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UNIT IV PASSIVE FILTERS Energy Transfer All the systems are designed to carryout the following jobs: 1.Energy generation. 2. Energy transportation. 3. Energy consumption. Here we are concerned with energy transfer. Electrons Electron is part of everything on earth. Electrons are the driving force for every activity on earth. Electron is a energy packet, Source of energy, capable of doing any work. Electron accumulation = Voltage Electron flow = current Electrons’ oscillation = Wave Electron transfer = Light Electron emission = Heat. No mass ; No inertia; Highly mobile; No wear and tear; No splitting of electron; No shortage; Excellent service under wider different conditions: Vacuum, gas, solid; Controlled by Fields : accelerated, retarded, change directions, increase and decrease of stream of electrons; instant reaction due to zero inertia. Energy = Electron - Wave Energy is transferred from place to by two means: 1.Current : conductors. Flow of electrons through 2. Wave : Wave propagation in space, using guiding systems or unguided system (free space). In this subject, except free space energy transfer, other means are discussed. Electron - waves Major Topics for discussion i) Circuit domain ( Filters ) ii) Semi Field domain (Transmission Line : VoltageCurrent – Fields) iii)More Field domain (Coaxial line) iv)Field domain : TEM waves ( Parallel plate guiding) v) Fully Field domain : TE-TM modes ( Waveguide ) Transmission Line – Waveguide Guided communication System Frequency Energy Flow Circuits LF, MF, HF Inside Conductor Transmission Lines VHF Outside Cond. Coaxial Lines UHF Outside Cond. Waveguides SHF Outside Cond. Optical Fiber 1015 Hz Inside Fiber Energy V = Voltage = Size of energy packet / electron. I = Current = Number of energy packet flow / sec Total energy flow / sec = V X I System Power Flow Medium Circuits P=VxI Conductor Transmission Lines P=ExH Free space Coaxial Lines P=ExH Free space Waveguides P=ExH Free space Optical Fiber P=ExH Glass Quantum of energy E = h f; h =6.626x10-34 J-s Quantum physics states the EM waves are composed of packets of energy called photons. At high frequencies each photon has more energy. Photons of infrared, visible, and higher frequencies have enough energy to affect the vibrational and rotational states of molecules and electrons in the orbits of atoms in the materials. Photons at radio waves do not have enough energy to affect the bound electrons in the materials. System Energy Flow Circuits Inside Conductor Transmission Lines TEM mode Coaxial Lines TEM mode Waveguides TE and TM modes Optical Fiber TE and TM modes Problems at high frequency operation 1.Circuits radiates and accept radiation : Information loss. Conductors become guides, current’s flow becomes field flow 2.EMI-EMC problems: Aggressor – Victim problems 3.Links in circuit behave as distributed parameters. 4. Links become transmission Line: Z0 , ρ, . 5.Delay – Phase shift-Retardation. 6. Digital circuits involves high frequency problems. 7. High energy particle behaviour. High Frequency Effects 1.Skin effect 2.Transit time – 3.Moving electron induce current 4. Delay 5. Retardation-.Radiation 6.Phase reversal of fields. 7.Displacement current. 8.Cavity High Frequency effects 1.Fields inside the conductor is zero. 2.Energy radiates from the conductors. 3.Conductor no longer behaves as simple conductor with R=0 4.Conductor offers R, L, G, C along its length. 