1. CAPACITANCE AND INDUCTANCE
Introduces two passive, energy storing devices: Capacitors and Inductors
CAPACITORS
Store energy in their electric field (electrostatic energy)
Model as circuit element
INDUCTORS
Store energy in their magnetic field
Model as circuit element

2. CAPACITORS Typical Capacitors Basic parallel-plates capacitor
CIRCUIT REPRESENTATION
NOTICE USE OF PASSIVE SIGN CONVENTION

3. 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

4. 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

5. Follow passive sign convention
A TIME VARYING MAGNETIC FLUX
INDUCES A VOLTAGE
Induction law
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

6. 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

8. AN INTRODUCTION With the switch on the left the capacitor receives
INDUCTORS AND CAPACITORS CAN STORE ENERGY. UNDER SUITABLE CONDITIONS THIS ENERGY
CAN BE RELEASED. THE RATE AT WHICH IT IS RELEASED WILL DEPEND ON THE PARAMETERS
OF THE CIRCUIT CONNECTED TO THE TERMINALS OF THE ENERGY STORING ELEMENT
With the switch on the left the capacitor receives
charge from the battery.
Switch to the right
and the capacitor
discharges through
the lamp

9. GENERAL RESPONSE: FIRST ORDER CIRCUITS
Including the initial conditions
the model for the capacitor
voltage or the inductor current
will be shown to be of the form
THIS EXPRESSION ALLOWS THE COMPUTATION
OF THE RESPONSE FOR ANY FORCING FUNCTION.
WE WILL CONCENTRATE IN THE SPECIAL CASE
WHEN THE RIGHT HAND SIDE IS CONSTANT AND to=0 SO THAT
Solving the differential equation
using integrating factors, one
tries to convert the LHS into an
exact derivative

10. EVOLUTION OF THE TRANSIENT AND INTERPRETATION OF
THE TIME CONSTANT
Tangent reaches x-axis in one time constant
Drops of initial
value in one time constant
With less than 2% error
transient is zero
beyond this point
A QUALITATIVE VIEW:
THE SMALLER THE THE TIME
CONSTANT THE FASTER THE
TRANSIENT DISAPPEARS

11. THE TIME CONSTANT
The following example illustrates
the physical meaning of time
constant
With less than 1%
error the transient
is negligible after
five time constants
Charging a capacitor
The model
The solution can be shown to be
transient
For practical purposes the
capacitor is charged when the
transient is negligible

12. ANALYSIS OF CIRCUITS WITH ONE ENERGY STORING ELEMENT
CONSTANT INDEPENDENT SOURCES
A STEP-BY-STEP APPROACH
THIS APPROACH RELIES ON THE KNOWN FORM OF THE SOLUTION
BUT FINDS THE CONSTANTS USING BASIC CIRCUIT
ANALYSIS TOOLS AND FORGOES THE DETERMINATION OF THE
DIFFERENTIAL EQUATION MODEL
**Go to handwritten notes**

13. THE STEPS
STEP 5: DETERMINE THE TIME
CONSTANT
STEP 1. THE FORM OF THE SOLUTION
STEP 2: DRAW THE CIRCUIT IN STEADY
STATE PRIOR TO THE SWITCHING AND
DETERMINE CAPACITOR VOLTAGE OR
INDUCTOR CURRENT
STEP 3: DRAW THE CIRCUIT AT 0+
THE CAPACITOR ACTS AS A VOLTAGE
SOURCE. THE INDUCTOR ACTS AS A
CURRENT SOURCE.
DETERMINE THE VARIABLE AT t=0+
STEP 6: DETERMINE THE
CONSTANTS K1, K2
STEP 4: DRAW THE CIRCUIT IN STEADY
STATE AFTER THE SWITCHING AND
DETERMINE THE VARIABLE IN STEADY
STATE.

14. LEARNING EXAMPLE
USE A CIRCUIT VALID FOR t=0+.
THE CAPACITOR ACTS AS SOURCE
USE CIRCUIT IN STEADY STATE
PRIOR TO THE SWITCHING
KVL
NOTES FOR INDUCTIVE CIRCUIT
(1)DETERMINE INITIAL INDUCTOR
CURRENT IN STEP 2
(2)FOR THE t=0+ CIRCUIT REPLACE
INDUCTOR BY A CURRENT SOURCE

15. USE CIRCUIT IN STEADY STATE
AFTER SWITCHING
NOTE: FOR INDUCTIVE CIRCUIT
ORIGINAL CIRCUIT