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Signal Path
VX
I
VOUT
t1 t2
VIN
VIN/R
VIN
VIN
C
R
VOUT
+ VC -
I
t1
t2
VX
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Signal Path
VX
I
VOUT
t1 t2
VIN
VIN/R
VIN
VIN
C
R
VOUT
+ VC -
I
t1
t2
VX
-VIN/R
-VIN
Path
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Pathto Ground
VIN C
RVOUT
+VC
-
I
t1
t2
VX VX
I
VOUT
t1 t2
Path
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Pathto Ground
VIN C
RVOUT
+VC
-
I
t1
t2
VX VX
I
VOUT
t1 t2
VIN
VIN/R
VIN
-VIN/R
Path
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Pathto Ground
VIN C
RVOUT
+VC
-
I
t1
t2
VX VX
I
VOUT
t1 t2
VIN
VIN/R
VIN
-VIN/R
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RC Networks - AC Signals
What happens when an AC input signal isapplied?C in Series with Signal Path C from Signal Path to Ground
VINC
R
VOUT
C
R
VOUT
t VIN t? ?
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Capacitors and AC signals
Capacitors act like frequency dependentresistor (capacitive reactance, XC)
Xc~1/(fC)
Instead of reactance, impedance (Z) is
used to characterize circuit elements:
Z=1/(2fC)
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Capacitors and AC signals
Act like frequency dependent resistor(capacitive reactance, XC)
Instead of reactance, impedance (Z) usedfor circuit elements.
Impedance1): The apparent opposition inan electrical circuit to the flow ofalternating current that is analogous to
the actual electrical resistance to a direct
C1
XCf
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Capacitors and AC signals
Act like frequency dependent resistor(capacitive reactance, XC)
Instead of reactance, impedance (Z) used forcircuit elements.
Impedance1): The apparent opposition in anelectrical circuit to the flow of alternatingcurrent that is analogous to the actual electricalresistance to a direct current.
The impedance of a circuit element represents
C1
XCf
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Capacitors and AC signals
Act like frequency dependent resistor (capacitivereactance, XC)
Instead of reactance, impedance (Z) used for circuitelements.
Impedance1): The apparent opposition in an electricalcircuit to the flow of alternating current that is analogous
to the actual electrical resistance to a direct current. The impedance of a circuit element represents its
resistive and/or reactive components
Besides the magnitude dependency between voltage
and current the impedance, Z, gives also information
C1
XCf
Capacitors Impedance
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Capacitor s ImpedanceMagnitude |ZC| vs. Frequency
C1
ZCf
|Z|= /(11fC) C= uF1
.11
.11
.11
.11
.11
.111
.111
.111
.111
.111
1 111 111 111 111 111
FREQUENCY (Hz)
|Z|(kohm)
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Capacitors and AC signals
VC,Max
iC,Max
t
iC,Max = VC,Max / |ZC|
= + /2 = + 90o The current leads the
volta e
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RC networks AC Signals
The capacitor acts as a frequency dependentresistor
It determines the current magnitude at a givenvoltage
It causes a 90 degree phase shift between thecapacitor current and voltage across theca acitor
C in Series with Signal Path C from Signal Path to Ground
VINC
R
VOUT
C
R
VOUT
t VIN t
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RC networks AC Signals
For high frequency signals:
The capacitor is low impedance
Signals can pass the capacitor For low frequency signals:
The capacitor is high impedance
Signals are blocked by the capacitor
C in Series with Signal Path C from Signal Path to Ground
VINC
R
VOUT
C
R
VOUT
t VIN t
|ZC|=1/(2fC)
C in Series with Signal Path
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C in Series with Signal PathHigh Pass Configuration
-1
-1
-1
1
1
1
1
1 .000 .111 .000 .111 .000
-1
-1
-1
1
1
1
1
1 .111 .111 .111 .111 .111
VIN
VOUT
VOUT/VINMAX
Low f 0.32
Medium f 0.76
High f 0.90
CRVIN
VOUT
|ZC|=1/(2fC)
C from Signal Path to Ground
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C from Signal Path to GroundLow Pass Configuration
VIN
VOUT
VOUT/VINMAX
Low f 0.96
Medium f 0.74
High f 0.39
C
RVIN
VOUT
-1
-1
-1
1
1
1
1
1 .000 .111 .000 .111 .000
-1
-1
-1
1
1
1
1
1 .111 .111 .111 .111 .111
|ZC|=1/(2fC)
Capacitor & Resistor
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Capacitor & ResistorNetworks Summary
Connected to DC voltages:
Capacitors will allow current to flow only until they arecharged
Once charged, they block future current flowFor AC signals:
Capacitors act similar to frequency dependent resistors
Low impedance at high frequencies
High impedance at low frequencies.
