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ElectromagneticsInductors and the Wave EquationECE 111 Introduction to ECEJake Glower - Week #8Please visit Bison Academy for correspondinglecture notes, homework sets, and solutions

InductorsInductors store energy in a magnetic field.Units are HenriesA coil of wire makes an inductorA long wire carrying current makes an inductorIMagnetic Field

Example 1: Find L and R100 windings of 36 gage wire copper wire ( 1.38 Ohms / meter)Cross sectional area 25mmLength 10mmAir core (relative permeability of 1.000)L µN 2 A/lL 4π 10 7 Hm (100) 2 (25 10 6 m 2 )/(0.01m)L 31.42µHm ΩR (100 windings) 0.02 winding1.38(m) R 2.77Ω

Example 2: Determine the inductance of a copper transmission line:Length 1km, radius 1cm, f 60HzFrom Wikipedia1µL 2π0 l(A B C)2 l l A ln r r 1 12.20 1B r l1 rl 1C 4 r2ρ ωµ2 1 0.1569givingL ductance formulas

VI Characteristics of InductorsInductors store energy in the magnetic field asE 12 L I 2JoulesThe power absorbed is the derivative ofenergyP dEdtWattsI VorP VI L I dIdtCanceling terms gives the VIrelationship for indictors:dIV L dtInductors act as differentiators.-Magnetic Field

Example: The current through a 1H inductor is shown below in red. Sketch thevoltage.Solution: Shown in blue. The derivative (times one) is the voltage.4CurrentSlope 33Slope 0Slope -121Voltage0-1-2-3-4012345Time (seconds)678910

Heat Equation (recap)One Element:V030V1One capacitorOne pot on a stove.V V0 VI 1 CV 1 030 1 1501 .V 1 4V 1 3.33V 0N Elements (heat equation).V i aV i 1 bV i aV i 1I10.01F10V -150

Wave Equation ( Mass-Spring )Coupled second-order differential equationsV̈ i aV i 1 bV i aV i 1Example: Mass-Spring SystemForce Mass * AccelerationForce (F)10 N/mF Kx Mẍ0.2kgXorẍ 50x 5F

Cascade N mass-spring systems togetherF 2 mẍ 210(x 1 x 2 ) 10(x 3 x 2 ) 0.2ẍ 2or10 N/m10 N/m0.2kgẍ 2 10x 1 20x 2 10x 3The same pattern holds for all N nodes.This is an important equation as it describesTransmission lines galloping on windy days,Cars stuck in traffic,Buildings swaying during an earthquake or a wind storm,Bridges oscillating during wind storms,etc.10 N/m0.2kgX10.2kgX2X3

Wave Equation Examples: Galloping Transmission Linehttps://www.youtube.com/watch?v GEGbYRii1d4

Wave Equation Example: Cars in Traffichttps://www.youtube.com/watch?v 19S3OdK6710&t 171s

Wave Equation Example: Tacoma Narrows Bridgehttps://www.youtube.com/watch?v j-zczJXSxnw&t 136s

Wave Equation Example: Swaying Skyscrapershttps://www.youtube.com/watch?v 2t2xxKMN-Ic&t 49s

Wave Equation ( LC Circuits )Also applies to LC CircuitsLong Transmission LinesL: Inductance of wiresC: Parallel plate capacitor (wire &ground)V1L1I1IcCircuit BoardsL: Inductance of circuit traceC: Trace to ground planeL2V2RI2CV3

Equations:.L1I 1 V1 V2.L2I 2 V3 V2.VCV 2 I 1 I 2 R2V1Differentiating. .V2CV̈ 2 I 1 I 2 RSubstituting the inductor equation.V VV VVCV̈ 2 1L 2 3L 2 R212or.1 11 1 1 CV̈ 2 L V 1 L L V 2 L V 3 R V 21122L1L2V2I1IcRI2CV3

Dynamic Response of the Wave Equation.1 2 1 1 V̈ 1 LC V 0 LC V 1 LC V 2 RC V 1.1 2 1 1 V̈ 2 LC V 1 LC V 2 LC V 3 RC V 2.1 1 1 V̈ 3 LC V 2 LC V 3 RC V 3or0.2H0.2HV1.V̈ 1 50V 0 100V 1 50V 2 0.1V 1.V̈ 2 50V 1 100V 2 50V 1 0.1V 2.V̈ 3 50V 2 50V 3 0.1V 3100100V30.1F0.1F0.1F -0.2HV2100100

