Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Engineering 43 Chp 6.2Inductors Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

Capacitance & Inductance • Introduce Two Energy STORING Devices • Capacitors • Inductors • Outline • 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 • Electronic Integration & Differentiation

Second of the Energy-Storage Devices Basic Physical Model: The Inductor Ckt Symbol • Details of Physical Operation Described in PHYS 4B or ENGR45 • Note the Use of the PASSIVE Sign Convention

Inductors are Typically Fabricated by Winding Around a Magnetic (e.g., Iron) Core a LOW Resistance Wire Applying to the Terminals a TIME VARYING Current Results in a “Back EMF” voltage at the connection terminals Physical Inductor • Some Real Inductors

From Physics, recall that a time varying magnetic flux, , Induces a voltage Thru the Induction Law Inductance Defined • Where the Constant of Proportionality, L, is called the INDUCTANCE • L is Measured in Units of “Henrys”, H • 1H = 1 V•s/Amp • Inductors STORE electromagnetic energy • They May Supply Stored Energy Back To The Circuit, But They CANNOT CREATE Energy • For a Linear Inductor The Flux Is Proportional To The Current Thru it

Inductors Cannot Create Energy; They are PASSIVE Devices All Passive Devices Must Obey the Passive Sign Convention Inductance Sign Convention

Recall the Circuit Representation Inductor Circuit Operation • Separating the Variables and Integrating Yields the INTEGRAL form • In a development Similar to that used with caps, Integrate − to t0 for an Alternative integral Law • Previously Defined the Differential Form of the Induction Law

From the Differential Law Observe That if iL is not Continuous, diL/dt → , and vL must also →  This is NOT physically Possible Thus iL must be continuous Inductor Model Implications • Consider Now the Alternative Integral law • If iL is constant, say iL(t0), then The Integral MUST be ZERO, and hence vL MUST be ZERO • This is DC Steady-State Inductor Behavior • vL = 0 at DC • i.e; the Inductor looks like a SHORT CIRCUITto DC Potentials

From the Definition of Instantaneous Power Inductor: Power and Energy • Time Integrate Power to Find the Energy (Work) • Subbing for the Voltage by the Differential Law • Units Analysis • J = H x A2 • Energy Stored on Time Interval • Again By the Chain Rule for Math Differentiation • Energy Stored on an Interval Can be POSITIVE or NEGATIVE

1 = 2 Li ( t ) 0 1 L 2 1 = 2 w ( t ) Li ( t ) L 2 Inductor: P & W cont. • In the Interval Energy Eqn Let at time t1 • Then To Arrive At The Stored Energy at a later given time, t • Thus Observe That the Stored Energy is ALWAYS Positive In ABSOLUTE Terms as iL is SQUARED • ABSOLUTE-POSITIVE-ONLY Energy-Storage is Characteristic of a PASSIVE ELEMENT

Given The iL Current WaveForm, Find vL for L = 10 mH Example • The Derivative of a Line is its SLOPE, m • Then the Slopes • And the vL Voltage • The Differential Reln

The Energy Stored between 2 & 4 mS Example  Power & Energy • The Value Is Negative Because The Inductor SUPPLIES Energy PREVIOUSLY STORED with a Magnitude of 2 μJ • The Energy Eqn • Running the No.s

Student Exercise • Let’s Turn on the Lights for 5-7 minutes to allow YOU to solve an INTEGRAL Reln Problem • Given The Voltage Wave Form Across L • Then Find iL For • L = 0.1 H • i(0) = 2A • t > 0

Given The Voltage Wave Form Across L , Find iL if L = 0.1 H i(0) = 2A Numerical Example • The PieceWise Function • A Line Followed by A Constant; Plotting • The Integral Reln

The Current Characteristic Numerical Example - Energy • The Initial Stored Energy • The “Total Stored Energy” • Energy Stored between 0-2 • The Energy Eqn • → Consistent with Previous Calculation • Energy Stored on Interval Can be POS or NEG

Example • Find The Voltage Across And The Energy Stored (As Function Of Time) • Notice That the ABSOLUTE Energy Stored At Any Given Time Is Non Negative (by sin2) • i.e., The Inductor Is A PASSIVE Element- • For The Energy Stored

L = 10 mH; Find the Voltage

L = 5 mH; Find the Voltage

Capacitors Standard Range 1 pF to 50 mF (50000 µF) Working (DC) Voltage Range  6.3-500 Vdc Standard Tolerances ± 5% ± 10% ± 20% C & L Specifications • Inductors • Standard Range 1 nH to 100 mH • Current Rating  10mA-1A • Standard Tolerances • ± 5% • ± 10% • As Wire Coils, Inductors also have a RESISTANCE Specification

VOLTAGE WAVEFORM dv = C i ( t ) C dt Cap SpecSensitivity • Given The Voltage Waveform Find the Current Variation Due to Cap Spec Tolerance 240 mA • Use

Given The Current Waveform Find the Voltage Variation Due to L Spec-Tolerance CURRENT WAVEFORM Ind SpecSensitivity • Use

WhiteBoard Work • Let’s Work This Problem • Find: v(t), tmax for imax, tmin for vmin

% Bruce Mayer, PE * 14Mar10 % ENGR43 * Inductor_Lec_WhtBd_Prob_1003.m % t = linspace(0, 1); % iLn = 2*t; iexp = exp(-4*t); plot(t, iLn, t, iexp), grid disp('Showing Plot - Hit ANY KEY to Continue') pause % i = 2*t.*exp(-4*t); plot(t, iLn, t, iexp, t, i),grid disp('Showing Plot - Hit ANY KEY to Continue') pause plot(t, i), grid disp('Showing Plot - Hit ANY KEY to Continue') pause vL_uV =100*exp(-4*t).*(1-4*t); plot(t, vL_uV) i_mA = i*1000; plot(t, vL_uV, t, i_mA), grid disp('Showing Plot - Hit ANY KEY to Continue') pause % % find mins with fminbnd [t_vLmin vLmin] = fminbnd(@(t) 100*exp(-4*t).*(1-4*t), 0,1) % must "turn over" current to create a minimum i_mA_TO = -i*1000; plot(t, i_mA_TO), grid disp('Showing Plot - Hit ANY KEY to Continue') pause [t_iLmin iLmin_TO] = fminbnd(@(t) -1000*(2*t.*exp(-4*t)), 0,1); t_iLmin iLmin = -iLmin_TO plot(t, vL_uV, t, i_mA, t_vLmin, vLmin, 'p', t_iLmin,iLmin,'p'), grid ByMATLAB

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege