RFIC Inductor Introduction

Dror Regev 2.07.08Dror Regev July 2008 1

Layout of a Spiral Inductor

•Most planar inductors are spirals.• Shape can be: Octagonal, Square or Rectangle.• Goal is to obtain the needed inductance within a minimal area while optimizing quality factor.

Total inductance of the spiral, include sum of all line self and mutual inductances.The Spiral has also serial resistance of the metal lines that limit the quality factor at every frequency. •We will first examine the inductor without Si substrate effects and later on add those effects.

∑∑∑ −+ −+= MMLL selfIND

Dror Regev July 2008 2

Dror Regev July 2008

Self and Mutual Inductance of a Micro strip

Mutual‐inductance of adjacent segments can be positive or negative depending on segment current direction

Self‐inductance of a segment l (cm) with a rectangular cross section w×t (cm × cm):

Mutual inductance between 2 filaments at distance d:

]/[10*4

])[1)1((ln2

70

220

mH

Hld

ld

dl

dllM

−=

+⎟⎠⎞

⎜⎝⎛+−+⎟

⎠⎞

⎜⎝⎛+=

πμ

πμ

Self‐inductance requires physical length and independent of Epsilon. 

][)3

5.02(ln2 Hnl

twtw

llL +++

+=

Dimensions in m

wt

l

d

d = w + ss

Inductance vs. line length W=10um, T=2um.

67.07

161.06

265.52

376.78

493.08

613.42

737.10

863.66

992.71

1124.00

38.60102.63

177.10

258.36

344.67

435.01

528.69

625.25

724.31

825.59

93.99

104.46

111.26

116.31

120.34

123.68126.55

129.06131.28

0

100

200

300

400

500

600

700

800

900

1000

1100

1200

100 200 300 400 500 600 700 800 900 1000

Trace length [um]

L, M

[pH

]

90

95

100

105

110

115

120

125

130

135

dL/d

l

“Edge” effect impact is stronger the shorter the line is. Longer  lines have higher inductances per unit length!. Hence, “corners” in the trace reduce inductance. 

∑∑∑ −+ −+= MMLL selfIND

3

Micro strip Metal Resistance

δ

δσω t

dc

skinser

e

tRtlwR

−−

≈1

),,,,(.

RF resistance:

Skin Depth:

Rseries of a100um line length with w=10um and t=1um

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Frequency [GHz]

R

AC current flowing in a Micro strip at RF frequencies crowds closer to RF Ground in a phenomena called “skin effect”.

Hence, effective current cross section may be smaller than the physical dimensions.

RF resistance increase with frequency

due to “skin effect”.

4Dror Regev July 2008

sδ t

Al skin depth

0.00

0.50

1.00

1.50

2.00

2.50

3.00

1 2 3 4 5 6 7 8 9 10

Frequency [GHz]

Skin

Dep

th [u

m]

fs ****2

μπρ

σμωδ ==

Dror Regev July 2008

Micro strip Series Quality Factor

Quality factor for 100um line length, t=2um, Al

0

2

4

6

8

10

12

14

16

18

1 2 3 4 5 6 7 8 9 10

Frequency

Qs w =14um

Qs w =10um

s

sselfSeries R

LQ ω=,

Serial Quality Factor:

L&R both depend on line length.

Rs – determines series Q factor and depends on w and t the thickness of the metal. Skin effect further increases Rs.

Increasing W seems to improve serial Q, but also reduces mutual inductance contribution from adjacent lines and increase capacitance to ground which has negative contribution to Q factor (as will be shown later).

Ls Rs

t

w

l

5

Inductor’s “Serial” Capacitance

In

Out

12

3 4

56

7

Cs “Equivalent” capacitance from inductor’s input to output originates in two major mechanisms:

“Metal Under path” capacitance

In

Out

Mutual capacitance

Capacitance depends on size and 

number of under paths.

Capacitance depends on distance 

between line centers.

Ls

nut

C_overlap

LsC_overlap

Dror Regev July 2008 6

Ls Rs

Cs

Inductor’s Series Quality Factor

01

111 2

| == VVIy

)1Re(

)1Im(

11

1111

y

yQ = I1I11

I2Ls Rs

CsZs

2

( )s

ss CjRLj

y ωω −

+= ||1

11

( )[ ]

( )s

sss

s

ssss

s

ss

RCLL

RCRCLL

ZZQ s

2

22

1

1)Re()Im(

ωω

ωω

−≅

−−==

Series quality factor: Q at low frequency follows ωLs / Rs value, but when approachingresonance it  reaches a maximum and than starts to decrease and zeroes at resonance. After resonance it behaves like a capacitor.