C in Series with Signal Path C from Signal Path to Ground
VINC
R
VOUT
C
R
VOUT
VIN
Characteristics and
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Characteristics andModeling
Introduction to Load Modeling
Introduction to Capacitors and RC
Networks Introduction to Inductors and RL
Networks
Example Load Models:Turning on an Incandescent Lamp
Switching a Relay
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Inductors
Physical object which can store a magnetic field (electriccurrent)
Consists of a conductive wire
Wire is typically a tightly wound coil around a center core
(toroid) Usually used for energy conversion and for filtering
purposes
The inductor type is usually defined by its core material
for example, air coil or ferrite coil inductors) Circuit symbol
Lor
Physical
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PhysicalProperties of Inductors
The inductance of a toroid, for instance, is given by:
L = Inductance;N = Number of turns of the coil;a = Coil cross section;
= Average field length;
0 = permeability of vacuum =410-7 V.s/(A.M)
r =relative permeability
Larger value inductors have more turns and bigger cross sectionin less volume. They can store more energy (and may be moreexpensive).
A
l
Wire
Core
L=0.rN2.a/l
l
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Inductance of a ToroidL = Inductance
Units of: H = V
s/AN = Number of turns of the coila = Coil cross section
Units of: m2
= Average field lengthUnits of: m
= Permeability = 0r ;Units of: H/m = V s/A m
0 = Permeability of free space = 410-7
Units of: H/m = V s/A mr = Relative permeability
Permeabilty1) : the property of a ferro-magnetic substance thatdetermines the degree in which it modifies the magnetic flux in theregion occupied by it in a magnetic field
1) acc. to Websters 9th edition
a
l
l
L=N2a/l
d
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Inductance
Inductance of a free air toroid (r = 1) with the crosssection of a=5cm2, average field length of =10cm, andN=100 turns is
Inductors in the H range are used in switchingregulators
Small relays, solenoids usually have mH values ofinductance
Inductors in general typically range from a few Henries(H) to micro Henries (H):
1 microHenry = 1H = 10-6H
l
( ) ( )11 1 1
1
1
( )( H / m) x m1 111 1111 11L ~ . x H11111
x m1111
=
Inductors -
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The change of magnetic field or coil flux ( ) in aninductor is proportional to the change of electriccurrent (I) flowing through the inductors windings:
~ I
The proportional factor between coil flux andcurrent is given by the inductance of the coil: =L I
I
N
Inductors Electrical Properties
Inductors -
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Inductors Electrical Properties
The change of magnetic field or coil flux ( ) inan inductor is proportional to the change ofelectric current (I) flowing through theinductors windings: ~ I
The proportional factor between coil flux andcurrent is given by the inductance of the coil: = L I
L = /I = 1 Vs / 2 A = 0.5 Henry (H)
Unit [L] = Vs/A = H
I = 2A
N
1Vs
I d t
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Inductance
Serial inductors Parallel inductors
L = L1 + L2 1L
1+
1L
1=
L
1
L1
L2
L
L1 L2
L
I d t E i t #1
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An ideal voltage source is connected toan inductor
V, I
t
Inductor Experiment #1
VIDEAL
IL
VL
tON
tON
L
IL
VIDEAL
VL
+
-
The constant voltage
causes the current
to increase through
the inductor.
