Ch1 RLC Load - Bridging Theory into Practice
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Transcript Ch1 RLC Load - Bridging Theory into Practice
Bridging Theory in Practice
Transferring Technical Knowledge
to Practical Applications
RLC Load
Characteristics and Modeling
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
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
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
Electromechanical
Power 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
(= )
Load Modeling
• “Power converters” (the loads) can be modeled by equivalent circuits
composed of simple RLC passive components
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
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
Capacitors:
Physical Properties
• The capacitance of a parallel plate capacitor is proportional to:
C~
C
a
d
a
d
= Capacitance;
= Area of each parallel plate;
= Distance between parallel plates;
d
a
• Larger value capacitors have larger plate areas and less spacing
between plates
• They can store more energy (and are more expensive)
Capacitors:
Physical 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)
·a
C=
d
a
Permittivity1) : the ability of a dielectric to store electrical potential
energy under the influence of an electric field
1) Webster’s 9th edition
d
Relative Size of Capacitance
• Capacitance of a free air (r = 1) parallel plate capacitor with
the dimensions of A=1m2 and d=1mm is:
1 8.854x1012 F / m (1m2 )
r 0 A
9
C
8.854x10
F
3
d
1x10 m
• 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
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
Parallel and Serial Capacitance
Parallel capacitors
C1
Serial capacitors
C2
C1
C
C2
C
C = C1 + C2
1 1
1
=
+
C C1 C2
Capacitor Experiment #1
• An ideal current source is connected to a capacitor
tON
IC
+
VC
-
IIDEAL
C
The constant current
causes the voltage
to linearly rise across
V, I
the capacitor.
IC
IIDEAL
Constant current source supplies
the current regardless of the
voltage drop across the load.
VC
tON
t
Capacitor Experiment #2
• An ideal current source is disconnected from a capacitor
tOFF
IC
+
VC
-
IIDEAL
If the constant current
C
source is removed,
the voltage across the
V, I
capacitor remains
IIDEAL
constant.
VC
IC
tON
tOFF
t
Capacitor Experiment #3
• An ideal current source is connected to a capacitor
tON
IC
+
VC
-
IIDEAL
C
The rate of voltage
change is proportional
to the current.
V, I
IC1
VC1
tON
t
Capacitor Experiment #3
• A variable ideal current source is connected to a capacitor
tON
IC
+
VC
-
IIDEAL
C
The rate of voltage
change is proportional
to the current.
V, I
IC2
VC2
IC1
VC1
tON
t
Capacitor Experiment #4
• A voltage source is connected to a capacitor through a
resistor
The peak current
tON
VIDEAL
+
-
R
IC
in the capacitor is
+
VC
-
C
limited by the
resistor.
The voltage across
V, I
VIDEAL/R
the capacitor will
IC
VC
VIDEAL
tON
reach VIDEAL
t
Ideal voltage source
supplies the voltage
regardless of the current
load.
