Design of Matching Network in Microwave FET...

Design of Matching Network in Microwave FET Amplifiers

Lucio Scucchia

17th ITSS - Pforzheim 2007

Dipartimento di Ingegneria ElettronicaUniversità degli Studi di Roma “Tor Vergata”Roma - ITALY

IntroductionSmith Chart Impedance Transformation with LEsTransmission Lines Narrow Band Matching with LEs and TLsBroad-Band Matching with LEs and TLsMatching at 2 frequencies with LEsDC Biasing, Stability and GainDesign of a Gain Amplifier at 2 frequenciesConclusions

Introduction (1)

Why impedance matching ?

Maximization of the return loss for Gain Amplifier.

Minimization of the Noise Figure for Low Noise Amplifier

Maximization of the output power for Power Amplifier.

Minimization of signal distortion in transmission lines, avoiding wavefront reflections and pulse superposition.

To avoid unnecessary power loss, lossless matching network are considered.

The Lossless Matching Network transforms ZL into 50 Ω or 50 Ω into ZL

Simple Matching Networks are usually more reliable than more complex Networks .Matching Network can be carried out with:

Matching

NetworkZL

Lumped Elements (LE)Transmission Lines (TL)

Introduction (2)

Smith Chartand

Impedance transformation with LEs

ˆZ R Zj ˆZ R ZΓ Γ Γ − −

= + = =+ +

1 1V R I2 21 1V R I2 2

jΓ Γ Γ

= ⋅ + ⋅

= ⋅ − ⋅

= + =r i

Incident Wave Voltage

Reflected Wave Voltage

Reflection Coefficient

One port characterization

Smith Chart (1)

The Smith Chart (SC) is a graphical tool useful to design high frequency circuits.

Every point on the SC represents a normalized impedance and the corresponding reflection coefficient.

The SC is achieved by plotting on the plane of the reflection coefficient the curves corresponding to constant resistance and reactance values.

10 1010 20 Γ= = −i

PRL log logP

|Γ| =1 , RL= 0 dB

No Reflection

TotalReflection

|Γ| =0 , RL= ∞ dB

Smith Chart (2)

Return Loss

To get curves corresponding to constant resistance values

( ) ( )2 2

2 22 2

1 1 21 1 1

ˆ ˆ ˆ ˆ ˆR - jX R - X Xj jˆ ˆR jX ˆ ˆ ˆ ˆR X R XΓ Γ Γ+ + ⋅

= = + = ++ + + + + +

2 2 2 11 1r i r

ˆ ˆR Rˆ ˆR R

Γ Γ Γ⋅ −+ − ⋅ =

+ +The equation describing a circle family with parameter is achieved R

If parameteris removed

R = 0 R = 1

Smith Chart (3)

X = 1X = 2

-1X =- 1

X = -2^ ^

To get circles corresponding to constant reactance values

( ) ( )2 2

2 22 2

1 1 21 1 1

ˆ ˆ ˆ ˆ ˆR - jX R - X Xj jˆ ˆR jX ˆ ˆ ˆ ˆR X R XΓ Γ Γ+ + ⋅

= = + = ++ + + + + +

2 2 22 1ir i r X

ΓΓ Γ Γ ⋅+ − ⋅ − = −

The equation describing a circle family with parameter is achievedX

If parameteris removed

Smith Chart (4)

1 1 1L L L

ˆZ R Y G G Y YˆZ R G Y Y

− − −= = = =

+ + ++

Placing the constant resistance circles on the constant reactance circles the SC is achieved.

Γ can be expressed as a function of the normalized admittance

So the SC of the admittance is obtained by a 180° rotation of the impedance Smith Chart

ˆ ˆR > 0 X > 0

ˆ ˆR > 0 X < 0

ˆ ˆG > 0 B < 0

ˆ ˆG > 0 B > 0

Γ Ŷ is equal to Γ -Z

Smith Chart (5)

Smith Chart (6)

( ) 2j1Zj1Y

+=−=

ˆˆ ( )

j1Z2j1Y

+=−=

( ) 2j1Zj1Y

−=+=

ˆˆ ( )

j1Z2j1Y

−=+=

jZjY−=

|Γ||Γ

j [S]L

j C [S]ω

− [Ω]ω

j L [ω Ω]

Smith Chart (7)

Transmission Lines

In microwave circuits the transmission lines are fundamentally used:

• to convey high frequency signals from one point to another point.

