Bridging Theory in Practice

1
Bridging Theory in Practice

• Transferring Technical Knowledge
• to Practical Applications

2
RLC Load Characteristics and Modeling
3
RLC Load Characteristics and Modeling

• Intended Audience
• Engineers with a basic knowledge of resistive
  circuits
• Engineers desiring a more intuitive understanding
  of capacitive and 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

4
RLC Load Characteristics 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

5
RLC Load Characteristics 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

6
ElectromechanicalPower Conversion

• Electrical power can be converted to mechanical
  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

( )
7
Load Modeling

• Power converters (the loads) can be modeled by
  equivalent circuits composed of simple RLC
  passive components

8
RLC Load Characteristics 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

9
Capacitors

• Physical object with the ability to store
  electric charge (i.e. electric voltage)
• Consists of two electrically isolated metal
  electrodes, typically two conductive parallel
  plates
• Is mostly used to store energy or for filtering
  purposes
• The isolating material the dielectric defines
  the type of capacitor e.g. tantalum or ceramic
  capacitor
• Circuit symbol

C
10
CapacitorsPhysical Properties

• The capacitance of a parallel plate capacitor is
  proportional to
• C Capacitance a Area of each parallel
  plate d Distance between parallel plates
  
• Larger value capacitors have larger plate areas
  and less spacing between plates
• They can store more energy (and are more
  expensive)

11
CapacitorsPhysical Properties

• 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 potential energy under the
  influence of an electric field
  1) Websters 9th edition

12
Relative Size of Capacitance

• Capacitance of a free air (?r 1) parallel plate
  capacitor with the dimensions of A1m2 and d1mm
  is
• Typically, capacitance values in the 1F range are
  uncommon
• Capacitances typically range from microFarads to
  picoFarads
• 1 microFarad 1mF 10-6F
• 1 nanoFarad 1nF 10-9F
• 1 picoFarad 1pF 10-12F

13
Capacitors Electrical Properties

• The stored electrical charge Q in a capacitor is
  proportional to the voltage V across the
  capacitor Q V
• The proportional factor between stored electrical
  charge and voltage difference is the capacitance
  value of the capacitor Q C ? V

Q 8 A?s 8 Coulombs
V 16V
C Q/V 8 A?s / 16V 0.5 Farad (F) Unit C
A ? s / V F
14
Parallel and Serial Capacitance
Parallel capacitors
Serial capacitors
C1
C2
C1
C
C2
C
C C1 C2
15
Capacitor Experiment 1

• An ideal current source is connected to a
  capacitor

IC
IIDEAL
C

• 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.

IC
IIDEAL
VC
tON
16
Capacitor Experiment 2

• An ideal current source is disconnected from a
  capacitor

IC
tOFF
IIDEAL

• If the constant current
• source is removed,
• the voltage across the
• capacitor remains
• constant.

C
IIDEAL
VC
IC
tON
tOFF
17
Capacitor Experiment 3

• An ideal current source is connected to a
  capacitor

IC
IIDEAL
C

• The rate of voltage
• change is proportional
• to the current.

V, I
IC1
VC1
t
tON
18
Capacitor Experiment 3

• A variable ideal current source is connected to a
  capacitor

IC
IIDEAL
C

• The rate of voltage
• change is proportional
• to the current.

V, I
IC2
VC2
IC1
VC1
t
tON
19
Capacitor Experiment 4

• A voltage source is connected to a capacitor
  through a resistor

The peak current in the capacitor is limited by
the resistor. The voltage across the capacitor
will reach VIDEAL Ideal voltage source supplies
the voltage regardless of the current load.
IC
tON
R

C
VIDEAL
-
V, I
IC
VIDEAL/R
VC
VIDEAL
t
tON
20
Capacitor Experiment 5

• A voltage source is connected through a variable
  resistor

21
Capacitor Experiment 5

• A voltage source is connected through a variable
  resistor

V, I
R R1
VIDEAL
R1
VC1
IC1
tON
t
22
Capacitor Experiment 5

• A voltage source is connected through a variable
  resistor

V, I
R1 gt R2
VIDEAL
VIDEAL
R1
VC1
R1
IC1
tON
t
23
Capacitor Experiment 5

• A voltage source is connected through a variable
  resistor

V, I
VIDEAL
IC2
R2
R1 gt R2
VIDEAL
VIDEAL
R1
VC1
R1
IC1
tON
t
24
Capacitor Experiment 5