Capacitor Experiment #5
• A voltage source is connected through a variable
resistor
tON
VIDEAL
+
-
R
IC
+
VC
-
C
Capacitor Experiment #5
• A voltage source is connected through a variable resistor
V, I
tON
VIDEAL
+
-
R
IC
VC
R = R1
+
C
-
VIDEAL
R1
VC1
IC1
tON
t
Capacitor Experiment #5
• A voltage source is connected through a variable resistor
V, I
tON
VIDEAL
VIDEAL
R1
+
-
R
IC
VC
R1 > R2
+
C
-
VIDEAL
VC1
R1
IC1
tON
t
Capacitor Experiment #5
• A voltage source is connected through a variable resistor
V, I
VIDEAL
R2
tON
IC2
VIDEAL
VIDEAL
R1
+
-
R
IC
VC
R1 > R2
+
C
-
VIDEAL
VC1
R1
IC1
tON
t
Capacitor Experiment #5
• A voltage source is connected through a variable resistor
V, I
VIDEAL
R2
tON
IC2
VIDEAL
VC2
VIDEAL
R1
IC
VC
R1 > R2
+
C
-
VIDEAL
VC1
IC1
tON
+
-
R
Capacitors are
charged faster
through smaller
resistors
t
Capacitor Experiment #5
• A voltage source is connected through a variable resistor
V, I
tON
VIDEAL
VIDEAL
R1
+
-
R
IC
VC
R1 < R3
+
C
-
IC1
VC1
tON
t
Capacitor Experiment #5
• A voltage source is connected through a variable resistor
V, I
tON
VIDEAL
VIDEAL
R1
IC1
+
-
R
IC
VC
R1 < R3
+
C
-
VC1
VIDEAL
R3
IC3
tON
t
Capacitor Experiment #5
• A voltage source is connected through a variable resistor
V, I
tON
VIDEAL
VIDEAL
R1
IC1
R3
IC3
tON
IC
VC
R1 < R3
+
C
-
VC1
VC3
VIDEAL
+
-
R
Capacitors are
charged faster
through smaller
resistors
t
Capacitor Experiment #6
• The rise time of the capacitor's voltage is monitored:
tON
VIDEAL
VC
+
-
R
IC
VC
R1
tC =<RC
R3
+
C
-
VIDEAL
0.63VIDEAL
0
tC
t
Capacitor Experiment #6
• The rise time of the capacitor's voltage is monitored:
tON
VIDEAL
VC
+
-
R
IC
VC
R1
tC =<RC
R3
+
C
-
0.95VIDEAL
0.87VIDEAL
0.63VIDEAL
0
tC
2tC
3tC
t
Development of Mathematical
Capacitor Model: IC vs. VC
• Current is defined as the amount of charge which is
transferred in a certain period of time: I = Q / t
dq
i
or dq i dt
dt
(1)
The relations above are derivatives for very small changes
differentials can be used for quasi linear changes:
i=Dq/Dt
or
Dq=i.Dt
(1a)
Development of Mathematical
Capacitor Model: IC vs. VC
• Current is defined as the amount of charge which is
transferred in a certain period of time: I = Q / t
dq
i
or dq i dt
dt
(1)
• Capacitance is defined as the stored charge on a capacitor
vs. the voltage across the capacitor, C = Q / V
dq
(2)
C
or dq C dv
dv
In differential form:
C=Dq/Dt or
Dq=C.Dv
(2a)
Development of Mathematical
Capacitor Model: IC vs. VC
• Current is defined as the amount of charge which is
transferred in a certain period of time: I = Q / t
dq
i
or dq i dt
dt
(1)
• Capacitance is defined as the stored charge on a capacitor
vs. the voltage across the capacitor, C = Q / V
dq
(2)
C
or dq C dv
dv
• Setting (2) equal to (1) results in:
i dt C dv
or
dv
iC
dt
Capacitors
IC
Voltage
across VIN
Capacitor
VC
Current
through
Capacitor
time
R
VIN
C
VC across
plates
Capacitor & Resistor Networks
• In general, there are two basic options for capacitor
placement:
C in Series with Signal Path
VIN
C
R
VOUT
C from Signal Path to Ground
VIN
R
C
VOUT
Capacitor & Resistor Networks
C in Series with Signal Path
C
VOUT
+ VC R
I
V
IN
C from Signal Path to Ground
R
VIN
VOUT
I
C
+
VC
-
• Initially a DC voltage is applied at the signal
input IN.
