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and Partial Inductance What's it all mean?

Bruce Archambeault, PhD

IEEE Fellow, IBM Distinguished Engineer

[email protected] Inductance

• Probably the most misunderstood concept in electrical engineering – Do not confuse ‘inductance’ with ‘’ • Common Usage – Self inductance – Loop inductance – Mutual inductance – Equivalent inductance – Partial inductance – Partial self inductance – Partial mutual inductance – Apparent inductance

Bruce Archambeault, PhD 2 Inductance

• Current flow through metal = inductance! • Fundamental element in EVERYTHING • Loop area first order concern • Inductive impedance increases with and is MAJOR concern at high

X L = 2πfL Bruce Archambeault, PhD 3 Current Loop = Inductance

Courtesy of Elya Joffe

Bruce Archambeault, PhD 4 Inductance Definition

• Faraday’s Law ∂B E ⋅dl = − ⋅dS ∫∫∫∂t

• For a simple rectangular loop

Area = A ∂B

V V = −A B ∂t

Bruce Archambeault, PhD 5 Given the Definition of Inductance

• Do these have inductance?

PCB Via “ Strap” SMT Not until return path for current is identified!

Bruce Archambeault, PhD 6 Self Inductance

• Isolated circular loop ⎛ 8a ⎞ ⎜ ⎟ L ≈ μ0a ln⎜ − 2⎟ ⎝ r0 ⎠ • Isolated rectangular loop

2μ a ⎛ p + 1+ p2 1 1 ⎞ L = 0 ln⎜ + −1+ 2 − 1+ p2 ⎟ π ⎜ p p ⎟ ⎝ 1+ 2 ⎠ length of side Note that inductance is directly influenced p = by loop AREA and less influenced by wire radius conductor size!

Bruce Archambeault, PhD 7 Mutual Inductance

Φ 2 = M 21I1 How much is induced in loop #2 from a Φ 2 current in loop #1? M 21 = I1

Loop #2 Loop #1 r Φ2 = B1(r)⋅nˆ dS2 ∫S 2

Bruce Archambeault, PhD 8 Flux from Current in Loop #1

Bruce Archambeault, PhD 9 Flux from Current in Loop #1

Bruce Archambeault, PhD 10 Change in mutual inductance with spacing 2

X: 24 Y: 1.835

1.5 The drops off rapidly, so then does

1 the mutual inductance X: 100 Y: 0.7312

Mutual Inductance (nH) Inductance Mutual 0.5

X: 500 X: 1000 Y: 0.02507 Y: 0.01955 0 0 200 400 600 800 1000 Spacing between the coils(mils)

Bruce Archambeault, PhD 11 Mutual Inductance

Loop #2 Loop #1 Less loop area in loop #2 means less magnetic flux in loop #2 and less mutual inductance

Loop #2 Less loop area perpendicular to Loop #1 the magnetic field in loop #2 means less magnetic flux in loop #2 and less mutual inductance

Bruce Archambeault, PhD 12 Partial Inductance

• We now know that a loop of current has inductance • We now know that there must be a complete loop to have inductance • But where do we place this inductance in a circuit?

Bruce Archambeault, PhD 13 Zero-to-One Transition Where’s the Inductance Go??

Power Supply

And how could you possibly calculate it?

Courtesy of Dr. Clayton Paul Bruce Archambeault, PhD 14 Total Loop Inductance from Partial Inductance

L total=Lp1+ Lp2 + Lp3 + Lp4 –2Mp1-3 –2Mp2-4

Lp2

Mp2-4 Mp1-3 L Lp1 p3

L p4 Courtesy of Dr. Clayton Paul Bruce Archambeault, PhD 15 Partial Inductance

• Simply a way to break the overall loop into pieces in order to find total inductance

L2

L1 L3

L4 L total=Lp11+ Lp22 + Lp33 + Lp44 -2Lp13 -2Lp24

Bruce Archambeault, PhD 16 Important Points About Inductance

• Inductance is everywhere • Loop area most important • Inductance is everywhere

Bruce Archambeault, PhD 17 Example Decoupling Capacitor Mounting

• Keep vias as close to capacitor pads as possible!

Via Separation Inductance Depends on Loop AREA Height above Planes

Bruce Archambeault, PhD 18 Via Configuration Can Change Inductance

SMT Capacitor

Via The “Good” Best Capacitor Pads

The “Bad” Better The “Ugly”

Really “Ugly”

Bruce Archambeault, PhD 19 High Frequency

• Myth or Fact?

Bruce Archambeault, PhD 20 What is ?

Q Q = CV C = V • Amount of charge • Capacitance is the stored is dependant ability of a structure to on the size of the hold charge capacitance (and (electrons) for a given ) voltage

Consider a capacitor as a bucket holding lot’s of electrons!

