European Master of Research on Information Technology Design and Analysis of RF and Systems

1.1 basic concepts: Introduction to narrow-band matching networks

March 2010

Francesc Torres, Lluís Pradell, Jorge Miranda

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Voltage and current in the transmission line V For any lossless transmission line: L IL V (z)  V  (z) V  (z) V(z V+(z) Z ≠ Z 1 Z0 I(z) - L 0 I(z)  V  (z) V  (z) V (z) Z0

-z z=0 where At z=0 Periodicity of V(z) and I(z): wavelength

   jz    (z  z)  z  n2 , z  n2 V (z)  V0 e V (0)  V0 V0  VL 1     jz I(0)  V V  I 2 V (z)  V0 e Z L z  n  n 0  Impedance: load Reflection coefficient: load V V  V  V    V  1  V  Z  Z Z  L  Z  Z L  Z L    L 0 L 0   0   0 L  I L V V V  LV 1 L V Z L  Z0

1 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

Reflection coefficient in the transmission line

At any point z of the transmission line the impedance is computed as:

V (z) 1 (z) Z(z)  Z0 Z(z)   Z0 (z)  Z0 I(z) 1 (z) Z(z)  Z0

At any point z of the transmission line the reflection coefficient is:

  V (z) V 2 jz 2 jz 2 jz (z)   e  (0)e  Le V  (z) V  2 jz Modulus (z)  Le  L Constant in z 180º 2z rad  2z deg Phase  2z  2 (z  z)  n2 , 2z  n2 Linear  n  n  Increasing with +z (towards load) z     n Periodicity: half a wavelength  2 2

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

The transmission line as impedance transformer

()  L  2 j

(z)  L  2 jz Z0, β ZL≠ Z0

|Γ(z) |=|ΓL|

i  () z   z=0

Example: Compute the Zi of a circuit formed by a transmission line of length is λ/8 and loaded with ZL=0 (sc).

2   2 j  j Z L  Z0 2 j  8 2 L   1 i  ()  Le  e  e  j Z L  Z0

1 i Zi  Z0  jZ0 1 i

2 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Standing waves in the transmission line (i)

V (z)  V e jz 1  e jL e2 jz  V  1  e jL e2 jz  L L   V  V  jz jL 2 jz jL 2 jz  I(z)  e 1 L e e   1 L e e  Z0 Z0

At any point z where the term (1+ΓL) is real, voltage and current are real an their magnitude is either maximum or minimum:

V  V (z)  V  1  max max L    V L  2zmax  2 rad I  I(z)  1   min min L   Z0 z  z  max min 4 V  V (z)  V  1  min min L  L  2zmin   rad  V  I  I(z)  1   max max L  Z0

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Standing waves in the transmission line (ii) At any point z where the voltage is maximum the impedance is real an maximum, if the voltage is minimum the impedance is real and minimum:

  jz V (zmax ) V e 1 L  1 L  Z(zmax )     Z0  Zmax I(zmax ) V  jz 1 L e 1 L Vmax  Imin (zmax ) Z0   jz Vmin  Imax (zmin ) V (zmin ) V e 1 L  1 L  Z(zmin )     Z0  Zmin I(zmin ) V  jz 1 L e 1 L Z0 The voltage standing wave ratio (VSWR or S) is defined as

Vmax 1 L S 1 0  L 1 S   , L   Vmin 1 L S 1 1 S  

Matched load: Z=Z0, Γ=0, SWR=1

3 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Standing waves in the transmission line (iii) Maximum and minimum impedances at in the transmission line:

λ/4 λ/4 V (z)

Vmax

Vmin

zmin zmax zmin z=0

Vmax 1 L Vmin 1 1 L Z0 Zmax   Z0  Z0S Zmin    Imin 1 L Imax Z0 1 L S

VSWR is easy to measure and it is widely used to specify mismatch

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

Power in the transmission line (i) At any point z of the transmission line the net power is computed as:  1 *  P(z)  e V (z)·I * (z)  e V e jz 1 (z)  V  e jz 1 * (z)          Z0   2  V 2  2 1 (z)  P 1 (z) e1 a1 a* 1 a 2 , a  r  jx Zo Since the modulus of the reflection coefficient is constant in z, the net transmitted power is constant at any z and equals the power delivered to the load:

 2  2   P(z)  P 1 (z)  P 1 L  P  P  PL

Where the power associated to the “positive” (incident) and “negative” (reflected) waves is: 2 2 V  V     2  P  , P  , P  L P Z0 Z0

4 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

Power in the transmission line (ii)

+ Example: P =1 Watt in transmission line of impedance Z0=50Ω j 2 1 1 Z  50  j100,   ,   , P  P  0.5W L L 1 j L 2 2 Return loss definition:

 P 2 RL    , RL  20·log (  ), P (dBW )  P (dBW )  RL(dB) P L 10 L In this case

RL  3 dB, P  0dBW  3dB  3dBW

The standing wave ratio (SWR) in the transmission line 1  S  L  5.8 1 L

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Matching networks If we have a mismatched load

+  2  P PL  P 1 L  P Z0 ZL≠ Z0 - • A fraction of the incident power P+ P ≠0 is not delivered to the load • A fraction of the incident power ΓL≠ 0 returns to the generator ZL≠ Z0   2 P  P  + L P Matching A matching network must be Z0 - Network •Simple (passive) P =0 •Lossless (L, C, Transformer,

transmission line, waveguide,…) Γi= 0 Zi= Z0

 If lossless: PL  Pi  P All power is delivered to the load

5 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Normalized impedances and admittances

An impedance can be normalized to a reference impedance Z0 Z Y 1 Z  L ; Z  0  P+ L Z L Y Y 0 L L Z0 ZL≠ 1 P-≠0

In this case, the reflection coefficient: ΓL≠ 0 ZL≠ 1 P+ Matching Z L  Z0 Z L 1 1YL Z0=1 L    P-=0 Network Z L  Z0 Z L 1 1 YL

Γi= 0 Zi= 1

Working with normalized impedances is equivalent to work with

reference transmission lines of Z0=1

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Example: lossless narrowband matching network (i)

Zx =jX

P+ Z0=1 YB =jB ZL=4-j2 P-=0

YL=0.2+j0.1 Zi= 1 Z1 In this case, the reflection coefficient: 1 1 1  Z1  jX Z1  1 jX   YL  jB 0.2  j(0.1 B) By equalling the real parts By equalling the imaginary parts

0.2 B  0.3  (0.1 B) X  2 1   1 X   1 2 2  2 2  0.2  (0.1 B) B2  0.5 0.2  (0.1 B) X 2  3

6 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Example: lossless narrowband matching network (ii) Solution 1 Solution 2

Zx =-j2 Zx =j2

YB =j0.3 Z0=1 ZL=4-j2 Z0=1 YB =-j0.5 ZL=4-j2

Zi= 1 Zi= 1

Sometimes a shunt-series solution does not exist and a series-shunt network must be used:

Z0=1 Z0=1 ZL

Zi= 1 Zi= 1

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems The Smith Chart Z(z) → Impedance at any point z of the transmission line ReZz  0,  ImZz , 

P.H. Smith, in 1939, developed a chart to represent any impedance Z(z) as a function of its related reflection coefficient ρ(z). This graphic tool is based in the fact that |ρ|≤1 which allows to represent all impedances in a finite area. The Smith chart is currently used as a universal tool to represent impedances.

