European Master of Research on Information Technology Design and Analysis of RF and Microwave Systems
1.1 Transmission line 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
jz (z z) z n2 , z n2 V (z) V0 e V (0) V0 V0 VL 1 jz 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 jz 2 jz 2 jz (z) e (0)e Le V (z) V 2 jz Modulus (z) Le L Constant in z 180º 2z rad 2z deg Phase 2z 2 (z z) n2 , 2z 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 jz Z0, β ZL≠ Z0
|Γ(z) |=|ΓL|
i () z z=0
Example: Compute the input impedance 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 jz 1 e jL e2 jz V 1 e jL e2 jz L L V V jz jL 2 jz jL 2 jz 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 2zmax 2 rad I I(z) 1 min min L Z0 z z max min 4 V V (z) V 1 min min L L 2zmin 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:
jz V (zmax ) V e 1 L 1 L Z(zmax ) Z0 Zmax I(zmax ) V jz 1 L e 1 L Vmax Imin (zmax ) Z0 jz Vmin Imax (zmin ) V (zmin ) V e 1 L 1 L Z(zmin ) Z0 Zmin I(zmin ) V jz 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 jz 1 (z) V e jz 1 * (z) Z0 2 V 2 2 1 (z) P 1 (z) e1 a1 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 1YL 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 ReZz 0, ImZz ,
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 rr11 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
01L|
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 100j 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
Zj0.8 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¯
Zj23 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
ZRmáx 270 máx S=3.6 ZR21 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 ZYY11 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 YYYLL0 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 Zj10.7 2’ 1 ZL x x ZZ11150 150 j 105 Z2 150j 105 Z 1.5 j Z 1' 100 e x
Zj2 181.80 09.9
ZZ22100 180 j 90
180j 90 Zj3.6 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 Zj12.1 x jXS j2.1 1 Z’ 2.1 Z C 0 Zj'1 2.1 0.234·λ
jXS '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 Yj12.4 x jBP j2.4 Y’ 1 x 2.4 Y0 Y L L 0.402·λ Yj'1 2.4
jBP '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 ReVILL Re VVY LLL V L G L 3 PL 20 10 GL 0.005 x 2 4 x VL
GL gGZLL0 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 antenna 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 Yj10.7 Y=1+j·B x
Ye=1 B = -Bc Bc>0 YL 0.25 Ye 1 x 75 1 ZYLL0.5 2 150 ZL
c / 310/ 8 2.2 r 0.506 m x f 400 106 Yj10.7 Yj10.7 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' ZYeeeL 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 amplifier
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?
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