Electromagnetic coupling in Si substrate - Basic Concepts of the Coupling Effect

CHAPTER 2 Basic Concepts of the Coupling Effect

2.2 Electromagnetic coupling in Si substrate

From Section 1.1, because of the concept of SOC, the integration of RF circuits in CMOS process is actively pursued in an effort to increase functionality and reduce cost. For the giga hertz frequency range, the electromagnetic coupling between RF circuit blocks through the lossy substrate is a concern. For example, the large RF signal power produced by a power amplifier will pass through the substrate and then affect the other sensitive circuit. Moreover, the on-chip inductors are often used in RF circuit designs, such as the low noise amplifier, band-pass filter, and mixer. As shown in Fig. 2.3 and Fig. 2.4, the electromagnetic coupling between two closely placed interconnects or inductors is also a way that induces the electromagnetic coupling in RFICs (Fig. 2.3). Therefore, if high-frequency coupling effects are not taken into account, the RF circuit performance will be degraded by these unwanted coupling

Signal transition

Sensitive node

Hot carrier Junction capacitance

Noise coupling

P-substrate

effects.

Fig. 2.3 Low noise amplifier suffering from the on-chip inductor coupling effect.

Fig. 2.4 Coupling effect of two adjacent inductors.

Substrate coupling effect

Lossy Substrate M2

Ls Lg

RFin

RFout

Coupling =?

VDD

Vbias

CHAPTER 3 Investigation of RF Spiral Inductor’s Coupling Effects in Lossy Substrate

In this chapter, the coupling effects of two adjacent coplanar spiral inductors are investigated. First, the manufacturing process of the inductor is provided by NCTU Nano Facility Center (國立交通大學奈米中心). The process is different from the standard CMOS process utilized by TSMC or UMC. Second, the inductor coupling structure is proposed and the simulation results are shown. Finally, the measurement data and proposed model of inductor coupling effect are also obtained.

3.1 Design consideration of a single inductor

3.1.1 The fabrication process of the inductor

As mention before, the process is provided by NCTU Nano Facility Center; the process is quite different from the commercial process, such as TSMC or UMC standard CMOS process. The wafer is the 4 inch P-type doped wafer with a substrate resistivity of 21~23 Ω/cm. In order to get a good electrical performance, a sandwiched type membrane consisted of 0.7um thermal oxide, 0.7um LPCVD Si3N4, and 0.7um TEOS oxide is employed (SiO20.7um/Si3N40.7um/SiO20.7um, [4]). The cross-section view of these oxide layers are shown in Fig. 3.1 and the advantages of the proposed sandwiched structure are illustrated as the follows:

(1) Low dielectric loss: Because of the low conductivity and loss tangent properties, these dielectric layers have the low electromagnetic energy loss.

(2) Stress issue: The Young’s modulus of Si3N4 is five times larger than that of SiO2

(Table 3.1); the Si3N4 layer has the much higher residual stress with the silicon substrate in comparison with SiO2. However, if proposed sandwiched structure is utilized, the nitride stress can be effectively released by using the double oxide layers.

Table 3.1 Young’s modulus.

Material

Parameter SiO2 Si3N4

Young’s modulus (GPa) 73 310

The copper is utilized to form the metal strip of inductor. The metal is deposited above oxide layer by using electroplating method with the H2SO4 plating bath and the copper anode. The total reaction of the plating procedure can be written as:

Cu e

Cu²⁺ +2 ⁻ → (3.1) The setup of the electroplating method is presented in Fig. 3.2.

Fig. 3.1 The cross-section view of the sandwich membrane.

Fig. 3.2 The setup of the electroplating method.

Cu²⁺

Copper anode

The cathode for the Cu²⁺

deposition DC power

H2SO4 solution Silicon substrate

(P-type) 0.7 um

0.7 um 0.7 um

TEOS oxide (SiO2) LPCVD nitride (Si3N2) Thermal oxide (SiO2)

3.1.2 Specification of spiral inductor

The size of the spiral inductor is shown in Fig. 3.3. Due to the process limitation, the space between two adjacent metal lines is 5um, the metal width and thickness are 15 um and 5 um, respectively. And the inner radius of the inductor is 60um. The graphical structure and design specifications are given in Fig. 3.3.

