inductance and partial inductance what's it all...

Bruce Archambeault, PhD

IEEE Fellow, IBM Distinguished Engineer

[email protected]

Inductance and Partial Inductance What's it all

mean?

Bruce Archambeault, PhD 2

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


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

fLX L π2=


Current Loop = Inductance

Courtesy of Elya Joffe


Inductance Definition

• Faraday’s Law∫ ∫∫ ⋅

∂∂

−=⋅ SdtBdlE

tBAV∂∂

−=V

B

Area = A

• For a simple rectangular loop


Given the Definition of Inductance

• Do these have inductance?

“Ground Strap” SMT Capacitor PCB Via

Not until return path for current is identified!


Self Inductance

• Isolated circular loop⎟⎟⎠

⎞⎜⎜⎝

⎛−≈ 28ln

00 r

aaL μ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+−+−+

+++

= 22

0 1121121

1ln2 p

ppppaL

πμ

• Isolated rectangular loop

Note that inductance is directly influenced by loop AREA and less influenced by conductor size!

radiuswiresideoflengthp =


Mutual Inductance

1

221

1212

IM

IMΦ

=

=Φ

Loop #1Loop #2

( )∫ ⋅=Φ2

212 dSˆrBS

nr

How much magnetic flux is induced in loop #2 from a current in loop #1?


Flux from Current in Loop #1


0 200 400 600 800 10000

0.5

1

1.5

2

X: 24Y: 1.835

Spacing between the coils(mils)

Mut

ual I

nduc

tanc

e (n

H)

Change in mutual inductance with spacing

X: 100Y: 0.7312

X: 500Y: 0.02507

X: 1000Y: 0.01955

The magnetic field drops off rapidly, so then does the mutual inductance


Mutual Inductance

Loop #1Loop #2 Less loop area in loop #2

means less magnetic flux in loop #2 and less mutual inductance

Loop #1Loop #2 Less loop area perpendicular to

the magnetic field in loop #2 means less magnetic flux in loop #2 and less mutual inductance


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?


Zero-to-One TransitionWhere’s the Inductance Go??

Power Supply

And how could you possibly calculate it?Courtesy of Dr. Clayton Paul


Lp4

Lp1Lp3

Lp2

Mp1-3Mp2-4

Total Loop Inductance from Partial InductanceL total=Lp1+ Lp2 + Lp3 + Lp4 – 2Mp1-3 – 2Mp2-4

Courtesy of Dr. Clayton Paul


Partial Inductance

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

L3

L4

L2

L1

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


Important Points About Inductance

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


ExampleDecoupling Capacitor Mounting

• Keep vias as close to capacitor pads as possible!

Height above Planes

Via Separation

Inductance Depends on Loop AREA


Via Configuration Can Change Inductance

Via

Capacitor Pads

SMT Capacitor

The “Good”

The “Bad”

The “Ugly”

Really “Ugly”

Better

Best


High Frequency Capacitors

• Myth or Fact?


What is Capacitance?

• Capacitance is the ability of a structure to hold charge (electrons) for a given voltage

VQC =

CVQ =

• Amount of charge stored is dependant on the size of the capacitance (and voltage)

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


Comparison of Decoupling Capacitor Impedance100 mil Between Vias & 10 mil to Planes

0.01

0.1

1

10

100

1000

1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10

Frequency (Hz)

Impe

danc

e (o

hms)

1000pF

0.01uF

0.1uF

1.0uF


0603 Size Cap Typical Mounting

Via Barrel 10 mils

60 mils

20 mils

10 mils*

9 mils9 mils

10 mils*

108 mils minimum

128 mils typical*Note: Minimum distance is 10 mils but more typical distance is 20 mils


0402 Size Cap Typical Mounting

Via Barrel 10 mils

40 mils

20 mils

10 mils*

8 mils8 mils

10 mils*

86 mils minimum

106 mils typical*Note: Minimum distance is 10 mils but more typical distance is 20 mils


3.2 nH3.7 nH4.2 nH100

3.0 nH3.5 nH3.9 nH90

2.8 nH3.2 nH3.6 nH80

2.6 nH3.0 nH3.4 nH70

2.3 nH2.7 nH3.1 nH60

2.1 nH2.5 nH2.8 nH50

1.9 nH2.2 nH2.5 nH40

1.6 nH1.9 nH2.2 nH30

1.3 nH1.6 nH1.8 nH20

0.9 nH1.1 nH1.2 nH10

0402 typical/minimum

(106 mils between via

barrels)

0603 typical/minimu

m(128 mils

between via barrels)

0805 typical/minimum

(148 mils between via

barrels)

Distance into board

to planes (mils)

Connection Inductance for Typical Capacitor Configurations


Connection Inductance for Typical Capacitor Configurations with 50 mils from Capacitor Pad to Via Pad

4.6 nH5.0 nH5.5 nH100

4.3 nH4.7 nH5.2 nH90

4.0 nH4.5 nH4.9 nH80

3.7 nH4.2 nH4.5 nH70

3.5 nH3.9 nH4.2 nH60

3.1 nH3.5 nH3.9 nH50

2.8 nH3.2 nH3.5 nH40

2.5 nH2.8 nH3.0 nH30

2.0 nH2.3 nH2.5 nH20

1.4 nH1.6 nH1.7 nH10

0402 (166 mils


0603 (188 mils


0805 (208 mils


Distance into board

to planes (mils)