• Current passes through the capacitor and the
voltage across the capacitor increases
Capacitor & Resistor Networks
C in Series with Signal Path
C
VOUT
+ VIN R
I=0A
VIN
C from Signal Path to Ground
R
VIN
VOUT
I=0A
C
+
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
Capacitor & Resistor Networks
C in Series with Signal Path
C
0V
+ VIN R
VIN
I=0A
C from Signal Path to Ground
R
VIN
I=0A
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 capacitor’s placement, the VOUT = 0V or
VOUT = VIN
Capacitance in Series with
Signal Path
VX
t1
VX
t2
+ VC -
VIN
VOUT
C
I
R
I
VOUT
t1
t2
Capacitance in Series with
Signal Path
VX
t1
VX
t2
+ VC -
VIN
VOUT
C
I
R
VIN
I
VIN/R
VOUT
VIN
t1
t2
Capacitance in Series with
Signal Path
VX
t1
VX
t2
+ VC -
VIN
VOUT
C
I
R
VIN
I
VIN/R
-VIN/R
VOUT
VIN
-VIN
t1
t2
Capacitance From Signal Path
to Ground
t1
t2
VIN
VX
VX
R
VOUT
I
C
+
VC
-
I
VOUT
t1
t2
Capacitance From Signal Path
to Ground
t1
t2
VIN
VX
VX
R
VOUT
I
C
VIN
+
VC
-
I
VIN/R
-VIN/R
VOUT
VIN
t1
t2
Capacitance From Signal Path
to Ground
t1
t2
VIN
VX
VX
R
VOUT
I
C
VIN
+
VC
-
I
VIN/R
-VIN/R
VOUT
VIN
t1
t2
RC Networks - AC Signals
• What happens when an AC input signal is applied?
C in Series with Signal Path
VOUT
C
VIN
t
R
?
C from Signal Path to Ground
VOUT
R
VIN
t
C
?
Capacitors and AC signals
• Capacitors act like frequency dependent resistor
(capacitive reactance, XC)
Xc~1/(fC)
• Instead of reactance, impedance (Z) is used to
characterize circuit elements:
Z=1/(2pfC)
Capacitors and AC signals
• Act like frequency dependent resistor (capacitive
1
reactance, XC)
X
C
fC
• 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.
Capacitors and AC signals
• Act like frequency dependent resistor (capacitive reactance,
XC)
1
XC
fC
• 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
Capacitors and AC signals
• Act like frequency dependent resistor (capacitive reactance, XC)
1
XC
fC
• 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.
Capacitor’s Impedance
Magnitude |ZC| vs. Frequency
|Z|=1/(2pfC) C=1uF
|Z| (kohm)
.
ZC
1
fC
18.0
16.0
14.0
12.0
10.0
8.0
6.0
4.0
2.0
0.0
0
100
200
300
FREQUENCY (Hz)
400
500
Capacitors and AC signals
iC,Max = VC,Max / |ZC|
VC,Max
iC,Max
= +p/2 = + 90o The current leads the voltage
t
RC networks – AC Signals
C in Series with Signal Path
VOUT
C
VIN
t
R
C from Signal Path to Ground
VOUT
R
VIN
t
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
RC networks – AC Signals
C in Series with Signal Path
VOUT
C
VIN
t
R
C from Signal Path to Ground
VOUT
R
VIN
t
C
• For high frequency signals:
– The capacitor is low impedance
– Signals can pass the capacitor
|ZC|=1/(2pfC)
• For low frequency signals:
– The capacitor is high impedance
– Signals are blocked by the capacitor
C in Series with Signal Path
High Pass Configuration
VOUT
VIN
VIN
6
4
C
2
R
0
-2
0
0.01
0.02
0.03
0.04
0.05
0
0.01
0.02
0.03
0.04
0.05
-4
|ZC|=1/(2pfC)
-6
VOUT
6
4
VOUT/VINMAX
Low f
0.32
Medium f 0.76
High f
0.90
2
0
-2
-4
-6
C from Signal Path to Ground
Low Pass Configuration
VOUT
VIN
VIN
6
4
R
2
C
0
-2
0
0.01
0.02
0.03
0.04
0.05
0
0.01
0.02
0.03
0.04
0.05
-4
|ZC|=1/(2pfC)
-6
VOUT
6
4
VOUT/VINMAX
Low f
0.96
Medium f 0.74
High f
0.39
2
0
-2
-4
-6
Capacitor & Resistor Networks
Summary
C in Series with Signal Path
VOUT
C
VIN
R
C from Signal Path to Ground
VOUT
R
VIN
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.