Fig. 3.3 Structure drawing of the inductor: (a) Top view. (b) Cross-section view.

3.2 Measurement results and modeling of a single inductor

3.2.1 Measurement setup

The S-parameters of a spiral inductor is measured by Agilent 8364B network analyzer and two GSG coplanar probes. The measurement setup is illustrated in Fig.

3.4 and the frequency range of measurement is from 0.1 GHz to 30 GHz.

S=5 um 2R

(a) (b)

W=15 um

Port 1 Port 2

Fig. 3.4 Measurement setup of the single inductor.

3.2.2 De-embedding procedure

To obtain a precise measurement of the inductor S-parameter, the paracitic effects of GSG pad are needed to be removed. Therefore, a process called

“de-embedding” should be done after the complete two-port calibration. A de-embedding procedure that contains the open circuit and short circuit method (OSD) [5] is used in the measurement of inductor S-parameter. The paracitics of GSG pads are illustrated in Fig. 3.5; the pad parasitic is lumped as yp and the parasitic due to pad and probe tip interface discontinuity is modeled as Zi. After the parasitics are defined, the Zi and yp are extracted by the short and open test pattern. Therefore, the S-parameters of single inductor without pad parasitics can be obtained by transmission matrix operation. The de-embedding procedure can be written as the follows:

(

inopen inshort

)

short in i

Z y Z

Z Z

, ,

= −

The transmission matrix (T) of DUT (Device Under Test) can be derived [6]:

GSG Probe

G G

S S

[ ] [ ]

Fig. 3.5 Parasitic of GSG pad and pad parastics extraction. (a) The equivalent circuit model of pad parasitic. (b) The extraction method of pad parasitic.

DUT yp

Zi Zi

Termination (50Ω)

Zin,open

(a)

(b)

3.2.3 Measurement results and equivalent circuit model of spiral inductor In reference to the micrograph of a 3.5 turns spiral inductor as shown in Fig.3.6 and the comparisons between measurement (Mea.) and simulation (Sim.) results using Ansoft HFSS are shown in Fig. 3.7 with a good agreement. Moreover, the effective inductance of 2.5 turns, 3.5 turns, and 4.5 turns inductor are 1.842 nH, 3.587 nH, and 6.528 nH, respectively, and the peak value of Q factor for 2.5 turns, 3.5 turns, and 4.5 turns inductor are 16.16, 12.566, and 10.2, respectively. The deviation of effective inductance and Q factor between simulation and measurement are within 8%.

Furthermore, the measured self resonant frequency (FSR) of 2.5 turns, 3.5 turns and 4.5 turns inductor are 22.3, 14.8, and 10.5 GHz, respectively. In comparison with simulated self resonant frequency is at 21.3 GHz, 13.9 GHz, and 9.9 GHz for 2.5 turns, 3.5 turns, and 4.5 turn inductor, respectively. The FSR of simulation is 1 GHz lower than measurement data. Finally, the measurement and simulation results are summarized in Table 3.2.

Fig. 3.6 Inductor micrograph (3.5 turns).

G S S

-6 -4 -2 0 2 4 6

0 5 10 15 20 25 30

Frequency(GHz)

L(nH)

-5 0 5 10 15 20

Mea.

Sim.

(a) 2.5 turns

-6 -4 -2 0 2 4 6 8

0 5 10 15 20 25 30

Frequency(GHz)

L(nH)

-5 0 5 10 15

Mea.

Sim.

(b) 3.5 turns

-6 -4 -2 0 2 4 6 8

0 5 10 15 20 25 30

Frequency(GHz)

L(nH)

-5 0 5 10 15

Mea.

Sim.

Fig. 3.7 Measurement (Mea.) and simulation (Sim.) results of inductor. (a) 2.5 turns.

Table 3.2 Summary of inductors (measurement and simulation).

