CAPACITANCE AND INDUCTANCE

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CAPACITANCE AND INDUCTANCE
Introduces two passive, energy storing devices
Capacitors and Inductors
LEARNING GOALS
CAPACITORS Store energy in their electric field
(electrostatic energy) Model as circuit element
INDUCTORS Store energy in their magnetic
field Model as circuit element
CAPACITOR AND INDUCTOR COMBINATIONS Series/paralle
l combinations of elements
RC OP-AMP CIRCUITS Integration and
differentiation circuits
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CAPACITORS
First of the energy storage devices to be
discussed
Typical Capacitors
Basic parallel-plates capacitor
NOTICE USE OF PASSIVE SIGN CONVENTION
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Normal values of capacitance are
small. Microfarads is common. For integrated
circuits nano or pico farads are not unusual
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Capacitors can be dangerous!!!
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Capacitors only store and release ELECTROSTATIC
energy. They do not create
LEARNING BY DOING
The capacitor is a passive element and follows
the passive sign convention
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If the voltage varies the charge varies and
there is a displacement current
One can also express the voltage across in terms
of the current
Or one can express the current through in terms
of the voltage across
Differential form of Capacitance law
Integral form of Capacitance law
The mathematical implication of the integral form
is ...
Implications of differential form??
A capacitor in steady state acts as an OPEN
CIRCUIT
Voltage across a capacitor MUST be continuous
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CAPACITOR AS CIRCUIT ELEMENT
LEARNING EXAMPLE
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CAPACITOR AS ENERGY STORAGE DEVICE
W
If t1 is minus infinity we talk about energy
stored at time t2.
If both limits are infinity then we talk about
the total energy stored.
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Energy stored in 0 - 6 msec
Charge stored at 3msec
LEARNING EXAMPLE
total energy stored? ....
total charge stored? ...
If charge is in Coulombs and capacitance in
Farads then the energy is in .
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FIND THE ENERGY
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LEARNING EXTENSION
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SAMPLE PROBLEM
Energy stored at a given time t
J
C
Charge stored at a given time
A
Current through the capacitor
Electric power supplied to capacitor at a given
time
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Energy stored over a given time interval
J
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If the current is known ...
SAMPLE PROBLEM
Current through capacitor
Voltage at a given time t
Voltage at a given time t when voltage at time
toltt is also known
V
C
Charge at a given time
Voltage as a function of time
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Electric power supplied to capacitor
V
J
Energy stored in capacitor at a given time
Total energy stored in the capacitor
J
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SAMPLE PROBLEM
Given current and capacitance
Compute voltage as a function of time
At minus infinity everything is zero.
Since current is zero for tlt0 we have
In particular
Charge stored at 5ms
Total energy stored
Before looking into a formal way to describe the
current we will look at additional questions that
can be answered.
Total means at infinity. Hence
Now, for a formal way to represent piecewise
functions....
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Formal description of a piecewise analytical
signal
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INDUCTORS
NOTICE USE OF PASSIVE SIGN CONVENTION
Circuit representation for an inductor
Flux lines may extend beyond inductor
creating stray inductance effects
A TIME VARYING FLUX CREATES A COUNTER EMF AND
CAUSES A VOLTAGE TO APPEAR AT THE TERMINALS OF
THE DEVICE
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LEARNING by Doing
INDUCTORS STORE ELECTROMAGNETIC ENERGY. THEY MAY
SUPPLY STORED ENERGY BACK TO THE CIRCUIT BUT
THEY CANNOT CREATE ENERGY. THEY MUST ABIDE BY THE
PASSIVE SIGN CONVENTION
Follow passive sign convention
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LEARNING EXAMPLE
FIND THE TOTLA ENERGY STORED IN THE CIRCUIT
In steady state inductors act as short circuits
and capacitors act as open circuits
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L10mH. FIND THE VOLTAGE
LEARNING EXAMPLE
ENERGY STORED BETWEEN 2 AND 4 ms
THE VALUE IS NEGATIVE BECAUSE THE INDUCTOR IS
SUPPLYING ENERGY PREVIOUSLY STORED
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L0.1H, i(0)2A. Find i(t), tgt0
SAMPLE PROBLEM
ENERGY COMPUTATIONS
Initial energy stored in inductor
Total energy stored in the inductor
Energy stored between 0 and 2 sec
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LEARNING EXAMPLE
FIND THE VOLTAGE ACROSS AND THE ENERGY STORED (AS
FUNCTION OF TIME)
NOTICE THAT ENERGY STORED AT ANY GIVEN TIME IS
NON NEGATIVE -THIS IS A PASSIVE ELEMENT-
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LEARNING EXAMPLE
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FIND THE CURRENT
LEARNING EXAMPLE
L200mH
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FIND THE POWER
NOTICE HOW POWER CHANGES SIGN
FIND THE ENERGY
ENERGY IS NEVER NEGATIVE. THE DEVICE IS PASSIVE
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LEARNING EXTENSION
L5mH FIND THE VOLTAGE
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CAPACITOR SPECIFICATIONS
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INDUCTOR SPECIFICATIONS
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IDEAL AND PRACTICAL ELEMENTS
CAPACITOR/INDUCTOR MODELS INCLUDING LEAKAGE
RESISTANCE
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SERIES CAPACITORS
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LEARNING EXAMPLE
DETERMINE EQUIVALENT CAPACITANCE AND THE INITIAL
VOLTAGE
OR WE CAN REDUCE TWO AT A TIME
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Two uncharged capacitors are connected as
shown. Find the unknown capacitance
LEARNING EXAMPLE
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PARALLEL CAPACITORS
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LEARNING EXTENSION
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SAMPLE PROBLEM
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SAMPLE PROBLEM
IF ALL CAPACITORS HAVE THE SAME CAPACITANCE VALUE
C DETERMINE THE VARIOUS EQUIVALENT CAPACITANCES
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Examples of interconnections
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SERIES INDUCTORS
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PARALLEL INDUCTORS
INDUCTORS COMBINE LIKE RESISTORS CAPACITORS
COMBINE LIKE CONDUCTANCES
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LEARNING EXTENSION
ALL INDUCTORS ARE 4mH
CONNECT COMPONENTS BETWEEN NODES
WHEN IN DOUBT REDRAW!
IDENTIFY ALL NODES
PLACE NODES IN CHOSEN LOCATIONS
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LEARNING EXTENSION
ALL INDUCTORS ARE 6mH
NODES CAN HAVE COMPLICATED SHAPES. KEEP IN MIND
DIFFERENCE BETWEEN PHYSICAL LAYOUT AND
ELECTRICAL CONNECTIONS
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RC OPERATIONAL AMPLIFIER CIRCUITS
INTRODUCES TWO VERY IMPORTANT PRACTICAL
CIRCUITS BASED ON OPERATIONAL AMPLIFIERS
THE IDEAL OP-AMP
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RC OPERATIONAL AMPLIFIER CIRCUITS -THE INTEGRATOR
IDEAL OP-AMP ASSUMPTIONS
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RC OPERATIONAL AMPLIFIER CIRCUITS - THE
DIFFERENTIATOR
IF R1 COULD BE SET TO ZERO WE WOULD HAVE AN IDEAL
DIFFERENTIATOR. IN PRACTICE AN IDEAL
DIFFERENTIATOR AMPLIFIES ELECTRIC NOISE AND DOES
NOT OPERATE. THE RESISTOR INTRODUCES A
FILTERING ACTION. ITS VALUE IS KEPT AS SMALL
AS POSSIBLE TO APPROXIMATE A DIFFERENTIATOR
DIFFERENTIATE
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ABOUT ELECTRIC NOISE
ALL ELECTRICAL SIGNALS ARE CORRUPTED BY EXTERNAL,
UNCONTROLLABLE AND OFTEN UNMEASURABLE, SIGNALS.
THESE UNDESIRED SIGNALS ARE REFERRED TO AS NOISE
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LEARNING EXTENSION
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LEARNING EXTENSION
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APPLICATION EXAMPLE
CROSS-TALK IN INTEGRATED CIRCUITS