• to obtain filters and matching networks.

GroundPlane

Conductor

Ground Plane

Conductor

MicrostripLine

CoplanarWaveguide

Transmission Lines (1)

For the structures shown, infinitesimal line lengths can be modeled using the following network :

-V(z+ dz)

R dz L dz

GdzCdz

I(z) I(z + dz)

Applying Kirchhof’s voltage and current laws and solving the 2 equations obtained for an infinitesimal line length:

G (S/m)R (Ω/m)L (H/m) C (F/m)

1(z) = + (z) = ( - )Z

- z z - z ze e e eγ γ γ γ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅+ - + -V V V I V V

+ ≅ ++ ≅ +

I( z dz ) I( z ) dI( z )V( z dz ) V( z ) dV( z )

The line length is infinitesimal so:

= ω L Cβ ⋅

L 1Z = YC Z

=Z0 = Characteristic impedance

β = Phase constant

To simplify the synthesis approaches the TLs used are supposed to be ideal.

( )+ - + -

1V(z)=V V I(z)= V VZ

- j z j z - j z j ze e e eβ β β β⋅ ⋅ ⋅ ⋅⋅ + ⋅ ⋅ ⋅ − ⋅

( )( )

L 00in

Z +j ZZ ( )=

Z j Zθ

⋅⋅

+ ⋅tan

Transmission Line Equation:

ZLZ0 β

V - j le β− ⋅

V j le β+ ⋅

= β θ⋅l = Electrical length

( )( )

L 0in 0

Y +j YY ( )=

Y j Yθ

⋅⋅

+ ⋅tan

G=0R=0

YL= Y0

Y0 θ YL

Quarter-Wave Transformer : 2πθ =

Yin=Y02/YL

Yin=YL

LinY Y=Y0 θ YL

inYY0 θ YL

L0Y Y≠

Short-circuited stub: YL= ∞ Open-circuited stub : YL=0

π/2θ3θ2θ1

inj YY

tan θ−=

Y0 θ( )0inY jY tan θ= ⋅

π/2θ3θ2θ1

Narrow Band Matching

With 2 lumped elements

Narrow Band Matching (1)

50 [Ω] 50[ Ω]

50 [Ω]

50[ Ω] 50 [Ω]

50 [Ω]Z*

50 [Ω]

The same reactive network: transforms 50 Ω Ztransforms Z* 50 Ω

50 [Ω]Z

50 [Ω]

The same reactive network: transforms Z to 50 Ωtransforms 50 Ω to Z*

Z *50Ω

Using a TL and a short-circuited stub

( ) ( )2 2 2 2 21 1

0.020.02L L

b B G Y YG

= ⋅ + + − +

( )( )

0.020.02

Y GArcTan

B b Gθ

⎡ ⎤−= ⎢ ⎥

⋅ − ⋅⎣ ⎦2

2YArcTanb

θ ⎡ ⎤= ⎢ ⎥⎣ ⎦

θ1 length TL

θ 2 len

YL= GL+ jBL

Y1 θ1

YC=YA+YB=0.02

YA=0.02+j b

Y2 θ2

YB=-j b

YC=YA+YB=0.02

Y1 θ1

YA=0.02- j b

Y2 θ2YB= j b

Using a TL and an open-circuited stub

( ) ( )2 2 2 2 21 1

0.020.02L L

b B G Y YG

− = − ⋅ + + − +

( )( )

0.020.02

Y GArcTan

B b Gθ

⎡ ⎤−= ⎢ ⎥

⋅ − ⋅⎣ ⎦2

bArcTanY

θ⎡ ⎤

= ⎢ ⎥⎣ ⎦

Narrow Band Matching (5)θ 1

θ 2len

gth st

Broadband Matching

In general broader bandwidth is achieved when the compensated load curve presents the maximum allowable RL at the high, low and medium frequencies.