Situation L (nH)@5GHz Qpeak FSR (GHz) Measurement 1.842 16.16@3GHz 22.3 2.5 turns

Simulation 1.974 15.648@2.4GHz 21.3

Measurement 3.587 12.566@3GHz 14.8 3.5 turns

Simulation 3.76 12.287@1.4GHz 13.9

Measurement 6.528 10.2@1GHz 10.5 4.5 turns

Simulation 6.723 10.817@0.6GHz 9.9

Qpeak: Peak Value of Q factor.

After getting the measurement data of inductor, a model is established and then used it to fit the measurement results. The traditional π model is presented in Fig. 3.8 and the meaning of the components for the equivalent circuit model is illustrated as the follows:

(1) Cp: Capacitance between adjacent metal lines.

(2) Rs: Resistance of metal line.

(3) Ls: Inductance of metal line.

(4) Cox1, Cox2: Capacitance between metal line and substrate.

(5) Csub1, Csub2: Parasitic capacitance of lossy substrate.

(6) Rsub1, Rsub2: Parasitic resistance of lossy substrate.

Fig. 3.8 Equivalent circuit model of the analyzed spiral inductor. (a) 3-D view. (b) Whole equivalent circuit model of the π model.

Furthermore, in order to extract these parameters of π model, a procedure is proposed to make the extraction more efficient. The procedure of parameter extraction is outlined in the following steps:

Step1. Rs and Ls: According to the equivalent model of the inductor, all the parasitic capacitances are opened as the frequency is low. Therefore, the Rs and Ls are extracted from the Y-parameters of the measurement results at the low

Rsub2

Csub2

Rsub1

Csub1

Cox1 Cox2

Cp Cp

Oxide layer

Lossy substrate

(a) Cp

Rs Ls

Cox1 Cox2

Csub1 Rsub1 Csub2 Rsub2

Port 1 Port 2

(b)

Rs= 











real y and Ls=

frequency y imag π× 2

) / 1 ( ₁₁

(3.2)

Where real= Real part and imag = imaginary part.

Step2. Cp: The parameter Cp is extracted from the y21 for the equivalent circuit model as shown in Fig. 3.9. From Fig. 3.9, the resonant frequency is given at

p sC π L 2

1 and the Cp is read as:

(

)

^, ²¹

1 f self resonant frequencyof y L

C f _SR

s SR

p = =

π (3.3)

Fig. 3.9 Equivalent circuit of y21.

Step3. Rsub1, Rsub2, Csub1, and Csub2: The parasitics of the lossy substrate start to affect the inductor as the frequency increases. Therefore, the Cox1 and Cox2 are neglected, and the equivalent circuit becomes Fig. 3.10. From Fig. 3.10, we can write:

( )

s s sub

p sub

in port j C C R j L R

Y = + + + +

ω ¹ ω ¹

) 1 (

1 (3.4) At the resonant point, we can obtain:

V1 Cp

Rs Ls

Cox1

Csub1 Rsub1

real Yin

Step4.Cox1 and Cox2: The parallel plate capacitor formula is utilized to derive the Cox1

and C . The parallel plate formula is written as:

plates parallel o

between tw distance

the d

plates parallel two

e between th filled

is that constant dielectric

plate parallel the

of area A ,

= d C εA

After the initial values of the model are gained, the optimal function of the simulation tool (Agilent ADS) is utilized to obtain the final values of the equivalent circuit model.

However, the traditional π model only can fit the measurement data up to 15 GHz. Hence, in order to get a wide band frequency response (up to 30 GHz), the high frequency parasitic of the lossy substrate are taken into account and some components(Rsub3, Csub3, Lsub1, Lsub2) are added to match the high frequency S-parameters of measurement data [7]-[9]. Therefore, the whole equivalent circuit of the wide band inductor model is shown in Fig. 3.11. As shown in Fig. 3.11, the Lsub1

and Lsub2 are used to model the inductive parasitic of the lossy substrate, and Rsub3 and Csub3 describe another signal path from port 1 to port 2 when the frequency goes to high. Therefore, if these components are added in the traditional π model, a good agreement between modeling and measurement results can be obtained in the frequency range from 0.2 GHz to 30 GHz.

Fig. 3.11 The broad band equivalent model of spiral inductor.

Furthermore, the effective inductance and Q factor of measurement and modeling results of a spiral inductor are presented in Fig. 3.12. Form Fig. 3.12, the effective inductance of proposed inductor model is almost identical to the measurement results. And the Q factor of the model is a little large than measurement.

Finally, Table 3.2 and Table 3.3 give the component list of wide band inductor model and a summary between measurement, HFSS simulation, and modeling results of the varying spiral inductors, respectively.

Cox1 Cox2

Csub1 Rsub1 Csub2 Rsub2

Lsub1 Lsub2

Rsub3

Csub3

Port 1 Port 2

-6 -4 -2 0 2 4 6

0 5 10 15 20 25 30

Frequency(GHz)

L(nH)

-5 0 5 10 15 20

Mea.

Model

(a) 2.5 turns

-6 -4 -2 0 2 4 6 8

0 5 10 15 20 25 30

Frequency(GHz)

L(nH)

-5 0 5 10 15

Mea.

Model

(b) 3.5 turns

-6 -4 -2 0 2 4 6 8

0 5 10 15 20 25 30

Frequency(GHz)

L(nH)

-5 0 5 10 15

Mea.

Model

Fig. 3.12 Measurement (Mea.) and modeling (Model) results of inductors. (a) 2.5 turns. (b) 3.5 turns. (c) 4.5 turn.

Table 3.3 Component list of the wide band inductor equivalent model.

2.5 3.5 4.5

Ls(nH) 1.73 3.232 5.422

Rs(Ω₎ 0.977 1.486 1.965

Cp(fF) 8.255 12.037 13.679

Cox1(fF) 242.824 313.848 426.241 Cox2(fF) 307.679 394.457 495.442

Rsub1(Ω₎ 1386.31 1110.61 978.768

Rsub2(Ω₎ 1209.78 930.016 708.827

Csub1(fF) 25.284 22.147 25.284

Csub2(fF) 25.312 33.263 41.723

Rsub3(Ω₎ 5218.28 4526.167 4340.981

Csub3(fF) 1.079 1.555 1.833

Lsub1(nH) 0.246 0.631 0.979

Lsub2(nH) 0.21 0.439 0.616

Size Turns

Table 3.4 Summary of the spiral inductor.

Situation L (nH)@5GHz Qpeak FSR (GHz) Measurement 1.842 16.16@3GHz 22.3

Simulation 1.974 15.648@2.4GHz 21.3

2.5 turns

Modeling 1.810 18.427@3.2GHz 22.1

Measurement 3.587 12.566@3GHz 14.8

Simulation 3.76 12.287@1.4GHz 13.9

3.5 turns

Modeling 3.609 13.327@1.8GHz 14.7

Measurement 6.528 10.2@1GHz 10.5

Simulation 6.723 10.817@0.6GHz 9.9

4.5 turns

Modeling 6.582 11.11@1.2GHz 10.7

FSR: self resonant frequency; Qpeak: peak value of Q factor.

3.3 Analysis of inductor coupling effects

In RF integrated circuits (RF ICs), such as LNA, mixer, and band-pass filter, the inductor is a key element for circuit designs. Unfortunately, because the substrate is lossy, the coupling effect in giga hertz frequency range is a big issue (Fig. 2.3 and Fig.

2.4). In this section, two types of inductor coupling situations are proposed (as shown in Fig.3.13) and their coupling mechanisms for different distance (d) are studied.

Fig. 3.13 Two type of inductor coupling situations.

3.3.1 Measurement setup and de-embedding method

The measurement instrument is the Agilent 8364B network analyzer. However, the inductor coupled pair is a four-port network, so a pair of GSGSG probes is utilized and the Agilent 8364B 4-port measurement function is used to get the 4× 4 S-parameters matrix of the structure. The measurement setup is shown in Fig. 3.14.

d d

Port1 Port2

Port 3

Port 4 Port 4 Port 3

Port1 Port2

Type A Type B

Fig. 3.14 Measurement setup of the inductor coupled pair.

As mention before, the de-embedding procedure must be done after measurement.