• A voltage source is connected through a variable
  resistor

V, I
VIDEAL
IC2
R2
R1 gt R2
VC2
VIDEAL
VIDEAL
R1
VC1
Capacitors are charged faster through
smaller resistors
IC1
tON
t
25
Capacitor Experiment 5

• A voltage source is connected through a variable
  resistor

V, I
R1 lt R3
VIDEAL
IC1
R1
VC1
tON
t
26
Capacitor Experiment 5

• A voltage source is connected through a variable
  resistor

V, I
R1 lt R3
VIDEAL
VC1
IC1
R1
VIDEAL
R3
IC3
tON
t
27
Capacitor Experiment 5

• A voltage source is connected through a variable
  resistor

V, I
R1 lt R3
VIDEAL
VC1
IC1
R1
VC3
Capacitors are charged faster through
smaller resistors
VIDEAL
R3
IC3
tON
t
28
Capacitor Experiment 6

• The rise time of the capacitor's voltage is
  monitored

VC
tC RC
VIDEAL
0.63VIDEAL
tC
0
t
29
Capacitor Experiment 6

• The rise time of the capacitor's voltage is
  monitored

VC
tC RC
0.95VIDEAL
0.87VIDEAL
0.63VIDEAL
3tC
tC
2tC
t
0
30
Development of MathematicalCapacitor Model IC
vs. VC

• Current is defined as the amount of charge which
  is transferred in a certain period of time I Q
  / t

(1)

• The relations above are derivatives for very
  small changes
• differentials can be used for quasi linear
  changes
• iDq/Dt or Dqi.Dt
  (1a)

31
Development of MathematicalCapacitor Model IC
vs. VC

• Current is defined as the amount of charge which
  is transferred in a certain period of time I Q
  / t
• Capacitance is defined as the stored charge on a
  capacitor vs. the voltage across the capacitor, C
  Q / V

(1)
(2)
In differential form CDq/Dt or
DqC.Dv (2a)
32
Development of MathematicalCapacitor Model IC
vs. VC

• Current is defined as the amount of charge which
  is transferred in a certain period of time I Q
  / t
• Capacitance is defined as the stored charge on a
  capacitor vs. the voltage across the capacitor, C
  Q / V
• Setting (2) equal to (1) results in

(1)
(2)
33
Capacitors
IC
Voltage across Capacitor
Current through Capacitor
VIN
VC
time
R
VIN
C
34
Capacitor Resistor Networks

• In general, there are two basic options for
  capacitor placement

C from Signal Path to Ground
C in Series with Signal Path
R
C
VIN
VIN
VOUT
VOUT
R
C
35
Capacitor Resistor Networks
C from Signal Path to Ground
C in Series with Signal Path
C
VOUT
VOUT
R
VC -
VC -
I
R
I
C
VIN
VIN

• Initially a DC voltage is applied at the signal
  input IN.
• Current passes through the capacitor and the
  voltage across the capacitor increases

36
Capacitor Resistor Networks
C from Signal Path to Ground
C in Series with Signal Path
C
VOUT
VOUT
R
VIN -
VIN -
I0A
R
I0A
C
VIN
VIN

• Initially a DC voltage is applied at the signal
  input IN.
• Current passes through the capacitor and the
  voltage across the capacitor increases
• When the voltage across the capacitor is equal to
  the input voltage the current stops

37
Capacitor Resistor Networks
C from Signal Path to Ground
C in Series with Signal Path
C
0V
VIN
R
VIN -
VIN -
I0A
R
I0A
C
VIN
VIN

• Initially a DC voltage is applied at the signal
  input IN.
• Current passes through the capacitor and the
  voltage across the capacitor increases
• When the voltage across the capacitor is equal to
  the input voltage the current stops
• Depending on the capacitors placement, the VOUT
  0V or VOUT VIN

38
Capacitance in Series with Signal Path
VX
VX
VOUT
t1
C
t2
VC -
I
R
I
VIN
VOUT
t1
t2
39
Capacitance in Series with Signal Path
VX
VIN
I
VIN/R
VOUT
VIN
t1
t2
40
Capacitance in Series with Signal Path
VX
VX
VIN
VOUT
t1
C
t2
VC -
I
R
I
VIN
VIN/R
-VIN/R
VOUT
VIN
-VIN
t1
t2
41
Capacitance From Signal Path to Ground
VX
VX
VOUT
t1
R
VC -
t2
I
I
C
VIN
VOUT
t1
t2
42
Capacitance From Signal Path to Ground
VX
VX
VOUT
t1
VIN
R
VC -
t2
I
I
C
VIN
VIN/R
-VIN/R
VOUT
VIN
t1
t2
43
Capacitance From Signal Path to Ground
VX
VX
VOUT
t1
VIN
R
VC -
t2
I
I
C
VIN
VIN/R
-VIN/R
VOUT
VIN
t1
t2
44
RC Networks - AC Signals

• What happens when an AC input signal is applied?