5.Signal gets delayed or phase shifted. Skin Effect Skin effect makes the current flow simply a surface phenomenon. No current that vary with time can penetrate a perfect conducting medium. Iac = 0 The penetration of Electric field into the conducting medium is zero because of induced voltage effect. Thus inside the perfect conductor E = 0 The penetration of magnetic field into the conducting medium is zero since current exists only at the surface. H=0. Circuits Radiate at high frequency opearation D →λ Skin Effect As frequency increases, current flow becomes a surface phenomenon. Conductor radiates at high frequencies Circuit theory Model OR Lumped Model ( 100s Km ); ( D << ) Is our scale • Frequency f Wavelength 50 Hz 3 KHz 30 KHz 300 KHz 3 MHz 30 MHz 300 MHz 3 GHz 30 GHz 300 GHz 6,000 Km 100 Km 10 Km 1 Km 100 m 10 m 1m 10 cm 1 cm 1 mm V= V0 sin (0 ) V= V0 sin (90) V= V0 sin (360) V= V0 sin (180) Circuit domain :Dimension << C= f x = 300,000 km/sec Given f = 30 kHz ; = 10 km Hence circuit dimensions << = 10 km Medium = Conducting medium. = Conductors in circuits. Electrons = Energy Packet Energy E = eV electron volts; W= V X I Circuit Theory Connecting wires introduces no drop and no delay. The wires between the components are of same potential. Shape and size of wires are ignored. [email protected] For f =3 KHz, = 10 Km R 0o 180o 360o = 10 Km At 3 KHz No Phase variation across the Resistor [email protected] D < ; D << • When circuit dimension is very small compared to operating wavelength ( D << ) , circuit theory approximation can be made. • No phase shift the signal undergoes by virtue of distance travelled in a circuit. • Circuit / circuit components/ devices/ links will not radiate or radiation is very negligible. Field domain : Dimension C= f x = 300,000 km/sec Given f = 3000 MHz ; = 10 cm Hence circuit dimensions = 10 cm Dielectric medium – Free space Waves = E/H fieldes Energy E = h.f joules Total radiated power W = EXH ds joules Lumped circuit Model • Electric circuits are modeled by means of lumped elements and Kirchhoff’s law. • The circuit elements R, L, C are given values in those lumped circuit models, for example R=10 K, L = 10 H c= 10 pf. • These models are physical elements and hence the element values depend on the structure and dimensions of the physical elements. For f =30 GHz, 0o = 1cm 360o 180o = 1cm Resistor 0o 180o 360o At 30GHz 360o Phase variation across the Resistor [email protected] Balanced transmission line opened out to form dipole radiator Reactive drop Transmission Line Voltage Variation along the line [email protected] Frequency dependent parasitic elements At high frequency operation all ideal components deviate from their ideal behavior mainly due to parasitic capacitance and parasitic inductance. Any two conductors separated by some dielectric will have capacitor between them. Any conductor carrying current will have an inductance. Reactance XC and XL Parasitic capacitance and parasitic inductance create reactance that varies with frequency j XC 2fC X L j 2fL At DC, capacitance impedance is infinity; an open circuit. The capacitive reactance decreases with frequency. At DC an inductive impedance is zero; a short circuit. The impedance of inductive reactance increase with frequency. Thus these real components behave different at high frequency operation. Cp =Parasitic capacitance due to leads of resistor, parallel to R. At high frequency it shunts the resistor reducing its value. Llead = Due to resistor and material of resistor. High value R are not recommended for high frequency operation. Caution: Minimize the lead size, Use surface mounted device. Llead = Lead inductance Rlead = Lead resistance RDC = Dielectric leakage RAC =Dielectric Frictional loss due to polarization. At high frequency operation, the component acts as L. Large values of C are not useful at high frequency operation. RL =Lead Resistance CL =Lead capacitance Rcore =Core loss resistance Phase Shift in Transmission Line Space Effect 0o 180o 360o Magnitude of C = f met = 300 MHz For f = 3 KHz, = 100 KM For f =3 GHz, = 10cm For f =30 GHz, = 1cm C=fx For f =3 KHz, = 10 Km R 0o 180o 360o = 10 Km At 3 KHz No Phase variation across the Resistor Circuit Theory Connecting wires introduces no drop and no delay. The wires between the