The de-embedding procedure is almost same as stated in Section 3.2.2. The only difference is that we expand the 2× matrix to 2 4× matrix. The de-embedding 4 formulas are derived as:

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]

then

T T T

T T T T T T

i measure i

p DUT

i p p

Z Z

Z y DUT y

Z measure

1 1

1 − −

= −

[ ]













4 3 2 1

d d d d

c c c c

b b b b

a a a a

T _DUT (3.7)

And the 4× S-parameters matrix becomes: 4 G

G S

S S

S ^{Port 1}

Port 2 Port 3

Port 4

[ ]

The formulas above are only the summarized results, and a complete derivation process can be referred to Appendix I.

3.3.2 Measurement results of inductor coupled pairs

The micrograph of inductor coupling pairs and measured S-parameters of type B inductor coupling pairs are presented in Fig. 3.15 and Fig. 3.16, respectively. The S41

of the inductor coupled pairs is taken as the amount of substrate coupling effect between two inductors. As shown in Fig. 3.16, the S41 is decreasing when d is increasing. And the measurement results conform to our prediction. Therefore, the inductor coupling effect will reduce if the distance between two inductors is increased.

After the measurement data is obtained, the comparison between the simulation and measurement results is done and found out the measurement data reasonable or not.

And the simulation tool is Ansoft HFSS, too. From Fig. 3.16, the measured S41 is lower then simulation and the maximum deviation of S41 is less than 1 dB. Therefore, the good agreement between measurement and simulation is obtained. Furthermore, to inspect the S41 value of 2.5 turns inductor coupled pairs and 3.5 turns inductor coupled pairs, it can be observed the S41 is slightly dependent on the frequency.

Moreover, the differences of S41 for different types of inductor coupled pairs are also observed. As presented in Fig. 3.17, the S14 of type A is smaller then type B. The

phenomenon is because that the inductor coupled pairs of type B has the longer metal lines in opposition to each other. As a result, the metal line induces the larger coupling effect between two inductors. Furthermore, the magnetic flux of type B is larger than that of type A; it is also a reason that the S41 value of type B is larger than that of type A.

Fig.3.15 Micrograph of inductor the coupling pair (3.5 turns typeB with d=30um).

G

G G G

G G S S

S S

-40 -35 -30 -25 -20

0 5 10 15 20

Frequency(GHz)

S41(dB) _30um(Mea.)

50um(Mea.) 75um(Mea.) 100um(Mea.) 30um(Sim.) 50um(Sim.) 75um(Sim.) 100un(Sim.)

(a)

-40 -35 -30 -25 -20

0 5 10 15 20

Frequency(GHz)

S41(dB)

30um(Mea.) 50um(Mea.) 75um(Mea.) 100um(Mea.) 30um(Sim.) 50um(Sim.) 75um(Sim.) 100un(Sim.)

(b)

Fig. 3.16 Measurement (Mea.) and simulation (Sim.) result of the inductor coupled pairs for the varying separate distance. (a) 2.5 turns, type B. (b) 3.5 turns, type B.

-45 -40 -35 -30 -25 -20

0 5 10 15 20

Frequency(GHz)

S4 1( dB )

3B3 3B5 3B7 3A3 3A5 3A7

Notes:

(1) 3.5 turns type A (2) 3.5 turns type B d=30um: 3A3 d=30um: 3A3 d=50um: 3A5 d=50um: 3A5 d=75um: 3A7 d=75um: 3A7

Fig.3.17 Comparison of different types for 3.5 turns inductor coupled pairs.

3.3.3 Discussion about the inaccuracy between measurement and simulation results

To compare the measurement and simulation results, the trend of simulation curve is similar to that of measurement, but the overall S41 values are about 1 dB larger than measurement results for different distances. Hence, we attempt to find the 1dB variation between simulation and measurement results. We suppose that the source of variation comes from the process we used. First, the oxide thickness is confirmed, and the oxide thickness is almost equal with 2.1 um after measurement.