PCB Example for Return Current Impedance

Trace

GND Plane

22” trace

10 mils wide, 1 mil thick, 10 mils above GND plane


PCB Example for Return Current Impedance

Trace

GND Plane

Shortest DC path

For longest DC path, current returns under trace


MoM Results for Current DensityFrequency = 1 KHz


MoM Results for Current DensityFrequency = 1 MHz


U-shaped Trace InductancePowerPEEC Results

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08

Frequency (Hz)

indu

ctan

ce (u

H)


Two Wires in Parallel

• Reduce inductance by factor of two? NO!

ppp

pppParallel MLL

MLLL

221

221

−+

−=

2

21

ppParallel

ppp

MLL

LLL+

=

==

Only if parallel wires are FAR APART!Courtesy of Dr. Clayton Paul


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


What Happens if a 2nd Decoupling Capacitor is placed near the First

Capacitor?

Observation Point

Via #1Via #2 Moved in arc around Observation point while maintaining 500 mil distance to observation point

500 mils

distance


Second Via Around a circle

theta: angle as shown in the figure in degree

d2: distance between Port 2 and Port 3 in mil

d1: distance between Port 3 and Port 1 in mil

d: thickness of dielectric layer in mil

r: radius for all ports in mil

R: distance between Port 1 and Port 2 in mil

Courtesy of Jingook Kim, Jun Fan, Jim Drewniak

Missouri University of Science and Technology

Port 2

Port 1 θ

( )yx,Port 3

( ) ( )( )

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛

++

−⎟⎟⎠

⎞⎜⎜⎝

⎛

+++

rrdrRrd

drdr

rdrRd2

12

23

21

2

ln

ln

4ln

4 πμ

πμ

2sin22

1

θRd

Rd

=

=

( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

++

= 3

4

)2/sin(2ln

4 rrRrRd

θπμ

1d

2d R

Port 2

Port 1 θ

( )yx,Port 3

( ) ( )( )

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛

++

−⎟⎟⎠

⎞⎜⎜⎝

⎛

+++

rrdrRrd

drdr

rdrRd2

12

23

21

2

ln

ln

4ln

4 πμ

πμ

2sin22

1

θRd

Rd

=

=

( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

++

= 3

4

)2/sin(2ln

4 rrRrRd

θπμ

1d

2d R

Lequiv


Effective Inductance for Various Distances to Decoupling CapacitorWith Second Capacitor (Via) Equal Distance Around Circle

Plane Seperation = 35 mil -- Via Diameter = 20 mil

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

0 50 100 150 200

Angle (degrees)

Indu

ctna

ce (p

H)

250 mil

500mil

750 mil

1000 mil




0

50

100

150

200

250

300

350

400

450

500

0 50 100 150 200Angle (degrees)

Indu

ctna

ce (p

H)

500mil

250 mil

750 mil

1000 mil




0

50

100

150

200

250

300

350

400

0 50 100 150 200Angle (degrees)

Indu

ctna

ce (p

H)

500mil

250 mil

750 mil

1000 mil


Understanding Inductance Effects and Proximity

1 via

10mm

2 via with degree 30°

2 via with degree 90° 2 via with degree 180°

20cm

20cm

10cm

10cm


Current DensityA/m2

[m]

[m]

A/m2

[m]

[m]

A/m2

[m]

[m]

A/m2

[m]

[m]


Current Density in Planes8

8

8

8

8

8

16

16

1616

1624

24

24

24

32

32

32

40

40

40

48

48

48

56

56

56 646464

6472

72

8080

80

80

0.08 0.0850.09 0.095 0.1 0.1050.11 0.1150.120.08

0.085

0.09

0.095

0.1

0.105

0.11

0.115

0.12

8

8

8

8 8

8

16

16

16

16

16

24

24

24

24

32

32

32

32

40

40

40

4040

40

4848

48

48

48

48

56

56

5656

5656

6464

64

64

727272

72

728080

0.08 0.0850.09 0.095 0.1 0.1050.11 0.1150.120.08

0.085

0.09

0.095

0.1

0.105

0.11

0.115

0.12

8

8

8

8

8

816

16

16

16

1616

24

24

24 2432

32

32 32

40

40

40

48

48 4856

56

56

56

64

64

6 4

64 72

72

72

72

80

80

80

80

0.08 0.0850.09 0.095 0.1 0.105 0.11 0.1150.120.08

0.085

0.09

0.095

0.1

0.105

0.11

0.115

0.12

8

8

8

8

8

8

16

16

16

16

1624

24

24

2432

32

32

3240

40

40

40

40

40

48

4848

48

48

48

5656

56

56

56

56

64

64

64

6472

72

72

80 80

0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.120.08

0.085

0.09

0.095

0.1

0.105

0.11

0.115

0.12


Effect of Plane width on Inductance

Case1 : 10 inches

Case2 : 5 inches

Case3 : 2 inches1 inch

Port1 Port2


Loop Inductance is Affected by Plane Width

~ 250pH

~ 330pH

~ 560pH

Case1 : 10 inches

Case2 : 5 inches

Case2 : 2 inches


Current Spreads in a Plane

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

parallel paths increases!


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


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

inductance and partial inductance what's it all...

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