C from Signal Path to Ground
C in Series with Signal Path
VOUT
VOUT
R
C
?
?
VIN
VIN
t
t
R
C
45
Capacitors and AC signals

• Capacitors act like frequency dependent resistor
  (capacitive reactance, XC)
• Xc1/(fC)
• Instead of reactance, impedance (Z) is used to
  characterize circuit elements

Z1/(2pfC)
46
Capacitors and AC signals

• Act like frequency dependent resistor (capacitive
  reactance, XC)
• Instead of reactance, impedance (Z) used for
  circuit elements.
• Impedance1) The apparent opposition in an
  electrical circuit to the flow of alternating
  current that is analogous to the actual
  electrical resistance to a direct current.
• 1) acc. to Websters 9th edition

47
Capacitors and AC signals

• Act like frequency dependent resistor (capacitive
  reactance, XC)
• Instead of reactance, impedance (Z) used for
  circuit elements.
• Impedance1) The apparent opposition in an
  electrical circuit 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

48
Capacitors and AC signals

• Act like frequency dependent resistor (capacitive
  reactance, XC)
• Instead of reactance, impedance (Z) used for
  circuit elements.
• Impedance1) The apparent opposition in an
  electrical circuit 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 about the phase shift between the
  two.

49
Capacitors Impedance Magnitude ZC vs.
Frequency
50
Capacitors and AC signals
iC,Max VC,Max / ZC
VC,Max
iC,Max
t
j p/2 90o ? The current leads the voltage
51
RC networks AC Signals
C from Signal Path to Ground
C in Series with Signal Path
VOUT
VOUT
R
C
VIN
VIN
t
t
R
C

• The capacitor acts as a frequency dependent
  resistor
• It determines the current magnitude at a given
  voltage
• It causes a 90 degree phase shift between the
  capacitor current and voltage across the capacitor

52
RC networks AC Signals
C from Signal Path to Ground
C in Series with Signal Path
VOUT
VOUT
R
C
VIN
VIN
t
t
R
C

• 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

ZC1/(2pfC)
53
C in Series with Signal PathHigh Pass
Configuration
VOUT
VIN
C
VIN
R
ZC1/(2pfC)
VOUT
VOUT/VINMAX Low f
0.32 Medium f 0.76 High f 0.90
54
C from Signal Path to GroundLow Pass
Configuration
VOUT
VIN
R
VIN
C
ZC1/(2pfC)
VOUT
VOUT/VINMAX Low f
0.96 Medium f 0.74 High f 0.39
55
Capacitor Resistor Networks Summary
C from Signal Path to Ground
C in Series with Signal Path
VOUT
VOUT
R
C
VIN
VIN
R
C

• Connected to DC voltages
• Capacitors will allow current to flow only until
  they are charged
• Once charged, they block future current flow
• For AC signals
• Capacitors act similar to frequency dependent
  resistors
• Low impedance at high frequencies
• High impedance at low frequencies.

56
RLC Load Characteristics 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

57
Inductors

• Physical object which can store a magnetic field
  (electric current)
• 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

L
or
58
Physical Properties 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
• u0 permeability of vacuum 4p10-7 V.s/(A.M)
• ur relative permeability
• Larger value inductors have more turns and
  bigger cross section in less volume. They can
  store more energy (and may be more expensive).

Lm0.mrN2.a/l
l
Core
Wire
59
Inductance of a Toroid
L Inductance Units of H V ? s/AN Number
of turns of the coil a Coil cross
section Units of m2 Average field
length Units of mm Permeability m0? mr
Units of H/m V ? s/A ? mm0 Permeability
of free space 4p10-7 Units of H/m V ? s/A
? mmr 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
LmN2a/l
60
Relative Size of Inductance

• Inductance of a free air toroid (mr 1) with the
  cross section of a5cm2, average field length of
  10cm, and N100 turns is
• Inductors in the mH range are used in switching
  regulators
• Small relays, solenoids usually have mH values of
  inductance
• Inductors in general typically range from a few
  Henries (H) to micro Henries (mH) 1 microHenry
  1mH 10-6H 1 milliHenry 1mH 10-3H 1
  Henry 1H