YL(fh)

RLmax=10 dB

YIIZII

YL(fl)YB(fh)

YB(fl)

The resulting load curve liesin a specified RL circle.

1 10 1

RL max

I IRL maxI

RL max

II IIRL maxII

Z R YZ

−= =

Broadband Matching (1)

With lumped elements

1. The first two elements are selected to position the mid frequency impedance point to accept RLmax (YI-ZI or YII-ZII)

2. The other two elements form a series or shunt resonant circuit which wraps the load curve into the required RLmax circle.

YL(fh)

RLmax=10 dB

YIIZII

YL(fl)YB(fh)

YB(fl)

2 sections are considered (4 elements).

SEC. 1

YA(f)YL(f )ZA(f )ZL(f )

ZL(fm)ZL(fh)

ZL(fl)

ZA(fm) =ZII

ZA(fh)

ZA(fl)

YL(fm)YL(fh) YL(fl)

YA(fh)

YA(fl)

YA(fm)=YI YII

ZA (f) ZB(f)

ZL(fm)ZL(fh)

ZL(fl)

ZA(fh)

ZA(fl)

ZB(fh)

ZB(fl)

YL(fm)YL(fh) YL(fl)

YA(fh)

YA(fl)

ZB(fh)

ZB(fl )

YA(f) ZB(f)

With distributed elements

The first section is a high impedance line. It must transform the load admittance so that the resulting conductances at fl and fhare equal.TLs with high characteristic impedance allow to obtain YA with smaller susceptance at fl and fh.

3 sections are considered.

SEC. 1

YL(fh)

YL(fl)

YA(fh)

YA(fl)

So that the resulting admittance line is wrapped. This result is obtained using short-circuted stub. The Y2 value, fulfilling the two equations, often is too high to be realized.

YB(fh)

YB(fl)YA(fh)

YA(fl)

The TL width relative to high Y2 values is a significant fraction of λ, so the position of the line is not well defined. To reduce Y2 value :

tan tanl h

Y YB B ff

θ θ= − = −

Two parallel short-circuted stubParallel short and open-circuted stubs Two parallel radial stubs

The second section is a shunt element which provides positive and negative susceptance values (Bh and Bl ) at fh and fl respespectively.

YA(f) YB(f)

θ2 Y2

YL(f) Y1 θ1

YA(f)YB(f)

Y2=0.02Sθ2a=59.2° (at 10GHz)θ2b=29.4° (at 10GHz)

Y2=0.041Sθ2=91.8° (at 10GHz)

freq (8.GHz to 12.GHz)

YA(12GHz)=0.021-j0.03

YA(8GHz)=0.021+j0.024

freq (8GHz to 12GHz)

YA(8GHz)=0.021+j0.024

YA(12GHz)=0.021-j0.03

The third section is a quarter wave transformer which centers the circle on the Smith Chart.To assure RL=12dB at 10GHz YII is fixed 0.0119S. YB(10GHz) is equal to 0.0592S so :

3 (10 ) 0.0265= ⋅ =II BY Y Y GHz

Y1θ1 θ2

Y2Yc(f)

θ1=90°(fm)

freq (8GHz to 12GHz)

YB(10GHz) YC(f)

8.5 9.0 9.5 10.0 10.5 11.0 11.58.0 12.0

freq, GHz

RL (dB)

Matching at 2 frequencies with LEs

The matching network is composed by three sections

SECTION

YLZLY3

Y3(fl) = Y3(fh) =G0Z3(fl) = Z3(fh) = R0

Matching at 2 frequencies with LEs (1)

Four events are considered

Re[Y ( )] Re[Y ( )] > G Event A

Re[Y ( )] Re[Y ( )] < G Event B

L1 YLY1

L1 ZLZ1

Re[Z ( )] Re[Z ( )] > R Event C

Re[Z ( )] Re[Z ( )] < R Event D

1R = =50 ΩG

Event A Event B

SeriesInductor

Event D Event C

ShuntInductor

Matching at 2 frequencies with Les (2)