In MATLABV0V1V2V3 100;0;0;0;dV1 0;dV2 0;dV3 0;V [];dt 0.01;for i 1:300ddV1 50*V0 - 100*V1 50*V2 - 0.01*dV1;ddV2 50*V1 - 100*V2 50*V3 - 0.01*dV2;ddV3 50*V2 - 50*V3- 0.01*dV3;dV1 dV1 ddV1*dt;dV2 dV2 ddV2*dt;dV3 dV3 ddV3*dt;V1 V1 dV1*dt;V2 V2 dV2*dt;V3 V3 dV3*dt;V [V; V1, V2, V3];endt [1:300]' * dt;plot(t,V);xlabel('Time (seconds)');ylabel('Voltage');

It's a little more fun to watch the voltages as the simulation runsV0V1V2V3 100;0;0;0;dV1 0;dV2 0;dV3 0;V [];t 0;dt 0.01;while(t 100)ddV1 50*V0 - 100*V1 50*V2 - 0.01*dV1;ddV2 50*V1 - 100*V2 50*V3 - 0.01*dV2;ddV3 50*V2 - 50*V3- 0.01*dV3;dV1 dV1 ddV1*dt;dV2 dV2 ddV2*dt;dV3 dV3 ddV3*dt;V1 V1 dV1*dt;V2 V2 dV2*dt;V3 V3 00,300]);pause(0.01);V [V; V1, V2, V3];end

30-Stage Transmission Line:.V̈ 1 50V 0 100V 1 50V 2 0.01V 1.V̈ 29V̈ 30. 50V 28 100V 29 50V 30 0.01V 29. 50V 29 50V 30 0.01V 30Creates a traveling waveV(i)Wave travels to the rightV0V30

Reflection Coefficients:Z ZΓ Z 1 Z 010whereZ0 is the impedance of the transmission line andZ1 is the impedance of the load (R30 in this case)Γ 1 : When R30 is large (100 Ohms)Wave Traveling RightReflection Traveling LeftV(20)V30

Γ 1 : R30 is small (0.01 Ohm)An infinite transmission line with a wave traveling to the rightA second wave (the reflection) with the opposite amplitude traveling to the leftWhen they hit, the amplitude at node 20 cancels and the point V(20) remains at zeroReflection Traveling LeftWave Traveling RightV(20)V(30)

Γ 0:Goldilocks R: "Just Right"No ReflectionWave Traveling RightReflection Traveling LeftV(20)V(30)

Matlab Simulation:Adjust R30 until the reflection is zeroReflection Coefficient for R30 0.1:Value in code is 1 / R30CV zeros(30,1);dV zeros(30,1);t 0;dt 0.01;while(t 1000)for i 1:3if (t 1.5) V0 100* ( sin(2*t). 2);else V0 0;endddV(1) 50*V0 - 100*V(1) 50*V(2) - 0.01*dV(1);for i 2:29ddV(i) 50*V(i-1) - 100*V(i) 50*V(i 1) - 0.01*dV(i);endddV(30) 50*V(29) - 50*V(30) - 100*dV(30);for i 1:30dV(i) dV(i) ddV(i)*dt;V(i) V(i) dV(i)*dt;endt t dt;endhold 0.01);end

Why Reflection Coefficients are ImportantWave equation models:Cars stuck in trafficBuildings swaying during an earthquake or a wind storm,Bridges oscillating during wind storms.Transmission lines galloping on windy days,Long electrical transmission lines, andCircuit tracesThe reflections create standing waves, which result inTraffic coming to a standstill, clogging the highwaysBuildings swaying and collapsing if the swaying becomes excessiveBridges oscillating and then collapsingTransmission lines galloping, hitting each other or snapping due to metal fatigue,Large power surges on the nations high-votlage grid, and

Solutions: SkyScrapersAdd mass dampers at the roofhttps://www.youtube.com/watch?v ebx5Y5qOmTMGo to 3:00

Circuits:Reflections can cause false triggersExample: Voltage on PIC board used in ECE 376Similar problem in industry: Higher clock frequency makes traces look like waveguides

Circuit Solutions:Dr. Braaten offers two fairly unique courses at NDSUECE 453. Signal Integrity.ECE 455. Designing for Electromagnetic Compatibility.These courses look at how to model circuits from the standpoint of electromagnetics(i.e. waveguides) and how to design them so they do not transmit radiation or havestanding waves.

ECE Shield RoomFrom physics, if you have a perfectly conductingsphere, there should be no electric fields insidethe sphere.Any fields you do measure must come frominside the sphere.With this room,Measure how much your circuit radiatesMeasure how much your circuit receivesTest before & after: how much techniquestaught in these courses help

SummaryHeat EquationCoupled 1st-order differential equationsRC circuitsWave EquationCoupled 2nd-order differential equationsRLC circuitsThe two behave very differentlyMatlab can be used to solve both types of equationsWhat's important is to get the differential equations rightSolving using numerical methods remains about the same.

Electromagnetics Inductors and the Wave Equation ECE 111 Introduction to ECE Jake Glower - Week #8 . Matlab can be used to solve both types of equations What's important is to get the differential equations right Solvi

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