61
Inductors -Electrical Properties

• The change of magnetic field or coil flux (y) in
  an inductor is proportional to the change of
  electric current (I) flowing through the
  inductors windings y I
• The proportional factor between coil flux and
  current is given by the inductance of the coil y
  L ? I

• ( N?F)

I
62
Inductors -Electrical Properties

• The change of magnetic field or coil flux (y) in
  an inductor is proportional to the change of
  electric current (I) flowing through the
  inductors windings y I
• The proportional factor between coil flux and
  current is given by the inductance of the coil y
  L ? I

• ( N?F)
• 1Vs

I 2A
L y/I 1 Vs / 2 A 0.5 Henry (H) Unit L
Vs/A H
63
Serial and Parallel Inductance
Serial inductors
Parallel inductors
L1
L2
L1
L
L2
L
L L1 L2
64
Inductor Experiment 1

• An ideal voltage source is connected to an
  inductor

IL
tON
The constant voltage causes the current to
increase through the inductor.
-
VIDEAL
L
VL
VIDEAL
IL
tON
65
Inductor Experiment 2

• An ideal voltage source is disconnected to an
  inductor

IL
tOFF
Vsrc
-

• If the constant voltage
• source is removed and
• the inductor is shorted
• the current through
• the inductor remains
• constant.

VIDEAL
L
VIDEAL
IL
VL
tON
tOFF
66
Inductor Experiment 3

• An ideal voltage source is connected to an
  inductor

IL
tON
-
-
VIDEAL
L

• The rate of current
• change is proportional
• to the voltage.

VL1
IL1
tON
67
Inductor Experiment 3

• An ideal voltage source is connected to an
  inductor

IL
tON
-
-
VIDEAL
L

• The rate of current
• change is proportional
• to the voltage.

VL2
IL2
VL1
IL1
tON
68
Inductor Experiment 4

• A voltage source is connected to an inductor
  through a resistor

The peak voltage across the inductor is
VIDEAL. The current through the inductor
will reach VIDEAL/R.
IL
tON
R

VIDEAL
L
-
VL
VIDEAL
IL
VIDEAL/R
tON
t
69
Inductor Experiment 5

• A voltage source is connected through a variable
  resistor

70
Inductor Experiment 5

• A voltage source is connected through a variable
  resistor

V, I
IL
VL
L
R1 gt R2
VIDEAL
VIDEAL/R1
IL1
VL1
tON
t
71
Inductor Experiment 5

• A voltage source is connected through a variable
  resistor

The smaller the resistor, the longer it takes
the current to become steady
V, I
IL
L
VL
L
R1 gt R2
VIDEAL/R2
IL2
VIDEAL
VIDEAL/R1
IL1
VL2
VL1
tON
t
72
Inductor Experiment 5

• A voltage source is connected through a variable
  resistor

The smaller the resistor, the longer it takes
the current to become steady
V, I
IL
VL
L
R1 lt R3
VIDEAL
VIDEAL/R1
IL1
VIDEAL/R3
IL3
VL1
VL3
tON
t
73
Inductor Experiment 6

• The rise time of the capacitor's voltage is
  monitored

IL
VL
L
VL
tC L/R
VIDEAL
0.37VIDEAL
tC
0
t
74
Inductor Experiment 6

• The rise time of the capacitor's voltage is
  monitored

IL
VL
L
VL
tC L/R
VIDEAL
0.37VIDEAL
0.14VIDEAL
0.05VIDEAL
tC
3tC
2tC
0
t
75
Development of MathematicalInductor Model IL
vs. VL

• The self induced coil voltage when exposed to an
  alternating magnetic field is proportional to the
  change of coil flux vs. time

76
Development of MathematicalInductor Model IL
vs. VL

• The self induced coil voltage when exposed to an
  alternating magnetic field is proportional to the
  change of coil flux vs. time
• The voltage v applied across an inductor is
  always directly opposed to the self induced
  voltage vind v -vind N?df/dt dy/dt (gt dy
  v?dt)

(1)
77
Development of MathematicalInductor Model IL
vs. VL

• The self induced coil voltage when exposed to an
  alternating magnetic field is proportional to the
  change of coil flux vs. time
• The voltage v applied across an inductor is
  always directly opposed to the self induced
  voltage vind v -vind N?df/dt dy/dt (gt dy
  v?dt)
• The inductance is defined as coil flux vs. coil
  current, Ly / IL, differentially expressed as

