Lecture 4 RF Amplifier Design. Johan Wernehag, EIT. Johan Wernehag Electrical and Information Technology

Lecture 4 RF Amplifier Design Johan Wernehag, EIT Johan Wernehag Electrical and Information Technology

Lecture 4 Design of Matching Networks Various Purposes of Matching Voltage-, Current- and Power Matching Design by Lumped Circuit Elements L, Pi and T Networks Design by Using the Smith Chart Design by Line Structures Transformation by a Transmission Line Quarter-Wave Transformer Line Section with Optimised Length and Z 0 Stubs Passive Components Lumped Components Resistors Capacitors Inductors Transformers Substrate and Conductor Materials Transmission Lines Coaxial Line Microstrip Stripline Discontinuities Bends, corners Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 3

The Purpose of Matching Minimum noise power Maximum output power Maximum transfer of power In most cases bandwidth and filtering are important parameters too Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 4

Maximum Transfer of Power Relative load signal level I L P L V L R S = 100W R L [ W] Current matching Voltage matching Power matching Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 5

Complex Conjugate Matching Maximum transfer of power when Z L = Z S * Þ ì ï í îï R L = R S X L =-X S resonance Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 6

Z or G Representation Impedance Z L = Z S * Þ ì ï í îï R L = R S X L =-X S G L = Z - Z L 0 = Z * S - Z 0 * Reflection coefficient =G Z L + Z 0 Z * S (Z 0 is resistive) S + Z 0 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 7

Available Power from Source, P AVS The maximum amount of power that can be delivered from the source is defined as: P AVS = V 2 L = V S R L æ è ç 2 R L Z S + Z L ö ø 2 1 * = V S Z R L =Z S L 4R S 2 NOTE! The unit of V S is here V RMS! Active power Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 8

The Need of a Matching Network If Z L and/or Z S are fixed a matching network is needed to ensure proper matching Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 9

Network Design by Lumped Circuits L network low-pass high-pass fixed circuit Q Pi and T network low-pass high-pass desired circuit Q Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 10

L Network, Four Different Variations Low-pass, R L > R S Low-pass, R L < R S High-pass, R L > R S High-pass, R L < R S - Series component in serie with the smallest impedance - Shunt component in with the largest impedance Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 11

Designing Reactive Circuit Elements in the Smith Chart Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 12

Experiment: Design a matching network by using the Smith chart and the VNA step 1 step 3 Add susceptance (connect an inductor in parallel) until you end up in Y = Yo = 1/50 mho. 1/wC= 0.47 50Ω 1 = 1.5 50Ω = 42 step 2 Subtract reactance (connect a capacitor in series) until you reaches the conductance circle g = 1/50 mho. 1/wC= 0.47 50Ω 55 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 13

Eight possible L-type Networks Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 14

Matching Networks by Line Structures Transformation by a serial transmission line line section with optimised length and Z 0 quarter-wave transformer matching by multiple sections Stubs short-circuited and open stubs symmetrical stubs Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 15

Impedance Transformation by a Single Serial Line Z in = Z 0 Z L + Z 0 tanhg l Z 0 + Z L tanhg l G in,z in G L Z 0 l a) lossless line b) lossy line Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 16

The Quarter-Wave Transformer For a line at the length l/4 is Z in = Z 2 0 Z L if Z L is resistive Z in G in Z L G L may be used for matching between arbitrary resistive source and load impedances. Z 0 = Z L Z in Useful to remember! Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 17

Quarter-Wave Transformer at Multiple Sections Transformation in several and minor impedance steps may provide a larger bandwidth If the characteristic impedances are distributed according to binomial coefficients maximum-flatness is achieved. R S G in Z in Z 01 Z 02 l 4 l 4 G in R L Ex. three sections: Binomial coefficients æ Z 01 = R S è ç æ Z 02 = Z 01 è ç æ Z 03 = Z 02 è ç R L R S R L R S R L R S ö ø 1 8 3 ö 8 æ R ö ø = L RS è ç ø 3 R S ö 8 æ R ö ø = L RS è ç ø R S 4 8 7 8 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 18

Quarter-Wave Transformer at Multiple Sections (cont.) Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 19

Impedance Transformation by Stubs and Serial Lines Short-circuited or open line sections may be used as reactive shunt elements. Combined with a serial line matching can be achieved between arbitrary loci in the Smith chart. l 1 Z S Y 1 = G in + jb 1 Z 01 Z in = Z S * Z 0stub Y stub = j(b in - B 1 ) l stub 1 = Y in = G in + jb in Z in Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 20

Length and Termination of the Stubs Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 21

Designing the Length of the Stub Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 22

Symmetrical Stubs Single stub Symmetrical stubs jb j B 2 j B 2 j B 2 l symm jb l single j B 2 l symm Note: 2 The length of the symmetrical stubs is designed to individually provide the half value of the requested susceptance. Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 23