componenets are of same potential. Shape and size of wires are ignored. For f =30 GHz, 0o = 1cm 360o 180o = 1cm Resistor 0o 180o 360o At 30GHz 360o Phase variation across the Resistor Filters Any complicated network with terminal voltage and current indicated A T network which may be made equivalent to the network in the box (a) A network equivalent to (b) and (a). The T section as derived from unsymmetrical L-sections, showing notation used in symmetrical network analysis The section as derived from unsymmetrical L-sections, showing notation used in symmetrical network analysis Examples of Transmission Line Transmission Line in communication carry 1)Telephone signals 2)Computer data in LAN 3)TV signals in cable TV network 4)Telegraph signals 5)Antenna to transmitter link TRASMISSION LINE • It is a set of Conductors used for transmitting electrical signals. • Every connection in an electrical circuit is a transmission line. • Eg: Coaxial line, Twisted-wire • Parallel wire pairs • Strip line , Microstrip A succession of n networks in cascade. Two types of transmission lines. Basic Transmission Line. A transmission line whose load impedance is resistive and equal to the surge impedance appears as an equal resistance to the generator. Infinite parallel plane transmission line. Transmission line is low pass filter Any complicated network can be reduced to T or network T and Network Resonant circuit and Filter Resonant circuits select relatively narrow band of frequencies and reject others. Reactive networks, called filters, are designed to pass desired band of frequencies while totally suppressing other band of frequencies. The performance of filter circuits can be represented in terms of Input current to output current ratios. Image Impedance Non-Symmetry Network Input impedance at the 1,1 terminalZ1in Z1i Z1in Z1 ( Z 2 Z 2 i ) Z1i Z Z1 Z 2 Z 3 Z 2i Likewise, the impedance looking into the 2,2 Z 2i to be terminal is required Z 3 ( Z1 Z1i ) Z 2i Z 2 Z1 Z 3 Z1i Upon solving for Z1i andZ 2i ( Z1 Z 3 )( Z1Z 2 Z 2 Z 3 Z 3 Z1 ) Z1i Z 2 Z3 ( Z 2 Z 3 )( Z1Z 2 Z 2 Z 3 Z 3 Z1 ) Z 2i Z1 Z 3 Z1oc Z1 Z 2 Z1sc Z 2Z3 Z1 Z 2 Z 2 Z 3 Z 3 Z1 Z1 Z 2 Z3 Z 2 Z3 Z1i Z1oc Z1sc Z 2i Z 2oc Z 2 sc If the image impedances are equal Z1i Z 2i then V1i V2o I1i I 2o Then the voltage ratios and current ratios can be represented by I1 V1 I 2 V2 (1) Performance of Unsymmetrical T & Networks Performance parameters of a Network (Active or Passive) 1. Gain of Loss of signal due to the Network in terms of Voltage or Current ratios. V1 A V2 I1 B I2 2. Delay of phase shift of the signal due to network. Performance of a N networks in cascade If several networks are used in succession as in fig., the overall performance may be appreciated V3a Vn1 V1 V1 V2 as V2 X V3 X V4 X ..... Vn Vn (2) Which may also me stated as A1 . A2 .1A3 . A4 A1 A2 A3 A4 Both the processes employing multiplication of magnitudes. In general the process of addition or subtraction may be carried out with greater ease than the process of multiplication and division. It is therefore of interest to note that e xe xe x.....e e b c n a b c .... n Is an application in which substituted for multiplication. addition is If the voltage ratios are defined as V1 a V2 b V3 c e ; e ; e ;.......etc V2 V3 V4 Eq. (2) becomes V1 a b c ........ n e Vn If the natural logarithm (ln) of both sides is taken, then V1 ln a b c d .......... n V2 (3) Thus it is common to define under conditions of equal impedance associated with input and output circuits. V1 I1 N e V2 I2 (4) The unit of “N” has been given the name nepers and defined as V1 I1 N ln nepers ln V2 I2 (5) Two voltages, or two currents, differ by one neper when one of them is “e” times as large as the other. Obviously, ratios of input to output power may also may also be expressed In this fashion. That is, P1 e2 N P2 The number of nepers represents a convenient measure of power loss or power gain of a network. Losses or gains of successive Transmission Line 1.It provided guided communication to distance with reasonable minimum attenuation 2.It overcomes the parasitic effects of lumped elements due to high frequency operation. 3. High frequency operation introduces distributed parameter effect. 4.Due to high frequency operation, energy carried by fields rather than voltage and currents. 5. Operation remains outside conductors. 6. Radiation and phase shift (delay) play important roles. 7. Radiation effects are much reduced or prevented by special arrangements. 8. Treating Tr.Line as infinite infinitesimal symmetrical networks, network theory analysis is adopted. Analysis of Transmission line ( N networks in cascade) based on basic symmetrical T and networks Transmission line is low pass filter Any complicated network can be reduced to T or network T and Network Resonant circuit and Filter Resonant circuits select relatively narrow band of frequencies and reject others. Reactive networks, called filters, are designed to pass desired band of frequencies while totally suppressing other band of frequencies. The performance of filter circuits can be represented in terms of Input current to output current ratios. Image Impedance Non-Symmetry Network Input impedance at the 1,1 terminalZ1in Z1i Z1in Z1 ( Z 2 Z 2 i ) Z1i Z Z1 Z 2 Z 3 Z 2i Likewise, the impedance looking into the 2,2 Z 2i to be terminal is required Z 3 ( Z1 Z1i ) Z 2i Z 2 Z1 Z 3 Z1i Upon solving for Z1i andZ 2i ( Z1 Z 3 )( Z1Z 2 Z 2 Z 3 Z 3 Z1 ) Z1i Z 2 Z3 ( Z 2 Z 3 )( Z1Z 2 Z 2 Z 3 Z 3 Z1 ) Z 2i Z1 Z 3 Z1oc Z1 Z 2 Z1sc Z 2Z3 Z1 Z 2 Z 2 Z 3 Z 3 Z1 Z1 Z 2 Z3 Z 2 Z3 Z1i Z1oc Z1sc Z 2i Z 2oc Z 2 sc If the image impedances are equal Z1i Z 2i then V1i V2o I1i I 2o Then the voltage ratios and current ratios can be represented by I1 V1 I 2 V2 (1) Performance of Unsymmetrical T & Networks Part-2 EC 2305 (V sem) Transmission Lines and Waveguides 24.7.13 Dr.N.Gunasekaran Dean, ECE Rajalakshmi Engineering College Thandalam, Chennai- 602 105 [email protected] Filters Filters -Resonant circuits Resonant circuits will select relatively narrow bands of frequencies and reject others. Reactive networks are available that will freely pass desired band of frequencies while almost suppressing other bands of frequencies. Such reactive networks are called filters. . Ideal Filter An ideal filter will pass all frequencies in a given band without (attenuation) reduction in magnitude, and totally suppress all other frequencies. Such an ideal performance is not possible but can be approached with complex design. Filter circuits are widely used and vary in complexity from relatively simple power supply filter of a.c. operated radio receiver to complex filter sets used to separate the various voice channels in carrier frequency telephone circuits. Application of Filter circuit Whenever alternating currents occupying different frequency bands are to be separated, filter circuits have an application. Neper - Decibel In filter circuits the performance Indicator is Performance Input current Output current If the ratios of voltage to current at input and output of the network are equal then I1 V1 I2 V2 (1) If several networks are used in cascade as shown if figure the overall performance will become V3 Vn 1 V1 V2 V1 X X X ..... V2 V3 V4 Vn Vn (2) Which may also me stated as A1 . A2 .1A3 . A4 A1 A2 A3 A4 Both the processes employing multiplication of magnitudes. In general the process of addition or subtraction may be carried out with greater ease than the process of multiplication and division. It is therefore of interest to note that e e e .....e e b c n a b c .... n is an application in which substituted for multiplication. addition is If the voltage ratios are defined as V1 a V2 b V3 c e ; e ; e ;.......etc V2 V3 V4 Eq. (2) becomes V1 a b c ........ n e Vn If the natural logarithm (ln) of both sides is taken, then V1 ln a b c d .......... n V2 (3) Consequently if the ratio of each individual network is given as “ n “ to an exponent, the logarithm of the current or voltage ratios for all the networks in series is very easily obtained as the simple sum of the various exponents. It has become common, for this reason, to define V1 I1 eN V2 I2 (4) under condition of equal impedance associated with input and output circuits The unit of “N” has been given the name nepers and defined as nepers N V1 I1 ln ln V2 I2 (5) Two voltages, or two currents, differ by one neper when one of them is “e” times as large as the other. Obviously, ratios of input to output power may also may also be expressed In this fashion. That is, P1 2N e P2 The number of nepers represents a convenient measure of power loss or power gain of a network. Loses or gains of successive networks then may be introduced by addition or subtraction of their appropriate N values. “ bel “ - “ decibel “ The telephone industry popularized a similar unit to the base 10, naming Alexander Graham Bell The “bel” is defined as power ratio, P1 number of bels = log proposed and has based on logarithm the unit “ bel “ for the logarithm of a P2 It has been found that a unit, one-tenth as large, is more convenient, and the smaller unit is called the decibel, abbreviated “db” , defined as P1 dB 10 log P2 (6) In case of equal impedance in input and output circuits, I1 V1 dB 20 log 20 log I2 V2 (7) Equating the values for the power ratios, e 10 2N dB 10 Taking logarithm on both sides 8.686 N = dB Or 1 neper = 8.686 dB Is obtained as the relation between nepers and decibel. The ears hear sound intensities on logarithmically and not on a linear one. a Performance parameters of a “series of identical networks”. 1.Characteristic Impedance Z0 2. Propagation constant For efficient propagation, the network is to be terminated by Z0 and the propagation constant should be imaginary. We should also attempt to express these two performance constants in terms of network components Z1 and Z2 . What is Characteristic impedance of symmetrical networks Symmetrical T section from L sections For symmetrical network the series arms of T network are equal Z1 Z 2 Z1 2 Symmetrical from L sections Z a Z c 2Z 2 Both T and networks can be considered as built of unsymmetrical L half sections, connected together in one fashion for T and oppositely for the network. A series connection of several T or networks leads to so-called “ladder networks” which are indistinguishable one from the other except for the end or terminating L half section as shown. Ladder Network made from T section Ladder Network built from section The parallel shunt arms will be combined For a symmetrical network: Z1i Z 2i the image impedance Z1i and Z 2 i are equal to each other and the image impedance is then called characteristic impedance or iterative impedance, . Z1i Z 2i Z it Z o That is , if a symmetrical T network is terminated in Z 0 , its input impedance will also be Z 0 , or the impedance transformation ration is unity. If ZR Z 0 then Z i Z 0 Zi Z0 Z R Z0 The term iterative impedance terminating impedance Z 0 the input impedance of a networks in which case Z 0 input to each network. is apparent if the is considered as chain of similar is iterated at the Z R Z 0 Z it Z in Characteristic Impedance of Symmetrical T section network For T Network terminated in Z 0 Z 1 Z ( Z0 ) 2 Z1 2 Z1in 2 Z1 Z Z 2 0 2 When Z1in Z 0 Z0 Z12 4 2 1 z1 z 2 z 2 z0 Z1 2 z 2 z0 Z Z Z1 Z 2 4 2 0 Z1 Z 0 (9) 2 Characteristic Impedance for a symmetrical T section Z 0T Z12 Z1 Z1Z 2 Z1Z 2 (1 4 4Z 2 (!0) Characteristic impedance Z 0 is that impedance, if it terminates a symmetrical network, its input impedance will also be Z 0 Z0 is fully decided by the network’s intrinsic properties, such as physical dimensions and electrical properties