+
-
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I d t E i t #3
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An ideal voltage source is connected toan inductor
Inductor Experiment #3
V, I
t
VIDEAL
IL1
VL1
tON
tON
L
IL
VL
+
-
+
- The rate ofcurrentchange is proportional
to the voltage.
+
-
I d t E i t #3
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An ideal voltage source is connected toan inductor
Inductor Experiment #3
V, I
t
VIDEAL
IL1
VL1
tON
tON
L
IL
VL
+
-
+
- The rate ofcurrentchange is proportional
to the voltage.
VL2
IL2
+
-
I d t E i t #4
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A voltage source is connected to an inductorthrough a resistor
Inductor Experiment #4
LVIDEAL
tON RIL
VL
+
-
+
-
tONt
VIDEALV
L
VIDEAL/RIL
The peak voltage
across the inductor
is VIDEAL.
The current through
the inductor will
reach VIDEAL/R.
I d t E i t #5
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Inductor Experiment #5
A voltage source is connected through avariable resistor
CVIDEAL
tON RIC
VC
+
-VC
+
-
+-
IL
VLL
I d t E i t #5
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CVIDEAL
tON R
IC
VC
+
-VC
+
-
+-
IL
VL L
Inductor Experiment #5
A voltage source is connected through avariable resistor
tON t
R1 > R2
IL1
V, I
VIDEAL
VIDEAL/R1
VL1
I d t E i t #5
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CVIDEAL
tON R
IC
VC
+
-VC
+
-
+- L
Inductor Experiment #5
A voltage source is connected through avariable resistor
tON t
R1 > R2
VL2
IL1
V, I
VIDEAL
VIDEAL/R1
IL2VIDEAL/R2
VL1
The smaller the
resistor, the longer
it takes the current
to become steady
IL
VL L
Ind ctor E periment #5
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CVIDEAL
tON R
IC
VC
+
-V
C
+
-
+
-
ILL
Inductor Experiment #5
A voltage source is connected through avariable resistor
tON t
R1 < R3
VL3
IL1
V, I
VIDEAL
VIDEAL/R1
IL3VIDEAL/R3
VL1
The smaller the
resistor, the longer
it takes the current
to become steady
IL
VL L
Ind ctor E periment #6
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Inductor Experiment #6
0 t
VL
VIDEAL
tC
The rise time of the capacitor's voltage ismonitored:
tC
= L/R
CVIDEAL
tON RIC
VC
+
-VC
+
-
+
-
ILL
IL
VL L
0.37VIDEAL
Inductor Experiment #6
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Inductor Experiment #6
0 t
VL
VIDEAL
tC
The rise time of the capacitor's voltage ismonitored:
0.37VIDEAL
0.05VIDEAL
3tC
tC= L/R
CVIDEAL
tON RIC
VC
+
-VC
+
-
+
-
ILL
IL
VL L
2tC
0.14VIDEAL
Development of Mathematical
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The self induced coil voltage when exposed toan alternating magnetic field is proportional tothe change of coil flux vs. time:
Inductor Model: IL vs. VL
ind
d d
v N dt dt
= =
Development of Mathematical
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The self induced coil voltage when exposed to analternating magnetic field is proportional to the changeof coil flux vs. time:
The voltage v applied across an inductor is alwaysdirectly opposed to the self induced voltage vind:
v = -vind = Nd /dt = d/dt (=> d = vdt)
Inductor Model: IL vs. VL
ind
d dv N
dt dt
= =
ind
d dv v N or d v dt
dt dt
= = = =
(1)
Development of Mathematicald d l
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The self induced coil voltage when exposed to analternating magnetic field is proportional to the changeof coil flux vs. time:
The voltage v applied across an inductor is alwaysdirectly opposed to the self induced voltage vind:
v = -vind = Nd /dt = d/dt (=> d = vdt)
The inductance is defined as coil flux vs. coil current,L= / IL, differentially expressed as:
Inductor Model: IL vs. VL
ind
d dv N
dt dt
= =
ind
d dv v N or d v dt
dt dt
= = = = (1)
dL or d Ldi
di
= =
(2)
Development of Mathematicald d l
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Inductor Model: IL vs. VL
Setting (1) equal to (2), the voltage -current relation for an inductor equalscan be found:
indd dv v N or d v dtdt dt
= = = = (1)
dL or d Ldi
di
= = (2)
div L
dt=
Inductors
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Inductors
time
Voltage
across
Inductor
Current
through
Inductor
VL
IL
IL,max=VIN/R
VIN RVL
VIN
Inductor & Resistor
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Networks
L in Series with Signal Path L from Signal Path to Ground
VINL
R VOUTL
RVIN VOUT
In general, there are two basic options forinductor placement:
ResistorNetworks
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ResistorNetworks
L in Series with Signal Path L from Signal Path to Ground
VIN
L
R
VOUT
L
RVIN
VOUT
Initially a DC voltage is applied at thesignal input IN.