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
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
or
L
Physical
Properties of Inductors
• The inductance of a toroid, for instance, is given by:
L=m0.mrN2.a/l
L
N
a
ul0
ur
A
= Inductance;
= Number of turns of the coil;
= Coil cross section;
= Average field length;
= permeability of vacuum =4p10-7 V.s/(A.M)
=relative permeability
Core
Wire
• Larger value inductors have more turns and bigger cross section in less volume.
They can store more energy (and may be more expensive).
Inductance of a Toroid
L
N
a
m
m0
mr
= Inductance
Units of: H = V · s/A
= Number of turns of the coil
= Coil cross section
Units of: m2
= Average field length
Units of: m
= Permeability = m0· mr ;
Units of: H/m = V · s/A · m
= Permeability of free space = 4p10-7
Units of: H/m = V · s/A · m
= Relative permeability
L=mN2a/l
Permeabilty1) : the property of a ferro-magnetic substance that
determines the degree in which it modifies the magnetic flux in the
region occupied by it in a magnetic field
1) acc. to Webster’s 9th edition
a
Relative Size of Inductance
• Inductance of a free air toroid (mr = 1) with the cross section of
a=5cm2, average field length of =10cm, and N=100 turns is
L~
(1)(4p 10 H/ m) 100 5x10 m
2
7
2
10x10 m
4
2
6.28x10
5
H
• 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
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 inductor’s windings: y ~ I
• The proportional factor between coil flux and current is given
by the inductance of the coil: y = L · I
y N·F
I
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 inductor’s windings: y ~ I
• The proportional factor between coil flux and current is
given by the inductance of the coil: y = L · I
I = 2A
y N·F
y1Vs
L = y/I = 1 Vs / 2 A = 0.5 Henry (H)
Unit [L] = Vs/A = H
Serial and Parallel Inductance
Serial inductors
Parallel inductors
L1
L1
L2
L
L2
L
L = L1 + L2
1 1 1
= +
L L1 L2
Inductor Experiment #1
• An ideal voltage source is connected to an inductor
tON IL
VIDEAL
+
-
+
VL
-
The constant voltage
L
causes the current
to increase through
the inductor.
V, I
VL
VIDEAL
IL
tON
t
Inductor Experiment #2
• An ideal voltage source is disconnected to an
inductor
tOFF
VIDEAL
+
-
Vsrc
IL
+
VL
-
If the constant voltage
L
source is removed and
the inductor is shorted
V, I
the current through
VIDEAL
the inductor remains
IL
constant.
VL
tON
tOFF
t
Inductor Experiment #3
• An ideal voltage source is connected to an inductor
tON IL
VIDEAL
+
VL
-
+
-
L
The rate of current
change is proportional
to the voltage.
V, I
VL1
IL1
tON
t
Inductor Experiment #3
• An ideal voltage source is connected to an inductor
tON IL
VIDEAL
+
VL
-
+
-
L
The rate of current
change is proportional
to the voltage.
V, I
VL2
VL1
IL2
IL1
tON
t
Inductor Experiment #4
• A voltage source is connected to an inductor through a
resistor
tON
VIDEAL
VIDEAL
+
-
R
The peak voltage
IL
+
VL
-
across the inductor
L
is VIDEAL.
The current through
VL
the inductor will
IL
VIDEAL/R
tON
reach VIDEAL/R.
