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by: Adaline Pollich


Adaline Pollich
GPA 3.81

Stephen Gedney

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Stephen Gedney
Class Notes
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This 39 page Class Notes was uploaded by Adaline Pollich on Friday October 23, 2015. The Class Notes belongs to EE 521 at University of Kentucky taught by Stephen Gedney in Fall. Since its upload, it has received 13 views. For similar materials see /class/228315/ee-521-university-of-kentucky in Electrical Engineering at University of Kentucky.

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Date Created: 10/23/15
Ladder Fillers Bullerworlh amp Chebyshev Fillers F iller Tables amp Freqnip Scaling Ladder Filter A network composed of alternating series and shunt reactive elements Low Pass gt gt The network is doubly terminated with the same source and load resistance Some observations The inductors and capacitors are chosen such that the lter has a cutoff frequency at we and it is matched to the load R Using a higherorder ladder network the rolloff of the fllter s response in the pass band is steeper than a traditional R L or RC type fllter Doubly terminated ladder filters have a lower sensitivity to component variations The reactive elements can be chosen in a manner to synthesize a desired response in the pass band and stop band EE521 pg 1 Ladder Fillers Bullerworlh amp Chebyshev Fillers F iller Tables amp Freqnip Scaling Types of Ladder Filters Four Basic Types 0 Maximally Flat or Butterworth Filters 0 Equal Ripple or Chebyshev Filters 0 Linear Phase Filters 0 Elliptic Filters Note that the topology of the lters are the same except Elliptic lters require mutually 3 L coupled elements However the lter s response will differ depending on the design Filter Response Loss Factor Pl average input power L P average output power Note that G Gain l L EE521 pg 2 Ladder Fillers Bullerworlh amp Chebyshev Fillers F iller Tables amp Freqnip Scaling Maximallv Flat Butterworth Low Pass Filters Choose the L and C values such that fc cutoff frequency Hz Example fc 1 Hz lUU I U 3 3 that LB2f If GBdb1 0 L356 9 10 J squot Stew 39 GBdb3t LB4f 39 39 GBdb4t3940 4539 I 1 01 1 10 60 f 01 1 10 f EE521 pg 3 Ladder F ilters Bullerworlh amp Chebyshev F ilters Filter Tables amp Freq1mg Scaling Equiripple CheszheV Low Pass Filters Choose the L and C values such that 2 LC l0Cn lI I I c Tcheblx 239 39 I a r1pple s1ze Tcheb2x r39 fl Cnx ChebysheV polynomial cf Abromowitz amp Stegun Elam 0 4 3 l I39 Tcheb4x R39sl quot l r Example fc le 1 dB r1pple a 0206 1 r J AK 391 0 1U V M X 20 0 GCdblt1tgt 95 0 cdblt2t U 2930 Elm 39 L99 Gcdblt4t W l 1 01 l 10 0U 01 l 10 EE521 pg 4 Ladder F ilters Bullerworlh amp Chebyshev F ilters Filter Tables amp Freq1mg Scaling Chebyshev Filter 1 dB ripple c39 wquot x 2 GCdb1f GCdb2f 39 I la GCdb4t 4 lll 01 1 10 Cheybshev Filter 05 dB ripple GCdb1 t GCdb3 t GCdb4 t EE521 pg 5 Ladder F ilters Bullerworlh amp Chebyshev F ilters F iller Tables amp Freq1mg Scaling LowPass Filter Design 0 Design Parameters 0 Filter Response 0 Maximally at Butterworth I Flat response through pass band close to 0 dB I 20n dB rolloff per decade rolloff in the stop band 0 Equiripple gt ChebysheV I Rippled response through pass band Level speci ed by design I Faster rolloff into the stop band just past cutoff I 20n dB per decade asymptotic response 0 Cutoff frequency f6 0 Order of the lter 0 Governs the rolloff in the