A voltage drops across the inductor andthe current through the inductorincreases
+ VL -
I
+VL-
I
ResistorNetworks
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ResistorNetworks
L in Series with Signal Path L from Signal Path to Ground
VIN
L
R
VOUT
L
RVIN
VOUT
Initially a DC voltage is applied at the signalinput IN.
A voltage occurs across the inductor and thecurrent through the inductor increases
When the current through the inductor is at itsmaximum and remains constant, the voltage
across the inductor e uals zero
+ 0V -
I
+0V-
I
ResistorNetworks
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ResistorNetworks
L in Series with Signal Path L from Signal Path to Ground
VIN
L
R
VIN
L
RVIN
0V
Initially a DC voltage is applied at the signal input IN.
A voltage drops across the inductor and the currentthrough the inductor increases
When the current through the inductor is at its maximumand remains constant, the voltage across the inductorequals zero
Depending on the inductors placement the steady state
final voltages are VOUT = VINor VOUT = 0V
+ 0V -
I
+0V-
I
Inductance in Series with
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Signal PathV
X
I
VOUT
t1 t2
VIN
L
R
VOUT
+ VL -
I
t1
t2
VX
Inductance in Series withi l h
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Signal PathV
X
I
VOUT
t1 t2
VIN
VIN/R
VIN
VIN
L
R
VOUT
+ VL -
I
t1
t2
VX
Inductance in Series withi l h
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Signal PathV
X
I
VOUT
t1 t2
VIN
VIN/R
VIN
VIN
L
R
VOUT
+ VL -
I
t1
t2
VX
Inductance From Signalh G d
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Path to Ground
VINL
R VOUT
+VL-
I
t1
t2
VXV
X
I
VOUT
t1
t2
Capacitance FromSi l P h G d
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Signal Path to Ground
VIN
R VOUT
I
t1
t2
VX
VX
I
VOUT
t1 t2
VIN
VIN/R
VIN
L
+VL-
Capacitance FromSi l P th t G d
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Signal Path to Ground
VIN
R VOUT
I
t1
t2
VX
VX
I
VOUT
t1 t2
VIN
VIN/R
VIN
L
+VL-
-VIN
RL Networks - AC Signals
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RL Networks - AC Signals
What happens when an AC input signal isapplied?
L in Series with Signal Path L from Signal Path to Ground
VIN L R
VOUT
L
R
VOUT
t VIN t? ?
Inductors and AC signals
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Inductors and AC signals
Act like frequency dependent resistor(inductive reactance, XL)
Instead of reactance, impedance (Z) usedfor circuit elements.
XL=2fL
Inductors and AC signals
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Inductors and AC signals
Act like frequency dependent resistor(inductive reactance, XL)
Instead of reactance, impedance (Z) usedfor circuit elements.
Impedance: The apparent opposition in
an electrical circuit to the flow ofalternating current that is analogous tothe actual electrical resistance to a direct
current.
XL=2fL
Inductors and AC signals
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Inductors and AC signals Act like frequency dependent resistor (inductive
reactance, XL)
Instead of reactance, impedance (Z) used for
circuit elements. Impedance: The apparent opposition in an
electrical circuit to the flow of alternating
current that is analogous to the actual electricalresistance to a direct current.