ω02ω

Q ( )neglectedeffectskinRLQs

straces ___

ω=

ωmax2

Qmax2

ω01ωmax1

Qmax1

Rs lowRs high 

Dror Regev July 2008 7

Spiral Inductor with Si Substrate - Simplified Pi Model

Cs

Ls Rs

Cox1

Rsub1

Cox2

Csub1 Csub2 Rsub2

ort1 ort2

Lumped Pi Model:

Ls – Inductance of Spiral Inductor

Rs – Metallization Series Resistance of Inductor

Cs – Input to Output Capacitance 

Cox – Capacitance of metal trace to substrate (through ILDs)

R sub – Substrate Resistance

C sub – Substrate Capacitance

Dror Regev July 2008 8

Performance Effect of the Si Substrate

Sp

Sp

ZZZZ

y +

•=

11

1

I1I1 I2Ls Rs

CsZs

Cox1

Rsub1 Csub1

1Zp

equivalent

I1I1 I2Ls Rs

CsZs

ZpRp Cp

1

( )( )ωω

pp

pp

RR

CC

=

=

( ) ( )

s

sp

p

s

ss

sps

sps

RLR

RR

CLLLRR

CLRL

y

yQ 22

2

22

2

11

1111

11

)1Re(

)1Im(

ωωω

ωωω

+

−=

+

−≅=

Inductor quality factor: with Zp the parallel RC impedance has two impacts:1. Resonance frequency decreases as total capacitance increases: C total=Cp + Cs2. There is a parallel dissipation path to substrate that further reduces Q3. Rp, the equivalent parallel  resistance to ground reduces with frequency.

Substrate losses at high frequencies

Series quality factorwith lower resonance frequency.

Dror Regev July 2008 9

Spiral Inductor Models with Higher Accuracy

Dror Regev July 2008 10

Cox

RsubCsub

Port 1 Port 2R1

R2

R3

L1

L2

R1

R2

R3

L1

L2

L ser L serK

Cox

Rsub Csub

Cox

Rsub Csub

Cs

“Skin Effect”

“Higher order model” with increased complexity and accuracy including modeling for skin effect 

Model vs. Test Data Comparison

Zs = ‐1/Y12

S‐Parameters

Q Factor

Inductor 2.5/60um Model vs. Test Performance

Dror Regev July 2008 11

Symmetrical Inductors and Differential Mode

Differential mode Common mode

+

‐

+

+

Virtual Ground Virtual Open

Dror Regev July 2008 12

“Differential” excitation of the inductor creates a virtual ground at the inductor’s center.  

Differential Inductor Q vs. Single Ended Q

For symmetric excitation we can use the same approximation:

s

sP

P

s

psss

RLR

RR

CCLLQ 22

2

11

)](1[ω

ωω

+

+−≅

I1I1 I2

1

Rp Cp

Ls/2 Rs/2

2Cs

Ls/2Rs/2

2Cs

I1I1 I2Ls Rs

Cs

1

Rp Cp Resonance frequencyfactor

s

sP

P

s

pss

s

RLR

RR

CCLLQ

2

)]2(2

1[22

2

11 ω

ωω

+

+−≅

Virtual Ground

Substrate losses at high frequencies

Dror Regev July 2008 13

Differential Inductor Q vs. Single Ended Q

So we found that symmetrical excitation both decreases substrate losses and increases resonance frequency !.

Symmetric: Single ended:

s

sP

P

s

psssse

RLR

RR

CCLLQ 22

2

11

)](1[ω

ωω

+

+−≅

s

sP

P

s

pss

s

diff

RLR

RR

CCLLQ

2

)]2(2

1[22

2

11 ω

ωω

+

+−≅

ω

Q

ω0 diffωdiff max

Qmax diff

ω0 seωse max

ωLs / R s

Qmax se

Dror Regev July 2008 14

Symmetrical Inductor Capacitance. Single Ended

SE Inductor – Coupling capacitor is “shorted” by low inductance.

Symmetric Inductor – Coupling capacitor is “shorted” by a larger inductance. Hence capacitance between turns is increased.

The larger the number of turns, the bigger the capacitance difference between symmetric and nonsymmetrical inductors.

Symmetrical InductorSingle ended Inductor

In Out

Dror Regev July 2008 15

Dror Regev July 2008 16