t
Inductor Experiment #5
• A voltage source is connected through a variable
resistor
tON
VIDEAL
+
-
R
IICL
+
V
VCL
-
C
L
Inductor Experiment #5
• A voltage source is connected through a variable resistor
V, I
tON
VIDEAL
+
-
R
R1 > R2
IIC
L
+
L -
C
L
VVC
VIDEAL
VIDEAL/R1
IL1
VL1
tON
t
Inductor Experiment #5
• A voltage source is connected through a variable resistor
V, I
VIDEAL/R2
VIDEAL
The smaller the
resistor, the longer
it takes the current
to become steady
tON
VIDEAL
+
-
R
IC
IL
+
VL -
C
VC
R1 > R2
LL
IL2
VIDEAL/R1
IL1
VL2
VL1
tON
t
Inductor Experiment #5
• A voltage source is connected through a variable resistor
V, I
The smaller the
resistor, the longer
it takes the current
to become steady
tON
VIDEAL
+
-
R
R1 < R3
IC
IILLV +
C
LL
VLC -
VIDEAL
VIDEAL/R1
IL1
VIDEAL/R3
IL3
VL1
VL3
tON
t
Inductor Experiment #6
• The rise time of the capacitor's voltage is monitored:
tON
VIDEAL
VL
+
-
R
IC
IILLV +
VL
C
tC = L/R
-
C
LL
VIDEAL
0.37VIDEAL
0
tC
t
Inductor Experiment #6
• The rise time of the capacitor's voltage is
monitored:
t
ON
VIDEAL
VL
+
-
R
IC
IILLV +
VL
C
tC = L/R
-
C
LL
VIDEAL
0.37VIDEAL
0.14VIDEAL
0.05VIDEAL
0
tC
2tC
3tC
t
Development of Mathematical
Inductor 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:
d
dy
vind N
dt
dt
Development of Mathematical
Inductor 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:
vind
d
dy
N
dt
dt
• The voltage v applied across an inductor is always directly opposed to
the self induced voltage vind:
v = -vind = N·d/dt = dy/dt (=> dy = v·dt)
v vind
d dy
N
dt
dt
or
dy v dt
(1)
Development of Mathematical
Inductor 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:
vind
d
dy
N
dt
dt
• The voltage v applied across an inductor is always directly opposed to
the self induced voltage vind:
v = -vind = N·d/dt = dy/dt (=> dy = v·dt)
v vind
d dy
N
dt
dt
or
dy v dt
(1)
• The inductance is defined as coil flux vs. coil current, L=y / IL,
differentially expressed as: dy
L
di
or
dy L di
(2)
Development of Mathematical
Inductor Model: IL vs. VL
v vind
d dy
N
dt
dt
dy
L
di
or
or
dy v dt
dy L di
• Setting (1) equal to (2), the voltage - current
relation for an inductor equals can be found:
di
v L
dt
(1)
(2)
Inductors
IL,max=VIN/R
VL
Voltage
across VIN
Inductor
IL
Current
through
Inductor
time
VIN
VL
R
Inductor & Resistor Networks
• In general, there are two basic options for inductor
placement:
L in Series with Signal Path
VIN
L
R
VOUT
L from Signal Path to Ground
VIN
R
L
VOUT
Inductor & ResistorNetworks
L in Series with Signal Path
L
VOUT
+ VL VIN
R
I
L from Signal Path to Ground
R
VIN
I
L
+
VL
-
VOUT
• Initially a DC voltage is applied at the signal input
IN.
• A voltage drops across the inductor and the current
through the inductor increases
Inductor & ResistorNetworks
L in Series with Signal Path
L
VOUT
+ 0V VIN
R
I
L from Signal Path to Ground
R
VIN
I
L
+
0V
-
VOUT
• 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
Inductor & ResistorNetworks
L in Series with Signal Path
L
VIN
+ 0V VIN
R
I
L from Signal Path to Ground
R
VIN
I
L
+
0V
-
0V
• 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 inductor’s placement the steady state
final voltages are VOUT = VIN or VOUT = 0V
Inductance in Series with
Signal Path
t1
VX
t2
+ VL -
VIN
VOUT
L
I
R
VX
I
VOUT
t1
t2
Inductance in Series with
Signal Path
t1
VX
t2
+ VL -
VIN
VOUT
L
I
R
VX
VIN
I
VIN/R
VOUT
VIN
t1
t2
Inductance in Series with
Signal Path
t1
VX
t2
+ VL -
VIN
VOUT
L
I
R
VXV
IN
I
VIN/R
VOUT
VIN
t1
t2
Inductance From Signal Path to
Ground
t1
t2
VIN
VX
R
VX
VOUT
I
L
+
VL
-
I
VOUT
t1
t2
Capacitance From
Signal Path to Ground
t1
t2
VIN
VX
VX
R
VOUT
I
L
VIN
+
VL
-
I
VIN/R
VOUT
VIN
t1
t2
Capacitance From
Signal Path to Ground
t1
t2
VIN
VX
VX
R
VOUT
I
L
VIN
+
VL
-
I
VIN/R
VOUT
VIN
-VIN
t1
t2
RL Networks - AC Signals
• What happens when an AC input signal is applied?