The impedance of a circuit element representsits resistive and/or reactive components
XL=2fL
Inductors and AC signals
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Inductors and AC signals
Act like frequency dependent resistor (inductivereactance, XL)
Instead of reactance, impedance (Z) used for circuit
elements. Impedance: The apparent opposition in an electrical
circuit to the flow of alternating current that is analogousto the actual electrical resistance to a direct current.
The impedance of a circuit element represents itsresistive and/or reactive components
Besides the magnitude dependency between voltageand current the impedance Z gives also information
about the phase shift between the two.
XL=2fL
Inductors ImpedanceMagnitude |Z | vs Frequency
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Magnitude |ZL| vs. Frequency
|ZL|(ohm)
frequency (Hz)
1
1
11
11
11
11
11
11
1 1111 1111 1111 1111 1111
|ZL|=2..f.L
Inductors and AC signals
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Inductors and AC signals
VL,Max
iL,Max
t
iL,Max = VL,Max / |ZL|
= - /2 = -90o The current lags the
volta e
RL networks AC signals
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RL networks AC signals
The inductor acts as a frequencydependent resistor
It determines the current magnitude at agiven voltage
It causes a 90 degree phase shift
between the inductor current and volta e
L in Series with Signal Path L from Signal Path to Ground
VINL
R
VOUT
L
R
VOUT
t VIN t
RC networks AC signals
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RC networks AC signals
For low frequency signals:
The inductor is low impedance
Signals can pass the inductor For high frequency signals:
The inductor is high impedance
Signals are blocked by the inductor
L in Series with Signal Path L from Signal Path to Ground
VINL
R
VOUT
L
R
VOUT
t VIN t
|ZL
|=2fL
L in Series with Signal PathL P C fi i
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-1
-1
-1
1
1
1
1
1 .000 .111 .000 .111 .000
-1
-1
-1
1
1
1
1
1 .000 .111 .000 .111 .000
Low Pass ConfigurationV
IN
VOUT
VOUT/VINMAX
Low f 0.96
Medium f 0.76
High f 0.38
LRVIN
VOUT
Z=2..f.L
L from Signal Path to GroundHi h P C fi ti
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High Pass ConfigurationV
IN
VOUT
VOUT/VINMAX
Low f 0.32
Medium f 0.74
High f 0.92
L
RVIN
VOUT
-1
-1
-1
1
1
1
1
1 .000 .111 .000 .111 .000
-1
-1
-1
1
1
1
1
1 .111 .111 .111 .111 .111
|ZL|=2..f.L
Inductor & ResistorNetworks Summary
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Networks Summary
Connected to DC voltages:
The voltage across an inductor changes as currentincreases
The voltage across inductor is 0V when current is constant
For AC signals:
Inductors act similar to frequency dependent resistors
Low impedance at low frequencies
High impedance at high frequencies.
