Inductance 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 ‘inductors’ • 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 frequency and is MAJOR concern at high frequencies 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 “Ground Strap” SMT Capacitor 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 magnetic flux 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 magnetic field 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 Capacitors • Myth or Fact? Bruce Archambeault, PhD 20 What is Capacitance? 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) 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 (ohms) 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 Current Density 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 Parallel • 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 Second Via Around a circle Port 3 ()x, y R: distance between Port 1 and Port 2 in d1 mil Port 1 θ r: radius for all ports in mil d 2 R d: thickness of dielectric 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 2 2 in mil theta: angle as shown in the figure in 2 ⎛ d1 + r ⎞ 2 2 ln ⎜ ⎟ degree μd ⎛ ()()R + r d + r ⎞ μd R + r ln⎜ 1 ⎟ − ⎝ ⎠ Courtesy of Jingook Kim, Jun 4π ⎜ r 3 ()d + r ⎟ 4π ⎛ d + r ⎞ ⎝ 2 ⎠ ln⎜ 2 ⎟ Fan, Jim Drewniak ⎝ r ⎠ Missouri University of Science μd ⎛ (R + r)4 ⎞ and Technology = ln⎜ ⎟ L ⎜ 3 ⎟ equiv 4π ⎝ ()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