1. 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/parallel combinations of elements
RC OP-AMP CIRCUITS
Integration and differentiation circuits

2. CAPACITORS Basic parallel-plates capacitor CIRCUIT REPRESENTATION
First of the energy storage devices to be discussed
Typical Capacitors
Basic parallel-plates capacitor
CIRCUIT REPRESENTATION
NOTICE USE OF PASSIVE SIGN CONVENTION

3. PLATE SIZE FOR EQUIVALENT AIR-GAP CAPACITOR
Normal values of capacitance are small.
Microfarads is common.
For integrated circuits nano or pico farads
are not unusual

4. Linear capacitors obey Coulomb’s law
Linear capacitor circuit representation
Basic capacitance law
Linear capacitors obey Coulomb’s law
C is called the CAPACITANCE of the device and has
units of
One Farad(F)is the capacitance of a device that can
store one Coulomb of charge at one Volt.
EXAMPLE
Voltage across a capacitor of 2 micro
Farads holding 10mC of charge V
Capacitance in Farads, charge in Coulombs
result in voltage in Volts
Capacitors can be dangerous!!!

5. LEARNING BY DOING 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
Linear capacitor circuit representation

6. Capacitance Law
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??
DC or steady state behavior
Voltage across a capacitor
MUST be continuous
A capacitor in steady state acts as an
OPEN CIRCUIT

7. CAPACITOR AS CIRCUIT ELEMENT LEARNING EXAMPLE
Ohm’s Law
The fact that the voltage is defined through
an integral has important implications...

8. CAPACITOR AS ENERGY STORAGE DEVICE
Instantaneous power W
Energy is the integral of power
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.”

9. LEARNING EXAMPLE Energy stored in 0 - 6 msec Charge stored at 3msec
“total energy stored?” ....
“total charge stored?” ...
If charge is in Coulombs
and capacitance in Farads
then the energy is in ….

12. FIND THE ENERGY

13. LEARNING EXTENSION

14. SAMPLE PROBLEM WHAT VARIABLES CAN BE COMPUTED?
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 W
Energy stored over a given time interval J

15. 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 to<t is also known V C
Charge at a given time
Voltage as a function of time
Electric power supplied to capacitor W V
Energy stored in capacitor at a given time J
“Total” energy stored in the capacitor J

16. SAMPLE PROBLEM Given current and capacitance
Compute voltage as a function of time
At minus infinity everything is zero. Since
current is zero for t<0 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....

17. Formal description of a piecewise analytical signal

18. 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

19. Follow passive sign convention
A TIME VARYING MAGNETIC FLUX
INDUCES A VOLTAGE
Induction law
LEARNING by Doing
FOR A LINEAR INDUCTOR THE FLUX IS
PROPORTIONAL TO THE CURRENT
DIFFERENTIAL FORM
OF INDUCTION LAW
THE PROPORTIONALITY CONSTANT, L, IS
CALLED THE INDUCTANCE OF THE COMPONENT
INDUCTANCE IS MEASURED IN UNITS OF
henry (H). DIMENSIONALLY
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

20. Power and Energy stored
Differential form of induction law
Integral form of induction law
A direct consequence of integral form
Current MUST be continuous
A direct consequence of differential form
DC (steady state) behavior
Power and Energy stored W J
Current in Amps, Inductance in Henrys
yield energy in Joules
Energy stored on the interval
Can be positive or negative
“Energy stored at time t”
Must be non-negative.
Passive element!!!

21. LEARNING EXAMPLE L=10mH. FIND THE VOLTAGE
THE DERIVATIVE OF A STRAIGHT LINE IS ITS
SLOPE
ENERGY STORED BETWEEN 2 AND 4 ms J
THE VALUE IS NEGATIVE BECAUSE THE
INDUCTOR IS SUPPLYING ENERGY
PREVIOUSLY STORED

22. Initial energy stored in inductor
SAMPLE PROBLEM
L=0.1H, i(0)=2A. Find i(t), t>0
ENERGY COMPUTATIONS
Energy stored on the interval
Can be positive or negative
Initial energy stored in inductor
“Total energy stored in the inductor”
Energy stored between 0 and 2 sec

23. LEARNING EXAMPLE FIND THE VOLTAGE ACROSS AND THE ENERGY
STORED (AS FUNCTION OF TIME)
FOR ENERGY STORED IN THE INDUCTOR
NOTICE THAT ENERGY STORED AT
ANY GIVEN TIME IS NON NEGATIVE
-THIS IS A PASSIVE ELEMENT-

24. LEARNING EXAMPLE

25. LEARNING EXAMPLE
FIND THE CURRENT
L=200mH

26. FIND THE POWER FIND THE ENERGY L=200mH NOTICE HOW POWER CHANGES SIGN
ENERGY IS NEVER NEGATIVE.
THE DEVICE IS PASSIVE
ENERGY