Resistors high-frequency model Z R p R Ideal resistor f SRF f f SRF = Self Resonance Frequency Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 24

Resistor Frequency characteristics - example Z R = 50W L = 0.25nH C = 200fF Resistive Self resonance f [Hz] Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 25

Capacitors high-frequency model Z R ESR capacitive f SRF inductive f R ESR = equivalent series resistance Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 26

Capacitor Frequency characteristics - example Z R = 2W L = 1nH C = 1pF C p = 100fF C R L C p f [Hz] Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 27

Inductors high-frequency model Z R P inductive f SRF capacitive f R P = equivalent parallel resistance Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 28

Inductor Frequency characteristics - example Z R = 3W L = 5nH C p = 100fF R L C f [Hz] Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 29

Inductor Q = X R S The Q-factor frequency dependence Q X T = wl R ~ w R ~ X T The losses increases rapidly versus the frequency X T = wl R = R DC SRF (self resonance frequency) X T = total reactance of the inductor Skin effect Parasitic capacitance ω C parasitic doesn t affect the Q if the inductor is used in a resonant circuit! Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 30

Transformer Windings on a ferrite rod or toroid core These materials provides a high permeability that unfortunately decreases at higher frequencies Usable at best up to a few GHz Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 31

Loss in Substrate Materials For dielectric materials often the loss tangent used to specify the losses: tand = s we for material with low loss tand» d where d is the loss angle The loss tangent is also related to the quality factor or Q-factor of the material: tand = 1 Q d Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 32

Properties of some Substrate Materials Material r tan Ceramic 10 0.004 Beryllium oxide 6 0.0003 Duorid 2.6 0.0001 Teflon fibre-glass 2.3 0.0015 Silicon 11.7 0.004 Quarz 3.8 0.0001 Epoxy fibre-glass 4 0.02 Aluminium oxide 10 0.0003 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 33

Properties of some Metals Metal [S/m] Aluminium 3.816 10 7 Copper 5.813 10 7 Gold 4.098 10 7 Iron 1.03 10 7 Silver 6.173 10 7 Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 34

Skin Effect The magnetic flux inside the conductor will effectively push the current to a narrow region close to the surface z Current density d i J 0 - - e J 0 J 2 Skin depth: d = the distance were the current density i = wms has decreased by a factor e A large circumference is more essential than a large cross section area of the conductor! Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 35

Example of Skin Depth d i [m] - - iron aluminium gold copper silver f [Hz] Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 36

Dimension of the Coaxial Cable Z 0 120 100 ε = 1 (air) ε D 80 60 50 40 20 0 1,0 1,2 1,4 1,6 1,8 2 2,4 3 4 5 6 æ è ç D d ö ø d Z 0 = 138 e r log 10 æ è ç = 60 æ ln D e r è ç d ö ø D d ö ø = Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 37

Microstrip Cross section of a microstrip structure W h H E å r ground plane Complex geometry and non-uniform fields makes a complicated flux image. e r in the substrate is therefore not usable for calculation of for example electrical length in the line structure. Instead the effective permittivity, e eff, based on the dimensions of the microstrip structure, may be used. Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 38

Design of Microstrip, Diagram 1 Determine the proper W for a specified Z 0 and h Z 0 1 2 4 6 8 10 12 16 e r - W/h Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 39

Design of Microstrip, Diagram 2 Determine the effective permittivity e eff e eff 16 12 10 8 6 e r 4 2 - W/h Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 40

Design of Microstrip, Summary Specified Z 0, substrate height h and permittivity ε r : Z 0 1. Select the width W in diagram 1 e r 2. Select the effective permittivity ε eff in diagram 2 e eff W/h 3. Calculate the effective wavelength l eff = l 0 e eff e r W/h Specified electrical length l el 4. Calculate the physical length l = l el l eff Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 41

Stripline W ground plane substrate e r conductor h ground plane Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 42

Discontinuities The design of circuits containing transmission lines must take into account the impact of junctions and transitions between different line widths. circuit diagram design Z 0,1 Z 0,1 l 1 l 1 l 2 l 3 l 3 Z 0,2 l 2 How to model such a geometry? Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 43

Discontinuities (cont.) T-junction, equivalent model Z 0, 1 Z 0, 1 Z 0, 2 Methods for manual calculation of the circuit elements in the equivalent model may be found in handbooks. Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 44

Discontinuities (cont.) Symmetrical step, an abrupt transition between two different line widths W 1 W Z 2 0, 1 Z 0, 2 Transmission line, open-circuit termination conductor C f ground plane e r e r Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 45

Discontinuities (cont.) Some other cases are corners and bends serial gap (to realize a serial capacitance) undesired coupling between nearby structures Crosstalk CAD tools are necessary for analysis and design of more complex circuits ADS and others In worst case a complete structure may be simulated by the finite element method and Maxwell s equations which is both time and memory consuming Johan Wernehag, EIT RF Amplifier Design ETIN50 - Lecture 4 46