stopband 0 Also affects the passband response 0 Impedance 0 Source and load impedance 0 Component design 0 Compute L and C for the lter to meet the design parameters EE521 pg 6 Ladder F ilters Bullerworlh amp Chebyshev F ilters F iller Tables amp Freq1mg Scaling LowPass Filter Protothe 0 Prototype lter 0 we 1 radsec R IQ R 32 a4 nn gt gt Vs a1 a3 R I I I R a1 33 n gt gt Vs a2 a4 R I I 0 Series avalues are normalized reactances 0 Parallel avalues are normalized susceptances Or EE521 pg 7 Ladder F ilters Bullerworlh amp Chebyshev F ilters F iller Tables amp Freq1mg Scaling Filter Tables From Table 51 Text Butterworth Filter Order a1 a2 a3 a4 a5 2 J5 J5 1 2 1 0765 1848 1848 0765 0618 1618 2 1618 0618 MLWNH Cheb sheV Filter 02 dB ripple order 6 1 a2 a3 a4 a5 1 0434 3 1228 1153 1228 5 1339 1337 2166 1337 1339 EE52 pg 8 Ladder F ilters Bullerworlh amp Chebyshev F ilters F iller Tables amp FreqImp Scaling Impedance and Frequency Scaling Given the order lter 11 of the desired prototype we can extract the normalized reactancessusceptances from the table Need to scale the design to the desired cutoff frequency f6 and sourceload resistance RS Frequency Scaling o Scale the reactance or the susceptance such that the impedance of the prototype remains unchanged at the cutoff frequency Impedance Scaling o Scale the reactance to RS 0 Normalize the susceptance by RS so the effective reactance scales by RS Formulas 0 Series inductance Li Rs Wjaiz Rs ai Rprololype we 272 0 Parallel capacitance Ci Rprozolype wciprololype ai i l ai RS we RS 2 7239 fc EE521 pg 9 Ladder F ilters Butterworth amp Chebzshev F ilters Filter T ables amp F reg1mg Sealing Example Design a Butterworth lter with j 8 MHz R 509 with a rejection loss of at least 23 dB at 14 MHz Solution LEW 1m if a z 3 a LEdE 3lAlEI6lIA7ZI ifx R5 5 4n 1 3 LEaE A141n6 19492 WWW 03 m LEdElSJAlU lZAZZ LZR552 LZ16E9gtltIE76 037958xm7m 1U Note atf 7 MHz L lOlogm 1 10 dB EE52I pg 10 Ladder F ilters Bullerworlh amp Chebyshev F ilters Filter Tables amp Freq1mg Scaling Scatterin Parameters of a 2 ort network R 6quot 15 Nesz R e 4 re ected 1 S input re ection coef cient 11 Pmczdenl 1 out 521 transmission coef cient 2 Pincz39dent 1 EE521 pg 11 Ladder F ilters Butterworth amp Cheb shev F ilters Filter T ables amp F re Im Sealin EE52I PUFFSimulation rrz Paints 1E1 Smith radius 1 i nu Hanhatta Mia file setup pg 12 Ladder F ilters Bullerworlh amp Chebyshev F ilters F iller Tables amp Freq1mg Scaling Creating the Circuit in Puff 0 Download DosBox from sourceforgenet see running puff under window s Xp on the web page 0 Place PUFF on you computer in the directory CPUFF 0 Open DOSBOX 0 Type the following commands in the DosBox window I mount P CPUFF I P I PUFF I Then hit any key to start puff o This will start up puff 0 Note that any time you want a print out you can make the puff window active Then selecting from your keyboard altPmtScrn will capture the puff window 0 Inside PUFF 0 Selecting the function keys F1 F2 F3 etc will move you to that window 0 We will generally not change at all the board window F4 Instead we will use typically in this order I F3 parts window I Fl layout window I F2 plot window EE521 pg 13 Ladder F ilters Bullerworlh amp Chebyshev F ilters F iller Tables amp Freq1mg Scaling Creating the PUFF circuit o In the parts window 0 Enter the component value of the circuit 0 Example I lumped 246pF a lumped 246pF capacitor I l 246pF same thing except 1 abbreviates lumped o For the unit micro use the key stroke