Bruce Archambeault, PhD 21 Comparison of Decoupling Capacitor Impedance 100 mil Between Vias & 10 mil to Planes 1000

1000pF 100 0.01uF 0.1uF 1.0uF 10

1 Impedance ()

0.1

0.01 1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10 Frequency (Hz) Bruce Archambeault, PhD 22 0603 Size Cap Typical Mounting

9 mils 9 mils 20 mils

10 mils* 10 mils* Via Barrel 10 mils

60 mils

108 mils minimum 128 mils typical *Note: Minimum distance is 10 mils but more typical distance is Bruce Archambeault, PhD20 mils 23 0402 Size Cap Typical Mounting

8 mils 8 mils 20 mils

10 mils* 10 mils* Via Barrel 10 mils

40 mils

86 mils minimum 106 mils typical *Note: Minimum distance is 10 mils but more typical distance is Bruce Archambeault, PhD20 mils 24 Connection Inductance for Typical Capacitor Configurations

Distance into 0805 0603 0402 board typical/minimum typical/minimu typical/minimum to planes (mils) (148 mils m (106 mils between via (128 mils between via barrels) between via barrels) barrels) 10 1.2 nH 1.1 nH 0.9 nH 20 1.8 nH 1.6 nH 1.3 nH 30 2.2 nH 1.9 nH 1.6 nH 40 2.5 nH 2.2 nH 1.9 nH 50 2.8 nH 2.5 nH 2.1 nH 60 3.1 nH 2.7 nH 2.3 nH 70 3.4 nH 3.0 nH 2.6 nH 80 3.6 nH 3.2 nH 2.8 nH 90 3.9 nH 3.5 nH 3.0 nH 100 4.2 nH 3.7 nH 3.2 nH Bruce Archambeault, PhD 25 Connection Inductance for Typical Capacitor Configurations with 50 mils from Capacitor Pad to Via Pad

0805 0603 0402 Distance into (208 mils (188 mils (166 mils board between via between via between via to planes (mils) barrels) barrels) barrels) 10 1.7 nH 1.6 nH 1.4 nH 20 2.5 nH 2.3 nH 2.0 nH 30 3.0 nH 2.8 nH 2.5 nH 40 3.5 nH 3.2 nH 2.8 nH 50 3.9 nH 3.5 nH 3.1 nH 60 4.2 nH 3.9 nH 3.5 nH 70 4.5 nH 4.2 nH 3.7 nH 80 4.9 nH 4.5 nH 4.0 nH 90 5.2 nH 4.7 nH 4.3 nH 100 5.5 nH 5.0 nH 4.6 nH

Bruce Archambeault, PhD 26 PCB Example for Return Current Impedance

Trace

GND Plane

22” trace 10 mils wide, 1 mil thick, 10 mils above GND plane

Bruce Archambeault, PhD 27 PCB Example for Return Current Impedance

Trace

GND Plane

Shortest DC path

For longest DC path, current returns under trace Bruce Archambeault, PhD 28 MoM Results for Frequency = 1 KHz

Bruce Archambeault, PhD 29 MoM Results for Current Density Frequency = 1 MHz

Bruce Archambeault, PhD 30 U-shaped Trace Inductance PowerPEEC Results

0.6

0.55

0.5

0.45

0.4

0.35

0.3 inductance (uH)

0.25

0.2

0.15

0.1 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08 Frequency (Hz) Bruce Archambeault, PhD 31 Two Wires in

• Reduce inductance by factor of two? NO!

2 Lp1Lp2 − M p LParallel = Lp1 + Lp2 − 2M p

Lp1 = Lp2 = Lp L + M L = p p Parallel 2 Only if parallel wires are FAR APART!

Courtesy of Dr. Clayton Paul Bruce Archambeault, PhD 32 Let’s Apply this to Decoupling Capacitors

• Equivalent inductance – Two capacitors vs one capacitor – Relative location of two capacitors – Use via between planes as ideal capacitor

Bruce Archambeault, PhD 33 What Happens if a 2nd Decoupling Capacitor is placed near the First Capacitor? Via #2 Moved in arc Via #1 around Observation point while distance maintaining 500 mil distance to observation point Observation Point 500 mils

Bruce Archambeault, PhD 34 Via Around a circle

Port 3 ()x, y R: distance between 1 and Port 2 in d1 mil Port 1 θ r: radius for all ports in mil

d 2 R d: thickness of layer in mil d1: distance between Port 3 and Port 1 in mil d = R 1 Port 2 d2: distance between Port 2 and Port 3 θ d = 2R sin μ 2 2 in mil π theta: angle as shown in the figure in 2 ⎛ d1 + r ⎞ 2 2 ln ⎜ ⎟ degree d ⎛ ()()R + r d + r ⎞ μ d R + r ln⎜ 1 ⎟ − ⎝ ⎠ Courtesy of Jingook Kim, Jun 4 ⎜ r 3 d()+ r ⎟ 4π ⎛ d + r ⎞ ⎝ 2 ⎠ ln⎜ 2 ⎟ Fan, Jim Drewniak μ ⎝ r ⎠ Missouri University of Science πd ⎛ (R + r)4 ⎞ and Technology = ln⎜ ⎟ L ()⎜ 3 ⎟ equiv 4 ⎝ 2R sin(θ / 2) + r r ⎠ Bruce Archambeault, PhD 35 Effective Inductance for Various Distances to Decoupling Capacitor With Second Capacitor (Via) Equal Distance Around Circle Plane Seperation = 35 mil -- Via Diameter = 20 mil 2100 2000 1900 1800 1700 250 mil 1600 500mil 1500 750 mil 1400 1000 mil 1300 1200