Event A Section 1

Series Inductance L1

L 1 L h 1

Re[Y ( )] Re[Y ( )]

1 1Re =Re1/Y ( ) + 2 L 1/Y ( ) + 2 L

f j f f j fπ π

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

L1 YLY1

If the Section 2 is a Parallel Resonant Cell

The Section 2 does not change Re[Y1]

The section 3 has to transform Re[Y1(f1)] and Re[Y1(fh)] into G0

Y2(fh)Y2(fl)

Y1(fl)

Y1(fh)

Y3(fl)=Y3(fh)=G0

Re[Y1(fl)] = Re[Y1(fh)]

Event A Section 3

j X3j B3Y3 Y2

L1YLY1

1= + C

2 L +1 Y ( )1

= + C 2 L +1 Y ( )

jj f f

⋅⋅

2 10 l l

h 3 h 3

1Re[Y ( )] = Re Re[Y ( )]

1/(G - 2 C )- 2 L

1Re[Y ( )] =Re Re[Y ( )]

1/(G - 2 C )- 2 Lh h

l lj f j f

j f j f

⋅ ⋅

⎡ ⎤=⎢ ⎥

⎣ ⎦⎡ ⎤

=⎢ ⎥⎣ ⎦

1 1= +

1 1 j 2 L + j 2 C Y ( )

1 1= +

1 1 j 2 L + j 2 C Y ( )

⋅⋅

1Re[Y ( )] = Re Re[Y ( )]

G -1/( 2 L ) 2 C

1Re[Y ( )] =Re Re[Y ( )]

G -1/( 2 L ) 2 C

j f j f

⋅ ⋅

⎡ ⎤⎢ ⎥

=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥

=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

Event A Section 3

Event A Section 2

Y ( ) Y ( )

2 1 22

1 Im[Y ( )]-Im[Y ( )] = 2 C -

Im[Y ( )]-Im[Y ( )] = C -2 L

l l ll

f f ff

Section 1

Section 3

L CSection 2

Y2=G2+j B2

Y1=G1+j B1

Event A

If the Section 2 is a Parallel Resonant Cell

The Section 2 does not change Re[Y1]

The Section 3 has to transform Re[Y1(fl)] and Re[Y1(fh)] into G0

Re[Y1(fl)] π Re[Y1(fh)]

L1 Y3(fl)= Y3(fh)=G0

Y2(fh)

Y2(fl)

Y1(fl)

Y1(fh)

Event B Section 1 - 3

j X3Y3 Y2

L1YLY1

Re[Y ( )] = Re[Y ( )] Re[Y ( )] = Re[Y ( )]

lf ff f

1 1Re = Re

1/Y ( )+ 2 L -

1 1Re = Re

1/Y ( )+ 2 L -

lf j f

⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

Y1=G1+j B1Y2=G2+j B2

1 1Re = Re

1/G - 2 L 1/Y ( )+ 2 L

1 1Re = Re

1/G - 2 L 1/Y ( )+ 2 L

j f f j f

⋅ ⋅

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Y1=G1+j B1Y2=G2+j B2

Event B Section 1 - 3

Event C

Z1=R1+j X1

Z2=R2+j X2

DC Biasing, Stability and Gain

To get the required perfomances the biasing point has to be selected

DC Block

To analyse stability and gain of a small-signal amplifier the transistor and the biasing networks can be represented by S-parameters.

(S11, S12, S21, S22)

0.9 Idss

Ids(mA)

0.15 Idss

0.5 Idss

3 6 9Vds( V )

Vgs= 0V

Vgs= -2V

Vgs= -3V

Vgs= -4V

Vgs= -1V GAIN

LINEARITY

DC Biasing, Stability and Gain (1)

Uncoditionallystable

12 2111

12 2122

ΓΓΓ

⋅ ⋅= +− ⋅

( ) 1 ( ) 1in L out S&Γ Γ Γ Γ< <

for (|ΓS| <1 & |ΓL|<1 )

ΓS Plane ΓL PlaneZS ZL

Zin= ZS* Zout= ZL

Simultaneous Conjugate matching

ZoutΓout

Outputnetwork

Inputnetwork

ZinΓin

Conditionally stable

for some values of (|ΓS|<1 , |ΓL|<1 )

1 1in L out S( ) and ( )Γ Γ Γ Γ< <

ΓS Plane

|Γout|=1ΓS stability circle

|Γout|>1Unstable Region

Zinπ ZS*

ΓL Plane|Γin|=1

ΓL stability circle |Γin|>1Unstable RegionZL

Zoutπ ZL*

The stability regions are defined by the stability circles on ΓS and ΓL.