altm 0 You can also put lumped elements in series or parallel 0 Example I Series 1 lnH lOOpF l nanoHenry in series in 100 picoFarad I Parallel l l6mH 20 pF symbol is the Altp 0 Key in the 3 parts for the lpf in the parts window as shown on the previous page 0 In the layout window 0 Move the cursor using shiftarrow 0 Move to your starting point near port 1 0 Enter the number 1 to connect to port 1 I This connects the circuit to port 1 with a transmission line I NOTE THAT PORT l HAS A SOURCE VOLTAGE WITH AN INTERNAL IIVIPEDANCE OF zd board window Default is 50 ohms o Press the a key to activate part a note it is highlighted in the parts window 0 Press the down arrow This places part a going down EE521 pg 14 Ladder F ilters Bullerworlh amp Chebyshev F ilters F iller Tables amp Freq1mg Scaling o Press the key This is a ground 0 Press the up arrow to go back to the top of part a o Press b to activate part b o Press the right arrow This places part b on the board 0 Press shiftleft arrow Note this deletes part b o Press rightarrow to put it back Next push c and the down arrow to place c o Press to ground c then arrow back up 0 Complete the circuit through a again 0 After placing the ground on the last part a on the righthandside Up arrow so the cursor is at the junction of parts b and a again on the righthandside Now enter 2 This terminates the circuit into port 2 0 NOTE PORT 2 IS TERMINATED BY A LOAD IMPEDANCE OF zd YOU DO NOT NEED TO TERMINATE THE CIRCUIT WITH A 50 OHIVI LOAD 0 You are now ready to simulate o Press F2 to go to the plot window 0 Use the arrow keys to go past S21 A blank S will show up Press 11 after this S so it reads 81 1 This will also plot the Sparameter 81 l 0 Use the arrow keys keep pushing down until the cursor goes into the plot area It will rst go to 0 then the 20 Change the 20 to 60 Arrow again to go to the horizontal aXis You can change the range of frequency in a similar way 0 Now simulate the circuit by pressing p If you did it correctly it will simulate the circuit and plot the frequency response EE52 pg 15 Ladder F ilters Bullerworlh amp Chebyshev F ilters F iller Tables amp Freq1mg Scaling 0 Note the cursor on the plot Use the pageup and pagedown keys to move the cursor to get speci c data points 0 You can change a component value Then press ctrlp to replot the response on top of the preVious response 0 Screen capture 0 Press AltPrntScreen to screen capture the puff window for your reports EE521 pg 16 Cgrtal Orcillatorr and Calm F ilterr Resonator Q o The Q of seriesparallel resonators built With lumped elements are limited 100 I This limits the realizable bandwidth of a BPF 0 Mechanical quart resonators can provide a Q on the order 50 000 gt 140000 Quartz Crystal 0 Piezoelectric material I Applying avoltage across the material creates mechanical stress I Applying mechanical stress creates an electrical voltage Tenslon Compresslon EE52I pg 1 C ital Oscillators and Cohn F fliers Cgstal Oscillator o The quartz crystal Will have a distinct resonant frequency at Which it Will oscillate 0 Mechanical resonance With subsequent electrical effects 0 Equivalent Circuit Model L C R i gl li 3 i C o L 7 the motional inductance C 7 the motional capacitance R 7 the mechanical loss 0 C a parallel capacitance due to packaging leads etc o Resonance Series resonator we ldLC 0 Series Reactznce f series resonance fp parallel resonance suf ciently large compared to 1 kHz that We neglect C EE52I pg 2 Canal Oscillators and Cohn F