27. LEARNING EXTENSION
L=5mH
FIND THE VOLTAGE

28. VOLTAGE WAVEFORM
CAPACITOR SPECIFICATIONS
LEARNING EXAMPLE
GIVEN THE VOLTAGE WAVEFORM
DETERMINE THE VARIATIONS IN
CURRENT

29. CURRENT WAVEFORM
INDUCTOR SPECIFICATIONS
LEARNING EXAMPLE
GIVEN THE CURRENT WAVEFORM
DETERMINE THE VARIATIONS IN
VOLTAGE

31. IDEAL AND PRACTICAL ELEMENTS
IDEAL ELEMENTS
MODEL FOR “LEAKY”
INDUCTORS
MODEL FOR “LEAKY”
CAPACITOR
CAPACITOR/INDUCTOR MODELS
INCLUDING LEAKAGE RESISTANCE

32. SERIES CAPACITORS Series Combination of two capacitors
NOTICE SIMILARITY
WITH RESITORS IN
PARALLEL

33. OR WE CAN REDUCE TWO AT A TIME
LEARNING EXAMPLE
DETERMINE EQUIVALENT
CAPACITANCE AND THE
INITIAL VOLTAGE
OR WE CAN REDUCE TWO AT A TIME
ALGEBRAIC SUM OF INITIAL VOLTAGES
POLARITY IS DICTATED BY THE REFERENCE
DIRECTION FOR THE VOLTAGE

34. Two uncharged capacitors are connected as shown.
LEARNING EXAMPLE
Two uncharged capacitors are connected as shown.
Find the unknown capacitance
SAME CURRENT. CONNECTED FOR THE SAME TIME PERIOD
SAME CHARGE ON BOTH CAPACITORS

35. PARALLEL CAPACITORS
LEARNING EXAMPLE

36. LEARNING EXTENSION

37. FIND EQUIVALENT CAPACITANCE
SAMPLE PROBLEM

38. SAMPLE PROBLEM IF ALL CAPACITORS HAVE THE SAME CAPACITANCE VALUE C
DETERMINE THE VARIOUS EQUIVALENT CAPACITANCES

39. Examples of interconnections
All capacitors are equal
with C=8 microFarads

40. SERIES INDUCTORS
LEARNING EXAMPLE

41. PARALLEL INDUCTORS LEARNING EXAMPLE INDUCTORS COMBINE LIKE RESISTORS
CAPACITORS COMBINE LIKE CONDUCTANCES

42. LEARNING EXTENSION ALL INDUCTORS ARE 4mH
CONNECT COMPONENTS BETWEEN NODES
WHEN IN
DOUBT…
REDRAW!
IDENTIFY ALL NODES
PLACE NODES IN CHOSEN LOCATIONS

43. LEARNING EXTENSION
ALL INDUCTORS ARE 6mH
NODES CAN HAVE COMPLICATED SHAPES.
KEEP IN MIND DIFFERENCE BETWEEN
PHYSICAL LAYOUT AND ELECTRICAL
CONNECTIONS
SELECTED LAYOUT

44. L-C

45. THE IDEAL OP-AMP RC OPERATIONAL AMPLIFIER CIRCUITS
INTRODUCES TWO VERY IMPORTANT PRACTICAL CIRCUITS
BASED ON OPERATIONAL AMPLIFIERS
THE IDEAL OP-AMP

46. RC OPERATIONAL AMPLIFIER CIRCUITS -THE INTEGRATOR
IDEAL OP-AMP ASSUMPTIONS

47. RC OPERATIONAL AMPLIFIER CIRCUITS - THE DIFFERENTIATOR
KVL
IDEAL OP-AMP ASSUMPTIONS
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

48. ABOUT ELECTRIC NOISE ALL ELECTRICAL SIGNALS ARE CORRUPTED BY
EXTERNAL, UNCONTROLLABLE AND OFTEN
UNMEASURABLE, SIGNALS. THESE UNDESIRED
SIGNALS ARE REFERRED TO AS NOISE
SIMPLE MODEL FOR A NOISY 60Hz SINUSOID
CORRUPTED WITH ONE MICROVOLT OF 1GHz
INTERFERENCE.
noise
signal
THE DERIVATIVE
noise
signal

49. LEARNING EXTENSION

50. LEARNING EXTENSION

51. LEARNING EXAMPLE SIMPLE CIRCUIT MODEL FOR DYNAMIC RANDOM ACCESS
MEMORY CELL (DRAM)
REPRESENTS CHARGE LEAKAGE
FROM CELL CAPACITOR
NOTICE THE VALUES OF THE
CAPACITANCES
CELL AT THE BEGINNING OF A MEMORY
READ OPERATION
THE ANALYSIS OF THE READ OPERATION
GIVES FURTHER INSIGHT ON THE
REQUIREMENTS
SWITCHED CAPACITOR CIRCUIT

52. 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

53. 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

54. 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