A B

variation of S41 is within 0.1 dB. As a result, the effective dielectric constant is not the main fact that can cause the S41 deviation. And the last, the conductivity of the wafer is checked. The conductivity=4 (Siemens/m) is used in the simulation. But when the conductivity is changed from 4 to 5 (Siemens/m), the simulation curve is more similar to the measurement one. Fig. 3.18 and Fig. 3.19 present the improved simulation results and compare them with measurement curve for the 2.5 turns and 3.5 turns inductor coupled pairs, respectively. Therefore, the variation of conductivity of the wafer is the main factor that causes the variation between measurement and simulation results.

-40 -35 -30 -25 -20

0 5 10 15 20

Frequency(GHz)

S41(dB)

2B3(Mea.) 2B3_C4(Sim.) 2B3_C5(Sim.)

(a)

-40 -35 -30 -25 -20

0 5 10 15 20

Frequency(GHz)

S41(dB)

2B5(Mea) 2B5_C4(Sim.) 2B5_C5(Sim.)

(b) Notes:

(1) 2.5 turns, type B

d=30 um, conductivity of wafer=4 (Seimens/m): 2B3_C4 d=30 um, conductivity of wafer=5 (Seimens/m): 2B3_C5 d=50 um, conductivity of wafer=4 (Seimens/m): 2B5_C4

d=50 um, conductivity of wafer=5 (Seimens/m): 2B5_C5

Fig. 3.18 Improved simulation (Sim.) and measurement (Mea.) result for the 2.5 turns

-40 -35 -30 -25 -20

0 5 10 15 20

Frequency(GHz)

S41(dB)

3B3(Mea.) 3B3_C4(Sim.) 3B3_C5(Sim.)

(a)

-40 -35 -30 -25 -20

0 5 10 15 20

Frequency(GHz)

S41(dB)

3B5(Mea.) 3B5_C4(Sim.) 3B5_C5(Sim.)

(b) Notes:

(1) 3.5 turns, type B

d=30 um, couductivity of wafer=4 (Seimens/m): 3B3_C4 d=30 um, couductivity of wafer=5 (Seimens/m): 3B3_C5 d=50 um, couductivity of wafer=4 (Seimens/m): 3B5_C4 d=50 um, couductivity of wafer=5 (Seimens/m): 3B5_C5

Fig.3.19 Improved simulation (Sim.) and measurement (Mea.) results for the 3.5 turns type B inductor coupled pairs. (a) d=30um. (b) d=50um.

3.3.4 Modeling of inductor coupled pairs

Having acquired the measurement data of inductor coupled pairs; a model is constructed to represent their noise coupling behavior. First, a RC parallel network (Rsub4, Rsub5, Csub4, and Csub5) is established to describe the electrical noise coupling effect in the lossy substrate. Second, the coupling coefficient (k) is exploited and it can express the magnetic coupling effect between two inductors. Finally, these two parts are combined with the wide band inductor model and the whole equivalent circuit model is presented in Fig. 3.20.

Fig. 3.21 shows the modeling results of 2.5 turns type B inductor coupling pair, and the meaning of 2B3, 2B5, and 2B7 are corresponding to d=30um, d=50um, d=75um, respectively. As observed in Fig. 3.21, the difference between modeling and measurement results is only within 0.3 dB. Moreover, the proposed model can represent the curve of coupling effect from 1 GHz~15 GHz and the component values of the coupling circuit is listed in Table 3.4.

Moreover, the scaling formulas of the inductor coupled pairs are also derived from the extracted parameters (Rsub4, Rsub5, Csub4, Csub5, and k). And the formulas are shown as the follows:

Rsub4=-0.129d2+57.778d+3630.7 (Ω) (3.10a) Rsub5=-0.3949d2+89.275d+2311.5 (Ω) (3.10b) Csub4=-0.0624d+7.352 (fF) (3.10c) Csub5=-0.0601d+6.9979 (fF) (3.10d) k=-0.0007d+0.2025 (3.10e) d=um

Therefore, the circuit designers can refer to these formulas and understand the

Fig. 3.20 Equivalent circuit of the inductor coupled pairs.