L in Series with Signal Path
VOUT
VIN
t
L
R
?
L from Signal Path to Ground
VOUT
R
VIN
t
L
?
Inductors and AC signals
• Act like frequency dependent resistor (inductive
XL=2pfL
reactance, XL)
• Instead of reactance, impedance (Z) used for circuit
elements.
Inductors and AC signals
• Act like frequency dependent resistor (inductive
XL=2pfL
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.
Inductors and AC signals
• Act like frequency dependent resistor (inductive reactance,
XL)
X =2pfL
L
• 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
Inductors and AC signals
• Act like frequency dependent resistor (inductive reactance, XL)
•
•
•
•
XL=2pfL
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.
Inductor’s Impedance
Magnitude |ZL| vs. Frequency
|ZL|=2.p.f.L
35
30
25
|ZL|
(ohm)
20
15
10
5
0
0
1000
2000
3000
frequency (Hz)
4000
5000
Inductors and AC signals
iL,Max = VL,Max / |ZL|
VL,Max
iL,Max
t
= -p/2 = -90o The current lags the voltage
RL networks – AC signals
L in Series with Signal Path
VOUT
L
VIN
t
R
L from Signal Path to Ground
VOUT
R
VIN
t
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
RC networks – AC signals
L in Series with Signal Path
VOUT
L
VIN
t
R
L from Signal Path to Ground
VOUT
R
VIN
t
L
• For low frequency signals:
– The inductor is low impedance
– Signals can pass the inductor |ZL|=2pfL
• For high frequency signals:
– The inductor is high impedance
– Signals are blocked by the inductor
L in Series with Signal Path
Low Pass Configuration
VOUT
VIN
VIN
6
4
L
2
R
0
-2
0
0.01
0.02
0
0.01
0.02
0.03
0.04
0.05
-4
Z=2.p.f.L
-6
VOUT
6
4
VOUT/VINMAX
Low f
0.96
Medium f 0.76
High f
0.38
2
0
-2
-4
-6
0.03
0.04
0.05
L from Signal Path to Ground
High Pass Configuration
VOUT
VIN
VIN
6
4
R
2
L
0
-2
0
0.01
0.02
0.03
0.04
0.05
0
0.01
0.02
0.03
0.04
0.05
-4
|ZL|=2.p.f.L
-6
VOUT
6
4
VOUT/VINMAX
Low f
0.32
Medium f 0.74
High f
0.92
2
0
-2
-4
-6
Inductor & Resistor
Networks Summary
L in Series with Signal Path
VIN
L
R
VOUT
L from Signal Path to Ground
VIN
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.