(1)
(2)
78
Development of MathematicalInductor Model IL
vs. VL

• Setting (1) equal to (2), the voltage - current
  relation for an inductor equals can be found

(1)
(2)
79
Inductors
IL,maxVIN/R
VL
Voltage across Inductor
Current through Inductor
VIN
IL
time
VIN
R
80
Inductor Resistor Networks

• In general, there are two basic options for
  inductor placement

L from Signal Path to Ground
L in Series with Signal Path
R
L
VIN
VIN
VOUT
VOUT
R
L
81
Inductor ResistorNetworks
L from Signal Path to Ground
L in Series with Signal Path
L
VOUT
VOUT
R
VL -
VL -
I
VIN
VIN
R
L
I

• Initially a DC voltage is applied at the signal
  input IN.
• A voltage drops across the inductor and the
  current through the inductor increases

82
Inductor ResistorNetworks
L from Signal Path to Ground
L in Series with Signal Path
L
VOUT
VOUT
R
0V -
0V -
I
VIN
VIN
R
L
I

• Initially a DC voltage is applied at the signal
  input IN.
• A voltage occurs across the inductor and the
  current through the inductor increases
• When the current through the inductor is at its
  maximum and remains constant, the voltage across
  the inductor equals zero

83
Inductor ResistorNetworks
L from Signal Path to Ground
L in Series with Signal Path
L
0V
VIN
R
0V -
0V -
I
VIN
VIN
R
L
I

• Initially a DC voltage is applied at the signal
  input IN.
• A voltage drops across the inductor and the
  current through the inductor increases
• When the current through the inductor is at its
  maximum and remains constant, the voltage across
  the inductor equals zero
• Depending on the inductors placement the steady
  state
• final voltages are VOUT VIN or VOUT 0V

84
Inductance in Series with Signal Path
VX
VX
VOUT
t1
L
t2
VL -
I
R
I
VIN
VOUT
t1
t2
85
Inductance in Series with Signal Path
VX
VX
VIN
VOUT
t1
L
t2
VL -
I
R
I
VIN
VIN/R
VOUT
VIN
t1
t2
86
Inductance in Series with Signal Path
VX
VX
VIN
VOUT
t1
L
t2
VL -
I
R
I
VIN
VIN/R
VOUT
VIN
t1
t2
87
Inductance From Signal Path to Ground
VX
VX
VOUT
t1
R
VL -
t2
I
I
L
VIN
VOUT
t1
t2
88
Capacitance From Signal Path to Ground
VX
VX
VOUT
t1
VIN
R
VL -
t2
I
I
L
VIN
VIN/R
VOUT
VIN
t1
t2
89
Capacitance From Signal Path to Ground
VX
VX
VOUT
t1
VIN
R
VL -
t2
I
I
L
VIN
VIN/R
VOUT
VIN
-VIN
t1
t2
90
RL Networks - AC Signals

• What happens when an AC input signal is applied?

L from Signal Path to Ground
L in Series with Signal Path
VOUT
VOUT
R
L
?
?
VIN
VIN
t
t
R
L
91
Inductors and AC signals

• Act like frequency dependent resistor (inductive
  reactance, XL)
• Instead of reactance, impedance (Z) used for
  circuit elements.

XL2pfL
92
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
  electrical resistance to a direct current.

XL2pfL
93
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
  electrical resistance to a direct current.
• The impedance of a circuit element represents its
  resistive and/or reactive components

XL2pfL
94
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
  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 about the phase shift between the
  two.

XL2pfL
95
Inductors Impedance Magnitude ZL vs. Frequency
ZL2.p.f.L
ZL (ohm)
frequency (Hz)
96
Inductors and AC signals
iL,Max VL,Max / ZL
VL,Max
iL,Max
t
j
j -p/2 -90o ? The current lags the voltage
97
RL networks AC signals
L from Signal Path to Ground
L in Series with Signal Path
VOUT
VOUT
R
L
VIN
VIN
t
t
R
L

• The inductor acts as a frequency dependent
  resistor
• It determines the current magnitude at a given
  voltage
• It causes a 90 degree phase shift between the
  inductor current and voltage across the inductor

98
RC networks AC signals
L from Signal Path to Ground
L in Series with Signal Path
VOUT
VOUT
R
L
VIN
VIN
t
t
R
L

• 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

ZL2pfL
99
L in Series with Signal PathLow Pass
Configuration
VOUT
VIN
L
VIN
R
Z2.p.f.L
VOUT
VOUT/VINMAX Low f
0.96 Medium f 0.76 High f 0.38
100
L from Signal Path to GroundHigh Pass
Configuration
VOUT
VIN
R
VIN
L
ZL2.p.f.L
VOUT
VOUT/VINMAX Low f
0.32 Medium f 0.74 High f 0.92
101
Inductor Resistor Networks Summary
L from Signal Path to Ground
L in Series with Signal Path
R
L
VIN
VIN
VOUT
VOUT
R
L

• Connected to DC voltages
• The voltage across an inductor changes as current
  increases
• 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.

