Inductance and Capacitance

Objectives 1. Find the current (voltage) for a capacitance or inductance given the voltage (current) as a function of time. 2. Compute the capacitance of a parallel-plate capacitor. 3. Compute the stored energy in a capacitance or inductance. 4. Describe typical physical construction of capacitors and inductors

Capacitors and Capacitance Capacitance – the ability of a component to store energy in the form of an electrostatic charge Capacitor – is a component designed to provide a specific measure of capacitance

Capacitors and Capacitance Capacitor Construction Plates Dielectric

Capacitor Charge Electrostatic Charge Develops Electrostatic Field Stores energy Insert Figure 12.2

Capacitor Discharge Insert Figure 12.3

Capacitors and Capacitance Capacity – amount of charge that a capacitor can store per unit volt applied where C = the capacity (or capacitance) of the component, in coulombs per volt Q = the total charge stored by the component V = the voltage across the capacitor corresponding to the value of Q

Capacitance Insert Figure 12.4

Capacitance Unit of Measure – farad (F) = 1 coulomb per volt (C/V) Capacitor Ratings Most capacitors rated in the picofarad (pF) to microfarad (F) range Capacitors in the millifarad range are commonly rated in thousands of microfarads: 68 mF = 68,000 F Tolerance Usually fairly poor Variable capacitors used where exact values required

Capacitors and Capacitance Physical Characteristics of Capacitors where C = the capacity of the component, in farads (8.85 X 10-12) = the permittivity of a vacuum, in farads per meter (F/m) or expressed as o r = the relative permittivity of the dielectric A = the area of either plate d = the distance between the plates (i.e., the thickness of the dielectric)

Capacitance of the Parallel-Plate Capacitor

Capacitance For DC It acts as a voltage source

Voltage in terms of Current , q(to) is the initial charge

Stored Energy

Series Capacitors Series Capacitors Where CT = the total series capacitance Cn = the highest-numbered capacitor in the string

Parallel Capacitors Connecting Capacitors in Parallel where Cn = the highest-numbered capacitor in the parallel circuit

Inductance Unit of Measure – Henry (H) Inductance is measured in volts per rate of change in current When a change of 1A/s induces 1V across an inductor, the amount of inductance is said to be 1 H Insert Figure 10.5

Inductance Induced Voltage where vL = the instantaneous value of induced voltage L = the inductance of the coil, measured in henries (H) = the instantaneous rate of change in inductor current (in amperes per second)

Inductance For DC It acts as a short circuit

Current in terms of Voltage

Stored Energy

Inductance Insert Figure 10.8

Connecting Inductors in Series Series-Connected Coils where Ln = the highest-numbered inductor in the circuit

Characteristic of Capacitor and Inductor Under AC Excitation

Connecting Inductors in Parallel Parallel-Connected Coils where Ln = the highest-numbered inductor in the circuit

Alternating Voltage and Current Characteristics AC Coupling and DC Isolation: An Overview DC Isolation – a capacitor prevents flow of charge once it reaches its capacity Insert Figure 12.6

AC Coupling and DC Isolation AC Coupling – DC offset is blocked Insert Figure 12.7

Capacitor Current where iC = the instantaneous value of capacitor current C = the capacity of the component(s), in farads = the instantaneous rate of change in capacitor voltage

Alternating Voltage and Current Characteristics Sine-Wave Values of reaches its maximum value when v = 0 Insert Figure 12.8

The Phase Relationship Between Capacitor Current and Voltage Current leads voltage by 90° Voltage lags current by 90°

Capacitive Reactance (XC) Series and Parallel Values of XC Insert Figure 12.18

Capacitive Reactance (XC) Capacitor Resistance Dielectric Resistance – generally assumed to be infinite Effective Resistance – opposition to current, also called capacitive reactance (XC) Insert Figure 12.15

Capacitive Reactance (XC) Calculating the Value of XC

Capacitive Reactance (XC) XC and Ohm’s Law Example: Calculate the total current below Insert Figure 12.17

The Phase Relationship Between Inductor Current and Voltage Sine-Wave Values of reaches its maximum value when i = 0 Insert Figure 10.9

The Phase Relationship Between Inductor Current and Voltage Voltage leads current by 90° Current lags voltage by 90°

Inductive Reactance (XL) Inductor Opposes Current Insert Figure 10.15

Inductive Reactance (XL) Inductive Reactance (XL) – the opposition (in ohms) that an inductor presents to a changing current Calculating the Value of XL

Inductive Reactance (XL) XL and Ohm’s Law Example: Calculate the total current below

Capacitive Versus Inductive Phase Relationships Voltage (E) in inductive (L) circuits leads current (I) by 90° (ELI) Current (I) in capacitive (C ) circuits leads voltage (E) by 90° (ICE)

Alternating Voltage and Current Characteristics Insert Figure 12.10

Euler’s identity In Euler expression, A cos t = Real (Ae j t ) Figure 4.23 In Euler expression, A cos t = Real (Ae j t )

( it is called the impedance of a capacitor) ( it is called the impedance of an inductor)

The impedance element Figure 4.29

Impedances of R, L, and C in the complex plane Figure 4.33

Figure 4.37

An AC circuit Figure 4.41

AC equivalent circuits Figure 4.44

Rules for impedance and admittance reduction Figure 4.45