fliers 0 Series Reactznce 5 7 fs m SUD F quotm5 sun 6 a a 42m 425m 491m 491m Amsm 4mm r r o In Problem 14 7 you Will measure R L and C R 1 m M w k o Sweepfofthe source and measure output Voltage across X o Sta1t around 491 MHz Scan With 10 to 100 Hz stepping till you nd a signi cant rise in the output 0 Fine tune With le increments to nd the resonant frequency and bandwidth EE52I pg 3 Canal Oscillators and Cohn F fliers Image Filters 0 The RF lter rejects an image at 28 MHz 0 2ndorder Butterworth BPF W lumped elements 0 Suf cient to reject the VFO image at 28 MHz M65 dB I Harmonics at 58 and 84 MHz also are rejected 0 IF Filter 0 Upper sideband signal is only 1240 Hz above the desired signal I Cannot get suf cient rejection With lumped lter 0 Q is too low 0 Introduce a Cohn lter design that uses Crystal oscillators EE52I pg 4 Crystal Oscillators and Calm Filters Impedance Inverters o Desire to use the crystal oscillators as the resonant elements of the branches of a bandpass lter 0 Problem 0 Crystal oscillators are series resonant elements 0 Cannot use directly for parallel resonant elements 0 Solution 1 Inverter o Impedance inverter yin 2 3 X E ZL ZL I Inverts a normalized impedance 0 A quarter wave transformer is an impedance inverter 2L g normalized load impedance Z in gt Z Z L 50 Z zin normalized input impedance 6 1 4 1 Z 20 zm or ZL Zo ZL 0 Dif culty quarter wave at 49 MHz is 153 m EE521 pg 5 Crystal Oscillators and Calm Filters Lumped Circuit Impedance Inverter JX JX Zin gt ZL 0 Input impedance 39X 39XZ 39 2 39 zinzjm zm 2 J LLJXZJX M 1 1 X ZL jX ZL jX jX ZL X2 ZL 0 Normalized inverter Zin X ZL o X is the impedance of the inverter 0 Note that the inverter is only valid at a single frequency 0 When applied to the narrow band of the Comb lter it can be assumed to invert over the pass band EE521 pg 6 Crystal Osczllatms and Cuhn leters Invenin the uanZC stal thY o Nonnauzeoreactance ofthe crystal oscillator 7 X jXCR X Jarjxgw yi1xrmr Z o This acts as apara11e1 resonator Jbt fbt o where b 5 b1 ch gr x amp b b o Note that at the resonant frequency X1 X5 o Resonant condition EE52 C ital Oscillators and Cohn F fliers Cohn Filter Desi n 7 The Conce t o The Cohn Filter topology Fig 516 HH il tf tll Hfc HEW a c We can add zero series reactance at the resonant frequency 49 MHz 0 7X jXL 0 Hw go Cb 0 Note that the center Tcircuit noW is our impedance inverter top of pg 6 EE52I pg 8 C ital Oscillators and Cohn F fliers 0 Replace the Tcircuit With an inverter c We add a capacitor to either end of the circuit so that noW our resonators are the crystal oscillators plus two capacitors o The capacitors are 270 pF The series capacitance ofthe resonator is lt 1 F Thus this will only slightly drop the series capacitance amp slightly increase the resonant frequency 0 At 49 MHZ XN 1209 c We can slide the inverters all the Way to the right 0 Ifa series branch elements has an odd number ofinVerters to the le ofit it Will become a parallel branch element 0 If an eVen number of inverters are to the le of it it Will remain a series branch EE52I pg 9 Canal Oscillators and Cohn F fliers 0 Final circuit o nquots d o This is a fourthorder bandpass lter jH l l l T Hm C10 C11 C9 7 C13 270 pF C12 0 The lter is centered about 4913 MHz It has a matching impedance of 200 Q EE52I pg 10 Canal Oscillators and Cohn F fliers Matching Network 0 The RF Mixer is matched to the IF lter Via a transformer more on that soon 0 The IF lter is matched to the product detector Via an