VOUT
Capacitor vs. Inductor
Unit Comparison
Capacitor Terms and Symbols
Units
Inductor Terms and Symbols
Units
Electrical Field Strength
E
V/m
Magnetic Field Strength
H
A/m
Charge
Q
As
Coil Flux (=N*F)
y
y
Vs
A
Voltage=-N(dF/dt)=–NDF/Dt
(negative rate of change of flux
times the number of turns)
V
V
L
Vs/A
Current: I=dQ/dt or DQ/Dt
(rate of change of charge)
I
Capacitance:
C As/V
Inductance:
Permittivity of Vacuum
e0=8.854.10-12
e0 As/Vm
Permeability of Vacuum:
m
m0=4p10-7
Energy Stored in a Capacitor:
E = C V2 / 2
E
Constant Current ( I )
Charging a Capacitor
V=It/C
C=Q/V
VAs
J
L= y /I
mm00 Vs/Am
Energy Stored in an Inductor:
E = L I2 / 2
E
Constant Voltage ( V )
Charging an Inductor
I=Vt/L
VAs
J
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
Lamp Experiment
• Turn on an incandescent light bulb and measure the
current
ton
I
1
14V
2
Lamp Experiment
• Turn on an incandescent light bulb and measure the
current
t
• Result:
I
on
1
~ 5.6A
14V
2
~ 600mA
ton
Developing a RC Load
Model For an Incandescent Light Bulb
14V
5.6A
600mA
ton
Light
Bulb
Developing a RC Load
Model For an Incandescent Light Bulb
R1
14V
5.6A
V 14V
R1= =
I 0.6A
600mA
R1= 23.3
ton
Developing a RC Load
Model For an Incandescent Light Bulb
23.3
14V
5.6A
I = I0 exp -t/RC
600mA
ton
Developing a RC Load
Model For an Incandescent Light Bulb
23.3
14V
5.6A
R2
V
23.3 R2 =
I
R2 = 2.80
600mA
ton
Developing a RC Load
Model For an Incandescent Light Bulb
C
23.3
14V
5.6A
2.8
600mA
C = 3.6mF
ton
Simulation of Lamp RC Model
6.0
1
Input Current (A)
5.0
ton
23.3
3.6mF
4.0
14V
2.80
3.0
2
2.0
1.0
0.0
0
50
ton
100
150
Time (ms)
200
250
300
350
Simulation of Lamp RC Model
6.0
1
Input Current (A)
5.0
ton
4.0
23.3
3.6mF
14V
3.0
2.80
2.0
2
1.0
0.0
0
50
ton
100
150
Time (ms)
200
250
300
350
A RC Load Model for
Incandescent 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
2.80
f(T)
2
23.3
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
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
Switching a Relay
• 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
VBattery
S
VR
Relay
VL
IL
Switching On a Relay
S
open
closed
VR
time
VBattery
time
S
VL decays over time
VL
time
+
VR
IL
time
IL = (VR-VL) / R
-
IL
+
VL
-
Switching Off a Relay (1)
S
IL
closed
open
time
VBattery
time
time
S
+
VR
time
-
IL
+
VL
-
Switching Off a Relay (2)
S
IL
VL
VR
closed
open
time
VBattery
time
IL cannot become
zero instantaneously!
VL becomes negative
to force the current to 0A
time
(VL = -L*di/dt)
For VL < 0V,
VR < 0V
S
+
VR
time
-
IL
+
VL
-
Switching Off a Relay (3)
S
closed
open
time
VBattery
Arcing
IL
VL
VR
time
IL cannot go to
zero instantaneously!
VL goes far below ground
to force the current to 0A
time
For VL < 0V,
VR < 0V (R~0)
S
+
VR
time
-
IL
+
VL
-
Switching Off a
Relay No Arcing (1)
S
closed
open
VBattery
time
IL
time
ID
S
+
time
VR
VL
time
ID
IL
VR
time
+
VL
-
-
Switching Off a
Relay No Arcing (2)
S
closed
open
VBattery
time
IL
time
ID
S
Diode turns on and
provides a current path
time
+
VR
VL
time
ID
IL
VR
time
+
VL
-
-
Switching Off a Relay No
Arcing (3)
S
closed
open
VBattery
time
IL
time
S
+
ID
time
VR
If R~0, VL ~ –VD
VL
time
ID
IL
If R~0, VR ~ -VD
VR
time
+
VL
-
-
Switching Off a
Relay No Arcing (4)
S
closed
open
VBattery
time
IL
time
ID
S
+
time
VR
VL
time
ID
IL
VR
time
+
VL
-
-
Switching Off a
Relay No Arcing (5)
S
IL
closed
open
VBattery
time
diL/dt = VL / L
time
S
+
ID
time
VR
If R~0, VL ~ –3VD
VL
time
ID
VR
If R~0, VR ~ -3VD
time
IL
+
VL
-
-
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
Thank you!
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