fundamentals of algorithm in general standard sensitivity...

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Algorithm in General

Standard Sensitivity . . .

Noise Sensitivity . . .

Noise-Figure . . .

Distributed Amplifier

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Fundamentals ofCommunications

(XE37ZKT), Part I

Sensitivity Analysis,Distributed Amplifier, MMIC

Josef Dobes

9th

http://radio.feld.cvut.cz

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1. Outline

• Algorithm in General

• Standard Sensitivity Analysis

• Noise Sensitivity Analysis

• Noise-Figure Sensitivity Analysis

• Distributed Amplifier

– Frequency Characteristic

– Sensitivity Analysis

– Assessing Results

• MMIC

– Impedance Matching

– Sensitivity Analysis

– Optimizing Noise Figure


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2. Algorithm in General

A parametric system of the circuit linear equations can be written inthe form

A(p) x(p) = b(p),

where p is one of the circuit parameters on which the sensitivities arerequested.


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2. Algorithm in General

A parametric system of the circuit linear equations can be written inthe form

A(p) x(p) = b(p),

where p is one of the circuit parameters on which the sensitivities arerequested.

The vector of the derivatives with respect to this parameter ∂x(p)/∂p

marked by x ′(p) can be obtained by differentiating

A ′(p) x(p) + A(p) x ′(p) = b ′(p),

which gives the basic system of the complex linear equations

A(p) x ′(p) = b ′(p) − A ′(p) x(p).


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3. Standard Sensitivity Analysis

The circuit contains an independent input source and no other internalsources in this case. Therefore, the first part of the right side is equalto zero and the system is simpler

A(p) x ′(p) = −A ′(p) x(p).


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3. Standard Sensitivity Analysis

The circuit contains an independent input source and no other internalsources in this case. Therefore, the first part of the right side is equalto zero and the system is simpler

A(p) x ′(p) = −A ′(p) x(p).

If simulator procedures are unable to determine the parametric deriva-tives ∂A.../∂p in a symbolic way, they must be computed numerically.For example, the simplest approximation of the derivatives

A ′(p) ≈ A(p + ∆p) − A(p)

∆p

can be used, and using this formula gives the final system

A(p) x ′(p) =b(p) − A(p + ∆p) x(p)

∆p.


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4. Noise Sensitivity Analysis

The circuit contains nn internal noise sources and no independent in-put source in this case. A jth output of the noise analysis is determinedby solving the system

A(p) jx(p) = jb(p), j = 1, . . . , nn,

which is of the same type as that in the standard analysis. Therefore,the complex LU factorization of A must be executed only once ∀f.


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A(p) jx(p) = jb(p), j = 1, . . . , nn,


Similarly, the sensitivity of the jth output is determined by solving

A(p) jx′(p) = jb

′(p) − A ′(p) jx(p).


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A(p) jx(p) = jb(p), j = 1, . . . , nn,


Similarly, the sensitivity of the jth output is determined by solving

A(p) jx′(p) = jb

′(p) − A ′(p) jx(p).

If the procedure is unable to compute the parametric derivatives∂A.../∂p in a symbolical way, they must be determined numericallyusing above approximation of A ′(p) and the analogy is used

jb′(p) ≈ jb(p + ∆p) − jb(p)

∆p, j = 1, . . . , nn,

which gives the final system

A(p) jx′(p) =

jb(p + ∆p) − A(p + ∆p) jx(p)

∆p, j = 1, . . . , nn.


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5. Noise-Figure Sensitivity Analysis

General formulae for computing the noise factor Fn and the noisefigure FdB

n are the following1

Fn =V2

n − V2n,Rload

A2V4kT0Rsource

, FdBn = 10 log (Fn) ,

where Vn, AV , k, Rsource, and T0 are the spectral density of the totaloutput noise voltage, voltage gain, Boltzman constant, internal resis-tance of the input source, and standard temperature (T0 = 290 K).


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5. Noise-Figure Sensitivity Analysis

General formulae for computing the noise factor Fn and the noisefigure FdB

n are the following1

Fn =V2

n − V2n,Rload

A2V4kT0Rsource

, FdBn = 10 log (Fn) ,

where Vn, AV , k, Rsource, and T0 are the spectral density of the totaloutput noise voltage, voltage gain, Boltzman constant, internal resis-tance of the input source, and standard temperature (T0 = 290 K).

However, the procedure should be improved in the following ways:

• The circuit must be matched in advance because the Friis defi-nition of the noise factor assumes available signal and noisepowers.

• The subtraction is performed manually for each frequency in theoriginal work. However, if the load resistor is created artificiallyas a current source controlled by its voltage, then the noise ofthe load resistor will not be generated.

1J. Ortiz and C. Denig, “Noise figure analysis using Spice,” Microwave Journal,pp. 89–94, Apr. 1992.