Lmatching network 0 Input impedance of the product detector mixer is a parallel 15 k resistance and a 3 pF capacitance L4 ISMI 1 mmhjngnmork 3 Figure 523 LC matchmg network for connecting the 11quot Filter o Lhc 0 Design 0 At 4913 MHZ ZL 1 472 7 12049 o RL1472gt RS 200 therefore use above design EE52I pg 11 Crystal Oscillators and Calm Filters 0 Load admittance 1 1 712 4 75 27239 3gtlt10 G B 2667gtlt10 926gtlt10 Z 1500 J f L JL J L 0 Solution using MathCad 5 99139196 1 X9 392i f31 12 91 1599 Zr 299 995 EL c ZL 14919 x 193 294423991 RL ReL39ZL Rl X9 XL 11921 XL 39E LwllRLE 5992 ZoRL Cl 3139 2 2 131 199599 5 19 3 RL XL 1 XLZo Zn E 91quot 59999195 H J B BL B RL XE 1 XCBID hf L 19651815911 5 91 11 mquot 2 c 529693 x 19 EE521 pg 12 Radio Waves Transmitting and Receiving Antennas The NORCAL Antenna ElUp until now the antenna of the NORCAL4OA has been treated as a 50 ohm load Antenna is the dipole antenna in the back of the lab The antenna appears as a 50 ohm load Note that the input impedance at 7 MHZ of the antenna alone is not 50 ohms There is a matching network a balun used to transform the actual antenna impedance to 50 ohms 39 50 ohm coaX cables connect the matched 50 ohm antenna to the antenna jack 39 Therefore we can assume that around 7 MHz Zant Rant annt z Q laNote The antenna is not a resistor that dissipates power The antenna is a radiator 39 Antenna radiates electromagnetic elds that carry power away from the antenna 39 By power conservation the electrical real power delivered to the antenna is equal to o The power radiated by the antenna plus 0 Any power dissipated by the antenna due to conduction losses EE521 pg 1 Radio Waves Transmitting and Receiving Antennas Radiated Power and Radiation Resistance EIAC power delivered from the NORCAL to the antenna hgz Rngz Rgizmijwr Pt Z Rant 1 l2 ORG can be expressed via the superposition RWampamp I R ohmic resistance of the antenna due to conduction losses I RV the radiation resistance 0 Effective resistance representing power radiated by the antenna I Note that in general RV gtgt R o The ef ciency of the antenna is de ned as 77 RV R By power conservation gzg mpf Neglecting ohmic losses I Pr Pl Therefore I Rant RV EE521 pg 2 J D 1quot in Am mm Radio Waves T Uniform Plane Waves OElectromagnetic Waves radiated by the antenna propagate power away from the antenna These Waves are received by remote antennas and can be down converted to our transmitted signal OThe Waves satisfy Maxwell s equations Vgtlt E jw oH V X 1 7 J39on I E the electric eld intensity V m 1 7 the magnetic eld intensity Am I 0 permeability of ee space 471gtlt10397 Hm I so permittivity of ee space 8854gtlt103912 Fm OSuf ciently far from the antenna the Wave can be approximated to be a E Uniform Plane Wave UPVI 1 By de nition I ELH Example E Ex HjHy I E amp 1 7 L direction of propagation Example zdirected ExHy 27 1 o 37673037 Q I Wave propagates With the speed oflight c lJ O o z 3gtlt108 ms EE52I Radio Waves Transmitting and Receiving Antennas IDExample of a UPW a A 7 a A E 7 szEOe koze w H y Oe koze w 770 39 k0 the wave number a 080 Z f 2 7 i wavelength 0 xi f EIPoynting Vector S E X H lbs 2 2 E0 2 mo 39 Note that the Poynting vector is directed along the direction of power ow ll Real power is carried by the UPW Re ReExH2EOZ 770 I3 is the power density vector Wmz EE521 pg 4 Radio Waves T 39 39 andReceivinQAntennas Transmitting Antennas OAntenna Coordinate System A Zenith Elevation Azimuth a bl i W OThe antenna does not radiate