P.H. SMITH

7 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Relation Z(z) - ρ(z)

An impedance can be normalized in relation to a reference impedance Zo as

Z 1  Z 1 j Zrjx  ej Z 1 ri Z0 1 

Mathematically, this correspond to a bilinear transformation which translates a circle in the impedance domain into a circle in the reflection coefficient domain.

x i r≥0  ≤1 Z  r r

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

Relation Z(z) - ρ(z)

Now, if ρ=ρr+jρi is substituted in the expression of the normalized impedance, the equations that relate the loci r and x constants as a function of the components ρr and ρi are obtained. This is a set of circumferences in the complex domain  :

22 2 r 2 1112  ri;1  ri  2 rr11  xx r 1 Constant resistance circle: CENTRE  ,0 RADIUS r 1 r 1

1 1 Constant reactance circle: CENTRE  1, RADIUS x x i x r=const x i x=const.

r  r r r

8 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems The reflection coefficient in the complex domain

ℓ= 360º ℓ= 180º ℓ= 90º ZL i z=-ℓ z=0 90º  e L Towards |=1 Ze load Rationale L   ZL L e Ze L|

L e 180º L 0º

ℓ r 4 jz()  ()z ee2 jz L  LL L|

e

j L LL(0)z e Towards 4 generator j(())    2 j  L  270º eLL()zee   

01L|

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Chart of impedances x = 0.5 x = 1

x = 2 x = ∞

Circles of constant resistance r = 0 r = 0.5 r = 1 r = 2 r = ∞ x = 0

Circles of constant x = - 2 reactance x = - 1 x = - 0.5

9 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems At a point placed in the transmission line of Z =50 0 x = +3 Ω we measure an impedance 100+j·150 Ω What is the value of ρ at this point? | ρ | = 0.75

r = 2 100j 150 Z 23j φ = 26º 50

  0.75 26º

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems If 1390º , what is its related normalized impedanceZ ? How does φ = 90º this impedanceZ change if the point is moved along the transmission line?  j  0.33 e 2 x = +0.6

Zj0.8 0.6 r = 0.8

As we move along a transmission line, the modulus of the reflection coefficient is constant:   0.33 The normalized impedance Z varies as given by this circle. | ρ | = 0.33

10 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems transformation: ρρ↔↔ Z¯

Zj23 26º

SWR = S (standing wave ratio) Z | ρ | x RET’ NNLOSSdB LOSS, dB = 20 log  2 REFL. COEFF. P =  φ REFL.COEFF, E OR I = 

  0.75 26º

S  7

Lret  2.6 dB

0.752.67

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems 0.45·λ

Input impedance

ZL= 60 –j·90 V V 1/S=0.28 S36S=3.6 máx mín Ze Z0 = 75 Ω ZL I I mín máx Z  3.6 l = 0.45·λ Z  0.28 máx mín x Z ZjL 0.8 1.2 e x ZL Zje 21.6

Zje 150 120

ZRmáx 270 máx S=3.6 ZR21 mín mín

11 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Admittance

Z RjX ZL x  YGjB 1 X    L B   L c   L 1 X  B   C C  C c ZYY11  1   Z 11YY  1 x Z  50  0 YL  ZjL 0.2 0.5 ZjL 10 25 Rotation: 180º YjL 0.7 1.7

YL YYYLL0  Y L  Z0

YjL 0.014 0.034

European Master of Research on Information Technology Design and Analysis 0.1·of RFλ and Microwave Systems Input admitance Z x e

ZL ZL= 10 + j·15 x

Ye Z0 = 50 Ω ZL

l = 0.1·λ

ZjL 10 15  x Z0 50  YL ZjL 020.20 03.3 x Ye

Yje 0.3 0.7 1 Y 0.02 0.1·λ 0 50

Yje 0.006 0.0145

12 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

ShortShort--circuitedcircuited line 0.1·λ

Ze, Ye s.c.

l = 0.1·λ x Short circuit Open circuit x Zje 0.73  Yje  1.4 

Open circuit line

Ze, Ye o.c

l = 0.15·λ 0.15·λ

Zje  0.73  Yje 1.4 

ChangeChange of reference reference European Master of Research Design and Analysis of RF and Microwave Systems on Information Technologyimpedance 0.15·λ