• REDUCE CROSSTALK BY
• Reducing C12
• Increasing C2

COST? EXTRA SPACE BY GROUND WIRE
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LEARNING EXAMPLE
SIMPLE CIRCUIT MODEL FOR DYNAMIC RANDOM
ACCESS MEMORY CELL (DRAM)
NOTICE THE VALUES OF THE CAPACITANCES
THE ANALYSIS OF THE READ OPERATION GIVES FURTHER
INSIGHT ON THE REQUIREMENTS
SWITCHED CAPACITOR CIRCUIT
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CELL READ OPERATION
IF SWITCH IS CLOSED BOTH CAPACITORS MUST HAVE THE
SAME VOLTAGE
ASSUMING NO LOSS OF CHARGE THEN THE CHARGE BEFORE
CLOSING MUST BE EQUAL TO CHARGE AFTER CLOSING
Even at full charge the voltage variation is
small. SENSOR amplifiers are required
After a READ operation the cell must be refreshed
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LEARNING EXAMPLE
FLIP CHIP MOUNTING
IC WITH WIREBONDS TO THE OUTSIDE
GOAL REDUCE INDUCTANCE IN THE WIRING AND REDUCE
THE GROUND BOUNCE EFFECT
A SIMPLE MODEL CAN BE USED TO DESCRIBE GROUND
BOUNCE
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MODELING THE GROUND BOUNCE EFFECT
IF ALL GATES IN A CHIP ARE CONNECTED TO A SINGLE
GROUND THE CURRENT CAN BE QUITE HIGH AND THE
BOUNCE MAY BECOME UNACCEPTABLE
USE SEVERAL GROUND CONNECTIONS (BALLS) AND
ALLOCATE A FRACTION OF THE GATES TO EACH BALL
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LEARNING BY DESIGN
POWER OUTAGE RIDE THROUGH CIRCUIT
CAPACITOR MUST MAINTAIN AT LEAST 2.4V FOR AT
LEAST 1SEC.
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DESIGN EXAMPLE
DESIGN AN OP-AMP CIRCUIT TO REALIZE THE EQUATION