Inductnace (pH) Inductnace 1100 1000 900 800

700 600 500 050100150200 Angle (degrees) Bruce Archambeault, PhD 36 Effective Inductance for Various Distances to Decoupling Capacitor With Second Capacitor (Via) Equal Distance Around Circle Plane Seperation = 10 mil -- Via Diameter = 20 mil 500

450

400

350

300

250

200 Inductnace (pH) Inductnace 500mil 150 250 mil 750 mil 100 1000 mil

50

0 050100150200 Angle (degrees) Bruce Archambeault, PhD 37 Effective Inductance for Various Distances to Decoupling Capacitor With Second Capacitor (Via) Equal Distance Around Circle Plane Seperation = 5 mil -- Via Diameter = 20 mil 400

350

300 500mil 250 mil 250 750 mil 1000 mil 200

Inductnace (pH) Inductnace 150

100

50

0 050100150200 Angle (degrees) Bruce Archambeault, PhD 38 Understanding Inductance Effects and Proximity 1 via 2 via with degree 30°

10cm 10mm

20cm

10cm 2 via with degree 90° 2 via with degree 180°

20cm

Bruce Archambeault, PhD 39 Current Density

[m] [m] A/m2 A/m2

[m] [m] [m] [m] A/m2 A/m2

[m] [m]

Bruce Archambeault, PhD 40 Current Density in Planes 0.12 0.12 8 8 0.115 6 0.115 8 4 1 1 8 6 2 6 4 6 8 2 6 2 1 1 4 4 1 24 0 65484 3 0 5 0.11 342 6 6 3 6 62 6 432 2 8 6 4 0 8 78 567 0 0.11 4 4 2 4 5 80 4 2 48 1 0 83 0 1 6 2 8 4 8 2 07 6 2 4 5 0.105 6 0.105 64 64 4 4 0 2 6 8 6 6 1 4 0.1 6 20 0.1 8 8 1 78 0 0 80 7 56484 8 7 5648 2 2 402 8 24 32 0.095 243 0.095 16 16 8 0.09 8 0.09 8

0.085 0.085

0.08 0.08 0.08 0.0850.09 0.095 0.1 0.1050.11 0.1150.12 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12

0.12 0.12

0.115 0.115 8 8 8 1 23426 24 5 4 56 1 0.11 60 480 0.11 4 480 4 6 4 6 4 6 67 4 3 487 85 2 1 0 562 64 2 64 6 8 8 1 6 0.105 0.105 1 3 8 4842 2 454 6 560 4 40 6 068 264 1 8 328 6 40 5 2 02 456 8 3 7 0 8 2 0 4 4 4 0.1 2 4 0.1 2 8 68 3 80 5 4 7 6 7 6 4 0 2 5 2 8 64 8 72 4320 72 48 16 6 0.095 24 0.095 1 16 8 428 0.09 0.09 4 6740 2 8 1 8 564 30 8 6 56 424 0.085 0.085 8

0.08 0.08 0.08 0.0850.09 0.095 0.1 0.1050.11 0.1150.12 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12

Bruce Archambeault, PhD 41 Effect of Plane width on Inductance

Case1 : 10 inches

Case2 : 5 inches

Port1 Port2 Case3 : 2 inches 1 inch

Bruce Archambeault, PhD 42 Loop Inductance is Affected by Plane Width Case2 : 5 inches

Case1 : 10 inches

~ 330pH

Case2 : 2 inches

~ 250pH ~ 560pH

Bruce Archambeault, PhD 43 Current Spreads in a Plane

Narrower planes means the multiple current paths are limited therefore effect of mutual inductance between parallel paths increases!

Bruce Archambeault, PhD 44 Observations

• Added via (capacitor) does not lower effective inductance to 70-75% of original single via case • Thicker dielectric results in higher inductance • Normalizing inductance to single via case gives same curve for all dielectric thicknesses

Bruce Archambeault, PhD 45 Summary • Inductance has meaning only for current loops • Size of the loop has the most impact on amount of inductance • Current density also impact inductance • Partial inductance is a very useful concept to understand which portions of the loop have the largest impact on loop inductance

Bruce Archambeault, PhD 46