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By differentiating, the requested sensitivity of FdBn on the parameter

p marked FdBn

′can be obtained:

FdBn

′∣∣∣Vn,Rload

= 0=

10

ln (10)

1

V2n

A2V4kT0Rsource

× 2VnV ′nA2

V4kT0Rsource − 2AVA ′V4kT0RsourceV

2n

(A2V4kT0Rsource)2

=20

ln (10)

(V ′

n

Vn

−A ′

V

AV

).


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By differentiating, the requested sensitivity of FdBn on the parameter

p marked FdBn

′can be obtained:

FdBn

′∣∣∣Vn,Rload

= 0=

10

ln (10)

1

V2n

A2V4kT0Rsource

× 2VnV ′nA2

V4kT0Rsource − 2AVA ′V4kT0RsourceV

2n

(A2V4kT0Rsource)2

=20

ln (10)

(V ′

n

Vn

−A ′

V

AV

).

Let us emphasize that the resulting formula is unusual—most of theCAD tools are not able to determine the sensitivity V ′

n and thereforeFdB

n′, too.


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6. Distributed Amplifier

R101/50

R2/50

R100/50

R1/50

V1/AC(1)

C4/47nL19/1n

V100/5V

Outp

ut

C3/47n

L101/100u

L9/1n

G8/CFY

11

L18/2n

L8/2n

G7/CFY

11

L17/2n

L7/2n

G6/CFY

11

L16/2n

L6/2n

G5/CFY

11

L15/2n

L5/2n

G4/CFY

11

L14/2n

L4/2n

G3/CFY

11

L13/2n

L3/2n

G2/CFY

11

L12/2n

L2/2n

G1/CFY

11

L11/1n

Input

L100/100u

C2/47n

L1/1n

C1/47n


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6.1. Frequency Characteristic

100000 1E+6 1E+7 1E+8 1E+9 1E+10

0

5

10

15

20

25

Frequencyÿ(Hz)

Am

plificat

ionÿ

(dB)


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6.2. Sensitivity Analysis

105 106 107 108 109 10103

4

5

6

∂∂C JS

∂∂C DS

−5×1012

−4×1012

−3×1012

−2×1012

−1012

0

1012

f (Hz)

∂|V

Outp

ut|/

∂C

JS,∂|V

Outp

ut|/

∂C

DS

(V/F)

|VO

utp

ut|

(for

unit

V1)

(V)


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105 106 107 108 109 10103

4

5

6

∂∂C JS

∂∂C DS

−5×1012

−4×1012

−3×1012

−2×1012

−1012

0

1012

f (Hz)

∂|V

Outp

ut|/

∂C

JS,∂|V

Outp

ut|/

∂C

DS

(V/F)

|VO

utp

ut|

(for

unit

V1)

(V)

Nominally CJS = 0.75 pF, which is the output for CJS = 0.751 pF?

• Using the sensitivity analysis: 4.098543 V,

• Using the direct computation: 4.098516 V.


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7. Monolithic Microwave Integrated Cir-cuit (MMIC)

6

7

V8/D

C(0

),A

C(1

)

R4/0.6

R1/200

R5/20

R2/4k

R3/440

L1/0.8

n

9

8

V6/D

C(5

),A

C(0

)

C2/10p

I7/D

C(0

),A

C(r

esist

(V7,V

0,3

00))

Q1/QRF6E20

Q2/QRF6E20

C1/100pRsource

Rload

LE

RE

RB

βF,rB,τF

βF,rB,τF


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7.1. Optimizing Noise Figure

Possible improvements of the noise figure:

1. Matching at the input and output

2. Updating a circuit parameter after the sensitivity analysis

0.6 0.8 1 1.2 1.4

1.8

2

2.2

2.4

2.6

2.8

3

Unmatched

f (GHz)

FdBn (dB)


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0.6 0.8 1 1.2 1.4

1.8

2

2.2

2.4

2.6

2.8

3

Unmatched

f (GHz)

FdBn (dB)

Matched


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0.6 0.8 1 1.2 1.4

1.8

2

2.2

2.4

2.6

2.8

3

Unmatched

f (GHz)

FdBn (dB)

Matched

Updated


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7.2. Impedance Matching

Zinp

−9.32

Zout

1.06

−20

−10

0

10

20.2

299

0.6 0.8 1 1.2 1.4

50

100

150

200

250

300

f (GHz)

|Zin

p,o

ut|

(Ω)

arg

( Zin

p,o

ut)

( )

As the noise figure is the worst problem at 1.5 GHz, the impedancematching at this frequency has been used.


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∂∂β F

∂∂r B

∂∂R B

∂∂R E

∂∂L E

∂∂τ F

0.0122

0.6 0.8 1 1.2 1.4

−0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

f (GHz)

(∂F

dB

n/∂p)(

p/100)

(dB),

p∈

βF,r

B,R

B,R

E,L

E,τ

F

The transit time τF of QRF6E20 was 28 ps. It is now possible touse the transistors with τF = 21 ps (i.e., − 25 %).


fundamentals of algorithm in general standard sensitivity...

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