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

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

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

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

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

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

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

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

EVOLUTION OF THE TRANSIENT AND INTERPRETATION OF THE TIME CONSTANT Tangent reaches x-axis in one time constant Drops 0.632 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

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

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**

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.

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

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