with equal power density in all directions Some antennas are designed to radiate equally along one angular direction typically azimuthal I Omnidirectional Antennas Other antennas are designed to focus radiation in specification directions I Directional antennas OLet P 9 Wm2 represent the power density at a given angle 9 ODe ne the gain of the transmitting antenna as G9 P9 R EE521 pg 5 Radio Waves 39F 39 39 4 Mm Am mm R the maximum power density from a reference antenna I The reference antenna is typically a ctitious isotropic radiator I An isotropic radiator radiates poWer uniform y in all directions I Approximated by o P P 39 y 47W2 O R total poWer delivered to the antenna O 47r2 the surface area ofa sphere ofradius r m2 OThe gain is thus Written as DG9 471r2PI9 R Gain is typically expressed in dBi I dB s relative to the isotropic radiator 0HalfWave dipole Max radiation at 19 90 G90 GquotEX 2 dBi E vznncnpmsw mama UNsluE D vzwncnmmz C soul punm EE52I pg 6 Radio Waves Transmitting and Receiving Antennas Receiving Antenna ElAn antenna acts as transmitter and a receiver QReceiving antenna collects electromagnetic signals Produces a current on the antenna and induces a voltage across the antenna terminals 39 Translates to real power collected from the illuminating EM wave Due to a principal known as reciprocity the receiving pattern characteristics of an antenna are equal to its transmitting pattern An antenna has an effective area The effective area is speci cally the effective area of the incident EM that is essentially collected by the receiving antenna 39 Complicated expression to derive By reciprocity Alta gtj70lta gt o Antenna formula EE521 pg 7 Radio Waves Transmittin and Receivin Antenna Dipole Antenna a Figure 1 54 Calculating m effective imgm 0 a hnruhpolr my and le Tlm39uun Lqun nlem uxcuu lt b V 0Consider a simple dipole antenna Defined by an effective length n oTerminal voltage of the dipole Va h E E amplitude of the incident wave Note that the current induced on the dipole is only due to the projection of the electric field onto the dipole axis recall E is a vector Thus 39 Va Esin E where 8 is the angle of the incident wave offthe dipole axis Thus 11 sin EE 521 Radio Wave r J p a 34 Am mm OThe effective length h can also be interpreted as a constant that determines the output voltage produced by a unit incident electric eld V l h sin 19 E 2 E o OThe terminal voltage then sees an equivalent circuit C RC re ects the antenna impedance R OThe current induced on the antenna will produce a received V 51 Sin 0 power quot 2 Received power 39 R A399gt P399gt o At9 the effective area ofthe dipole 0c 112 o Pt9 power density of the transmitting antenna 0 1997 are the relative angles of the axes of the transmitting and receiving antennas o R the power received N EE52I Radio Waves Transmitting and Receiving Antennas FriisFormula EIWe have now been given IbPr A6 P6 12 A6 EG6 EIAlso from the equivalent circuit 2 2 2 2 322 JMiJMWI 8Ram 8Ram 8R ant Assuming the incident eld has the power density of a UPW then P0945 2 E0 or E0 2 22770P6 770 Therefore 2 2 Bzwzm anZanw 8Rant 4Rant Finally since Pr A6 P6 then lhlz 770 I A 67 4Rant EE521 pg 10 Radio Waves Transmitting and Receiving Antennas GFor the transmitting antenna we can derive P G6 gt 3 9 Z ZS 47z39r2 OThus for the receiving antenna IbPr A6 P6 A6 W Friis Formula 727 22 Also given that A6 4 G 7239 93 Pr Pl Gr6r7 r39Gz627 z j EE521 pg 11


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