The system stability is guaranteed if ZS and ZL are chosen outside the stability circles.

Simultaneous Conjugate matching is not possible.

The stability can be studied only on the GS plane if the condition Zout =ZL

* is realized.

The MSG Circle has to be defined.

The stability is guaranteed if ZS is chosen outside the MSG circle.

The device is unconditionally stable if :

2 2 211 22

11 22 12 21

k Rollet Stability Factor' s

S SS S

S S S S

− − += >

= ⋅ − ⋅ <

|Γin|=1ΓL Plane

Zout = ZL*

|Γout|=1ΓS Plane

MSGCircle

Zinπ ZS*

If Simultaneous Conjugate Matching (S.C.M.) is verified the Gains are maximized and GTmax= GAmax = G max= MAG

If the System is conditonally stable the S.C.M. is not possible and a set of couples (Zs , ZL) provide the obtainable GTmax= GAmax = G max= MSG

( )2211

MAG k kS

= ± −

( , , )

LT S L

Outputnetwork

Inputnetwork

Unconditionally Stable Device

ΓS PlaneGA = MAGGA1< MAGGA2 < GA1GA3 < GA2GA4 < GA3

Conditionally Stable Device

ΓS Plane

GA = MSGGA1< MSGGA2 < GA1GA3 < GA2

Design of a Gain Amplifier at 2 frequencies

Design Procedure at a Given Frequency

1) Choice of the device2) Biasing network design3) Stability condition 4) Choice of Zs5) Input Network Design6) Computation of Zout

7) Output Network Design

Inputnetwork

R0OutputnetworkVS

ZSMSG Zout

ZL= Zout*Zin

Design of a Gain Amplifier at 2 frequencies (1)

The matching technique described can be used considering 2 frequencies. The used device is a low-noise GaAs FET with a 4x50μm gate periphery by Alenia. The operating frequency is 5 GHz (fl = 4 GHz, fh = 6 GHz ).

Choice of ZS(fl) and ZS(fh) so that GA(fl) = GA(fh) and stability condition is satisfied.Imput Network design.Computation of Zout (fl) and Zout (fh).Output network design for the conjugate matching.

ZS(fl)(

ZS(fh)(

Zin(fl) (Zin(fh)

Zout(fl) (Zout(fh)

R0Inputnetwork

OutputnetworkVs

3.6 nH

1.6 n F

1.1 pF

0.2 p F

V ggV dd

0.5 pF

0.2 pF

5.8 nH

2.3 nH

Matching network element values

C2: MICAP3W=10 mmL=10 mmGe=10 mmL=10 mmNP=2Wt=20mmWF=20mm

InputNetwork

L1: MRIND N=2.25L1=200 mmL2=224 mmW=10 mmL=10 mm

L2: MRINDN=3.25L1=280 mmL2=305 mmW=10 mmL=10 mm

C3: TFC W=61mmL=74mmT=0.3mmEr=7.8DO=10mm

Output Network

C2: TFC W=49 mmL=49 mmT=0.3 mmEr=7.8DO=10 mm

C3: TFC W=61 mmL=68 mmT=0.3 mmEr=7.8DO=10mm

1 2 3 4 5 6 7 8 9 10

freq, GHz

0|S(2,2)| [dB]

lumped

1 2 3 4 5 6 7 8 9 10

freq, GHz

|S(2,1)| [dB]

lumped

Conclusions

Some fundamental points about matching have been examined.

A systematic approach for matching network design has been described.

In order to demonstrate the usefulness of the proposed approaches a gain amplifier design has been carried out.

Design of Matching Network in Microwave FET...

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