0.1·λ Ze 50 Ω 100 Ω 150 Ω ZL 1’ 1 2’ 2 0.15·λ Z1 Z 0.1·λ x 1’ 0.15·λ x 75 ZL 0.5 150 x Z Zj10.7 2’ 1 ZL x x ZZ11150  150  j 105 Z2 150j 105 Z  1.5 j Z 1' 100 e x

Zj2 181.80 09.9

ZZ22100  180  j 90

180j 90 Zj3.6 1.8 0.15·λ 2' 50

Z3 0.28j  0.52

Ze Z3 50 14j  26

13 MATCHING.MATCHING. Transmission Transmission European Master of Research Design and Analysis of RF and Microwave Systems lineon Informationline plus Technology plus reactance reactance in in 0.114·λ series series

x Z j·XS Z x ZL L

¿ l ?

Ze=1 Z=1+j·X XS = -X Ze=1

ZjL 0.2 0.5 Zj12.1 x jXS j2.1 1 Z’ 2.1 Z  C 0 Zj'1 2.1 0.234·λ

jXS '2.1 j

 LZ2.1 0

MATCHING.MATCHING. Transmission European Master of Research Design and Analysis of RF and Microwave Systems 0.289·λ lineon InformationTransmission plus Technology reactance line inplus reactance in parallel parallel ZL x

ZL j·BP x YL Y

¿ l ? Y=1+j·B Ye=1 BP = -B Ye=1 ZjL 0.4 1.4

YjL 0.2 0.65 Yj12.4 x jBP j2.4 Y’ 1 x 2.4 Y0 Y   L L 0.402·λ Yj'1 2.4

jBP '2.4 j

 CY2.4 0

14 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

Example A)

In a Smith chart referenced

to an impedance Z0 =50 Ω, represent the ROE=2 following sets of x R>100 impedance loci: a) The impedances that cause a standing wave ratio SWR=2 b) Impedances with real part larger than 100 

100 r 2 50

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems c) Set of impedances that disipate 20 mW when a x x voltage of 2 V rms is applied YL + g =0.25 r =4 V =2 V L L L Z L x x PL=20 mW -

ZL ***2 PL ReVILL Re VVY LLL  V L G L 3 PL 20 10 GL  0.005 x 2 4 x VL

GL gGZLL0 0.25 Y0 d) Impedances that have the same real and imaginary parts r=x

15 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

Example B) A generator at f=300 MHz feeds an unknown load ZL by menas of a transmission line of impedance Z0=70 . In the line it is measured |Vmax|=5.2 V and |Vmin|=1.1 V. Furthermore when the load is substituted by a short circuit the positions of all minimum voltages move 15 cm towards the load. ¿What is the impedance ZL?

V 1 v ROES max 4.72, Z  0.212,f  300MHz  p 1 m VROEfmin min S 0.15  |V(z)| Vmin Vmin

|V|max

ZL ZL ZL ZL

|V|min z=-ℓ 0.15  z=0 z( 0 3   -1.5 -1 -0.5 0 2 Zmin Zmin 2 |V(z)| Vmin Vmin |V|max

c.c c.c c.c c.c.

z=-ℓ z=0 z( 0 -1.5 -1 -0.5 0

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

Example B)

Zmin  0.212

z  0.15

Zmin x

Z L 0.52j 1.2

Z   j121.2 L 0.52 ZZZLL·0 36.4 j 84

0.15

16 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

Example B) The same result is obtained if the position V moves 35 cm V 1 v p min ROES max 4.72, Z  0.212, f  300 MHz  1 m towards the generator VROEfmin min S 0.35  0.15  |V(z)| Vmin Vmin

|V|max

ZL ZL ZL ZL

|V|min z=-ℓ 0.35  0.15  z=0 z( 0 3   -1.5 -1 -0.5 2 Zmin Zmin 2 |V(z)| Vmin Vmin |V|max

c.c c.c c.c c.c.

z=-ℓ z=0 z( 0 -1.5 -1 -0.5 0

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Example B

Zmin  0.212

z  0.35

Zmin x

Z L 0.52j 1.2

Z   j121.2 L 0.52 ZZZLL·0 36.4 j 84

0.150.35 

17 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

Example C An has and input impedance of 75  at 400 MHz. It is fed by means of a parallel wires

transmission line with an impedance Z0=150 . a) Designamatching network compounded of transmission line plus a shunt capacitor. The dielectric

constant of the transmission line is εr=2.2.