102
Capacitor vs. InductorUnit Comparison
y
?0
?0
103
RLC Load Characteristics 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

104
Lamp Experiment

• Turn on an incandescent light bulb and measure
  the current

105
Lamp Experiment

• Turn on an incandescent light bulb and measure
  the current
• Result

5.6A
600mA
ton
106
Developing a RC Load Model For an Incandescent
Light Bulb
14V
Light Bulb
5.6A
600mA
ton
107
Developing a RC Load Model For an Incandescent
Light Bulb
R1
14V
5.6A
600mA
ton
108
Developing a RC Load Model For an Incandescent
Light Bulb
23.3?
14V
5.6A
600mA
ton
109
Developing a RC Load Model For an Incandescent
Light Bulb
23.3?
14V
5.6A
R2
600mA
ton
110
Developing a RC Load Model For an Incandescent
Light Bulb
C
23.3?
14V
5.6A
2.8?
600mA
ton
111
Simulation of Lamp RC Model
6.0
5.0
4.0
Input Current (A)
3.0
2.0
1.0
0.0
250
350
300
50
150
100
200
0
ton
Time (ms)
112
Simulation of Lamp RC Model
6.0
5.0
4.0
Input Current (A)
3.0
2.0
1.0
0.0
250
350
300
50
150
100
200
0
ton
Time (ms)
113
A RC Load Model forIncandescent Light Bulbs

• The model for this lamps is represented by the
  network below
• When a lamp initially turns on, the filament is
  cold and has a relatively low resistance BUT as
  the filament warms up, the resistance increases
  dramatically

1
3.6mF
23.3?
2.80?
f(T)
2
114
Lamp Experiment

• When a lamp initially turns on, the filament is
  cold and has a relatively low resistance
• As the filament warms up, the resistance
  increases dramatically

5.6A
600mA
115
RLC Load Characteristics 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

116
Switching a Relay
VBattery

• To the right a high side switching application
  is shown
• The switch itself is modeled as a simple
  mechanical switch
• The relay can be modeled as a low ohmic resistor
  and inductor connected in series

S
VR
Relay
VL
IL
117
Switching On a Relay
S
VBattery
open
time
closed
VR
time
S
VL decays over time
VL
time
IL
IL
time
IL (VR-VL) / R
118
Switching Off a Relay (1)
S
VBattery
closed
time
open
IL
time
S
time
IL
time
119
Switching Off a Relay (2)
S
VBattery
closed
time
open
IL
time
IL cannot become zero instantaneously!
S
VL becomes negative to force the current to 0A
VL
time
(VL -Ldi/dt)
For VL lt 0V, VR lt 0V
IL
VR
time
120
Switching Off a Relay (3)
S
VBattery
closed
time
open
Arcing
IL
time
S
IL cannot go to zero instantaneously!
VL goes far below ground to force the current to
0A
VR -
VL
time
For VL lt 0V, VR lt 0V (R0)
VL -
IL
VR
time
121
Switching Off a Relay No Arcing (1)
S
VBattery
closed
time
open
IL
time
S
VR -
ID
time
VL
time
ID
VL -
IL
VR
time
122
Switching Off a Relay No Arcing (2)
S
VBattery
closed
time
open
IL
time
S
Diode turns on and provides a current path
VR -
ID
time
VL
time
ID
VL -
IL
VR
time
123
Switching Off a Relay No Arcing (3)
S
VBattery
closed
time
open
IL
time
S
VR -
ID
time
If R0?, VL VD
VL
time
ID
VL -
IL
If R0?, VR -VD
VR
time
124
Switching Off a Relay No Arcing (4)
S
VBattery
closed
time
open
IL
time
S
VR -
ID
time
VL
time
ID
VL -
IL
VR
time
125
Switching Off a Relay No Arcing (5)
S
VBattery
closed
time
open
diL/dt VL / L
IL
time
S
VR -
ID
time
If R0?, VL 3VD
VL
time
ID
VL -
IL
If R0?, VR -3VD
VR
time
126
RLC Load Characteristics 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

127

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