L in Series with Signal Path L from Signal Path to Ground
VINL
R VOUTL
RVIN VOUT
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Characteristics andModeling
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Modeling
Introduction to Load Modeling
Introduction to Capacitors and RC
Networks Introduction to Inductors and RL
Networks
Example Load Models:Turning on an Incandescent Lamp
Switching a Relay
Lamp Experiment
-
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Lamp Experiment
Turn on an incandescent light bulb andmeasure the current
1
2
14V
Iton
Lamp Experiment
-
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p p
Turn on an incandescent light bulb andmeasure the current
Result:
~ 600mA
~ 5.6A
ton
1
2
14V
Iton
Model For an Incandescent LightBulb
-
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Bulb
600mA
5.6A
ton
14V Light
Bulb
Model For an Incandescent LightBulb
-
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Bulb
600mA
5.6A
ton
14V
R1
V V11R = =1
I . A11
R = .0 000
Model For an Incandescent LightBulb
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Bulb
600mA
5.6A
ton
14V
23.3
[ ]1I =I exp -t/RC
Model For an Incandescent LightBulb
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Bulb
600mA
5.6A
ton
14V
23.3
R2
V. R =111 1I
R = .0 000
Model For an Incandescent LightBulb
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Bulb
600mA
5.6A
ton
14V
23.3
2.8
C
C= . mF11
Simulation of Lamp RC
-
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Model
Time (ms)
InputC
urrent(A)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
500 150100 250200 350300ton
1
1
.000
.000
. mF00
V00
ton
Model
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Time (ms)
InputC
urrent(A)
0.0
1.0
2.0
3.0
4.0
5.0
6.0
500 150100 250200 350300ton
1
1
.000
.000
. mF00
V00
ton
A RC Load Model forIncandescent Light Bulbs
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Incandescent Light Bulbs
The model for this lamps is represented by the networkbelow
When a lamp initially turns on, the filament is cold andhas a relatively low resistance BUT as the filamentwarms up, the resistance increases dramatically1
2
23.3
2.80
3.6mF
f(T)
Lamp Experiment
-
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When a lamp initially turns on, the filament is cold andhas a relatively low resistance
As the filament warms up, the resistance increasesdramatically
~ 600mA
~ 5.6A
Characteristics andModeling
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Modeling
Introduction to Load Modeling
Introduction to Capacitors and RC
Networks Introduction to Inductors and RL
Networks
Example Load Models:Turning on an Incandescent Lamp
Switching a Relay
Switching a Relay
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Switching a Relay
To the right a highside switchingapplication is shown
The switch itself ismodeled as a simplemechanical switch
The relay can bemodeled as a lowohmic resistor andinductor connected in
VBattery
VR
VL
IL
S
Relay
Switching On a Relay
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g y
timeVR
time
S
timeVL
timeIL
VBattery
+
VR
-
+
VL-
IL
S
open closed
VL decays over time
IL = (VR-VL) / R
Switching Off a Relay (1)
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time
S
timeIL
time
time
g y
VBattery
IL
S
closedopen
+
VR
-
+
VL-
Switching Off a Relay (2)
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time
S
timeILIL cannot become
zero instantaneously!
timeVR
For VL < 0V,
VR < 0V
timeVL
VL becomes negative
to force the current to 0A
(VL
= -L*di/dt)
g y
VBattery
IL
S
closedopen
+
VR
-
+
VL-
Switching Off a Relay (3)
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time
S
timeIL
timeVR
timeVL
g y
VBattery
IL
S
closedopen
Arcing
+
VR
-
+
VL-
IL cannot go to
zero instantaneously!
For VL < 0V,
VR < 0V (R~0)
VL goes far below ground
to force the current to 0A
Switching Off aRelay No Arcing (1)
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time
S
time
IL
timeVR
timeVL
Relay No Arcing (1)V
Battery
IL
S
closed open
ID +
VL-
+
VRtime
ID
Switching Off aRelay No Arcing (2)
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time
S
time
IL
timeVR
timeVL
Relay No Arcing (2)V
Battery
IL
S
closed open
ID
Diode turns on andprovides a current path
+
VL-
+
VRtime
ID
Switching Off a Relay NoArcing (3)
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time
S
time
IL
timeVR
timeVL
Arcing (3)VBattery
IL
S
closed open
ID +
VL-
+
VRtime
ID
If R~0, VL ~ VD
If R~0, VR ~ -VD
Switching Off aRelay No Arcing (4)
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time
S
time
IL
timeVR
timeVL
Relay No Arcing (4)closed open
timeID
VBattery
IL
S
ID +
VL-
+
VR
Switching Off aRelay No Arcing (5)
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time
S
time
IL
timeVR
timeVL
Relay No Arcing (5)closed open
timeID
If R~0, VL ~ 3VD
If R~0, VR ~ -3VD
VBattery
IL
S
ID +
VL-
+
VR
diL/dt = VL / L
Characteristics andModeling
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Modeling
Introduction to Load Modeling
Introduction to Capacitors and RC
Networks Introduction to Inductors and RL
Networks
Example Load Models:Turning on an Incandescent Lamp
Switching a Relay
-
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127/127
Thank you!
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