ZL j·Bc YL

ℓ Y=1+j·B

Ye=1 B = -Bc Bc>0

¿ ℓ, C ?

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

ZL j·Bc YL

ℓ Yj10.7 Y=1+j·B x

Ye=1 B = -Bc Bc>0 YL 0.25  Ye  1 x 75 1 ZYLL0.5 2 150 ZL

c /  310/ 8 2.2  r 0.506 m x f 400 106 Yj10.7 Yj10.7 ℓ=0.088  Solution with capacitor

ℓ=0.088  ℓ=4.45 cm BC  0.7

0.348  0.7 / Z C 0 1.85 pF 2 f

18 European Master of Research Design and Analysis of RF and Microwave Systems on Information Technology x Compute the length of a Y  j 0.7 transmission line, ended in c short-circuit, that can substitute the capactior.

CC

x

ZL Yc=j·0.7 YL

ℓ Y=1-j·0.7

Ye=1 ℓ=0.347 

ℓ=0.347 =cm

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems b) Matching network formed by a quarter wavelength transformer .

Z 0’0 ZL

ℓ=

Ze= Z0

In the case of a quarter wavelength transformer

2 ZZe 0' Z 0' ZYeeeL   Ze ZZ0' L ZL

ZZZ0'  eL

ZZe  0 150

Z0'  106.07    12.65cm 4

19 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

Matching networks utilities and examples

Lecturer: Francesc Torres ([email protected])

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

There are a number of on-line tools for RF design and/or educational purpose

20 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

21 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

22 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

23 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

24 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

25 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

26 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems EXERCISES: www.amanogawa.com:

1) Microstrip impedance computation: Glass Substrate thickness: 1.59 mm Dielectric permittivity: 4.15 Strip thickness: 0.1 mm Copper conductivity: 5.8 107 S/m Substrate conductivity: 2.3 10-4 S/m

a) Calculate the substrate width W (mm) in order to have Zo=50 Ω at f=2.5 GHz b) In the previous case, compute the return loss (RL) referred to Zo=50 Ω at f=1 GHz and f=5 GHz: RL degradation due to frequency dependence Zo(f) c) Compute the return loss (RL) referred to Zo=50 Ω at f=2.5 Ghz if the strip thickness is neglected (t=0)

2) Narrowband matching networks

Select the narrow band matching structure that provides the best bandwidth

(VSWR<1.5) to adapt an impedance ZL=100+j120 Ω, referred to Zo=50 Ω at f=10 GHz, εr=2.4. • Quarter wavelength adapter • Double stub adapter • Single stub (short/open) adapter

Gives line length in mm.

27 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

Layout and picture of a microstrip two-stage

Interstage matching network: coupled lines (DC block) TRT2 Ouput matching network TRT1 GND IN OUT

λ/4 λ/4 o.c.

λ/4 s.c. GND Bias resistor

Input matching network Bias network Low frequency s.c.

Gate bias voltage Drain bias voltage

European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems

Other utilities: http://www.hp.woodshot.com/ •Simple tool for transmission line calculations, bias circuits, smith chart,..

28 European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems Exercises 1) Derive the expression of the input impedance of a transmission line of

impedance Zo, length λ/4 and loaded with an impedance ZL. 2) Demonstrate that |Γ|≤1 for any load Z=R+jX if R≥0 3) What is the return loss of a load ZL=75Ω when connected to a transmission line of Z0=50Ω ? What fraction (%) of the incident power is delivered to the load? 4) What is the tolerance (±x Ω) of a resistor of nominal impedance R=50Ω

when connected to a transmission line of impedance Z0=50Ω if VSWR ≤1.1?

29