Cox2

Rsub3

Csub4

Cox1

Rsub2

Csub2

Rsub1

Csub1

Rsub1

Lsub1 Lsub1

Lsub2 Lsub2

Rsub3

Csub3

Rsub4

Csub3

Rsub5

Csub5

Cox2

Cox1

Coupling coefficient k

Port 1 Port 3

Port 2 Port 4

(1) The definition of ports is the same with Fig.3.12

(2) Csub4||Rsub4: The electrical noise coupling between Port 1and Port 2 (3) Csub5||Rsub5: The electrical noise coupling between Port 3and Port 4 (4) Coupling coefficient (k): Magnetic coupling between two inductors

-40 -35 -30 -25 -20

0 3 6 9 12 15

Frequency(GHz)

S41(dB)

Mea.

Model

(a)

-40 -35 -30 -25 -20

0 3 6 9 12 15

Frequency(GHz)

S41(dB)

Mea.

Model

(b)

-40 -35 -30 -25 -20

0 3 6 9 12 15

Frequency(GHz)

S41(dB)

Mea.

Model

(c)

(d) Note:

(1) 2.5 turns type B d=30um: 2B3 d=50um: 2B5 d=75um: 2B7 d=100um: 2B1

Fig. 3.21 Measurement (Mea.) and Modeling (Model) results. (a) 2B3. (b) 2B5. (c) 2B7. (d) 2B1.

-40 -35 -30 -25 -20

0 3 6 9 12 15

Frequency(GHz)

S41(dB)

Mea.

Model

Table 3.5 Component list of the equivalent circuit of 2.5 turns type B inductor coupled pairs.

d(um) Component

30 50 75 100

Rsub4(Ω₎ 5281.7 6111.07 7312.56 8090.65

Rsub5(Ω) 4609.75 5849.66 6730..65 7307.25

Csub4(fF) 5.34 4.492 2.549 1.12

Csub5(fF) 5.165 4.113 2.225 1.09

k 0.1818 0.1643 0.1493 0.13

3.4 Comparison with other works

When the measurement and modeling results is done, we try to compare the results with the publication data. First, from [10] and [11], the measurement method is the two-port measurement case with the other terminals being open or floating (as depicted in Fig.3.22). On the contrary, our measurement data are the four-port S-parameters matrix with all ports at the 50 Ω terminations. Those measurement data are more useful for RF circuit designers to apply them for the post-simulation procedure before the chip tape out.

Moreover, in the part of proposed equivalent circuit of inductor coupled pairs, the frequency range is up to 15 GHz with the deviation of 0.3 dB. According to [11], its frequency range of equivalent model is only about 10 GHz and the deviation between measurement and model is larger than 1 dB. As a result, the proposed model can explore the coupling effect of inductor coupled pairs more accurately.

Fig. 3.22 Sketch of the inductor coupling pair in [10].

Fig. 3.23 Sketch of the inductor coupling pair in [11].

Table 3.6 Comparison with other papers.

This work Ref.[10] Ref.[11]

Process

Home-made process on 21-23Ω-cm P-type substrate

0.25um, 5-level metal, CMOS process

on 10Ω-cm p-type substrate

0.25um, 4-level metal, logic-CMOS process on 10Ω-cm P-type substrate Measurement Full four ports Two ports

(other ports floating)

Two ports (other ports floating)

Modeling 1 GHz -15 GHz N/A 1 GHz -10 GHz

Error

(Sim. and Mea.) About 1dB N/A N/A

Error (Mea. and

Model.)

0.3 dB About 1dB N/A

N/A: Not available.

Port 1 floating

Port 4 floating Port 2

Port 3

Port 1 d Port 2

3.5 Summary and contributions

In Section 3.2, the wide band model for spiral inductor is proposed and the improved model represents the good agreement from 0.2 GHz to 30 GHz. Moreover, the coupling effect of two inductors is discussed in Section 3.3. The simulation method can predict the coupling effect precisely with the error of 1 dB between the simulation and measurement results. Utilizing the simulation method can understand the coupling effect between inductors. Thus, the circuit designers can use the method to avoid the unwanted coupling effect during the circuit design procedure.

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