of network. Characteristic Impedance section Z 0 Z1in 2Z 2 Z 0 Z1 ( 2 Z Z ) 2 Z 2 2 0 2Z 2 Z 0 Z1 2Z 2 Z 0 2Z 2 When Z1in Z 0 , for symmetrical Characteristic Impedance Z1 Z 2 Z 1 1 4Z 2 Z 0 (11) Z1 Z1oc Z oc Z 2 2 Z1Z 2 Z1 2 Z1sc Z sc 2 Z1 Z 2 2 Z12 Z oc Z sc Z1Z 2 Z 02T 4 2 2 4Z Z1 2 Z 0c Z sc Z 0 Z1 4Z 2 Z0 Zoc Z sc (13) (12) propagation constant V1 I1 eN V2 I2 The magnitude ratio does not express the complete network performance , the phase angle between the currents being needed as well. The use of exponential can be extended to include the phasor current ratio. I1 e I2 (14) Where is a complex number defined by j Hence If (15) I1 j e e I2 I1 A I2 I1 A e I2 e j With Z0 termination, it is also true, V1 e V2 The term has been given the name propagation constant = attenuation constant, it determines the magnitude ratio between input and output quantities. = It is the attenuation produced in passing the network. Units of attenuation is nepers = phase constant. It determines the phase angle between input and output quantities. = the phase shift introduced by the network. = The delay undergone by the signal as it passes through the network. = If phase shift occurs, it indicates the propagation of signal through the network. The unit of phase shift is radians. If a number of sections all having a common Z0 the ratio of currents is I1 I 2 I 3 I1 ........ I 2 I3 I 4 In 1 2 3 from which e e e ........ e n and taking the natural logarithm, 1 2 3 4 .................. n (16) Thus the overall propagation constant is equal to the sum of the individual propagation constants. Z 0 and of symmetrical networks and the introduction of Use the definition of e as the ratio of currents for a Z 0 termination leads to useful results The T network in figure is considered equivalent to any connected symmetrical network terminated in a Z 0 termination. From the mesh equations the current ratio can be shown as I1 I2 Z1 2 Z2 Z0 Z2 e (30) Where the characteristic impedance is given 2 as Z Z 02 1 4 Z1 Z 2 (32) Eliminating Z 0 Z1 cosh 1 2Z 2 (33) Z1 sinh 2 4Z 2 (36) The propagation constant can be related to network parameters by use of (10) for Z OT In (30) as Z1 e 1 2 2Z 2 Z1 2 Z1 ( ) 2Z 2 Z2 Taking the natural logarithm Z1 ln 1 2Z 2 Z1 Z1 2Z Z 2 2 2 For a network of pure reactance it is not difficult to compute. The input impedance of any T network terminated in any impedance ZR , may be written in terms of hyperbolic functions of . Writing Z in 2 12 Z Z11 Z 22 It is reduced to Z R cosh Z0 sinh Z in Z 0 Z0 cosh ZR sinh For short circuit, Z R = 0 Z SC Z 0 tanh For a open circuit Z R Z lim Z Z0 tanh (39) (40) (41) From these these two equations it can be shown that Z SC tanh ZOC (42) Z 0 Z OC Z SC Thus the propagation constant and the characteristic impedance Z0 can be evaluated using measurable parameters Z SC and Z OC Filter fundamentals Pass band – Stop band: The propagation constant is j For = 0 or I1 I 2 There is no attenuation , only phase shift occurs. It is pass band. when ve; I1 I 2 , attenuatio n occurs; - Stop band Is conveniently studied by use of the expression. Z1 sinh 2 4Z 2 It is assumed that the network contains only pure reactance and thus Z1 4Z will be real 2 and either positive or negative, depending on the type of reactance used for Z1 and Z2 Expanding the above expression j sinh sinh ( ) 2 2 2 sinh cos j cosh 2 2 It contains much information. 2 sin 2 If Z1 and Z2 are the same type reactances then Z1 0 or the ratio is positive and real. 4Z 2 This condition implies a stop or attenuation band of frequencies. The attenuation will be given by 2 sinh 1 Z1 4Z 2 If Z1 and Z2 are opposite type reactances then Z1 0 or the radical is imaginary. 4Z 2 This results in the following conclusion for pass band. Z 1 1 4Z 2 0 The phase angle in this pass band will be given by 2 sin 1 Z1 4Z 2 Another condition for stop band is given as follows: Z1 when 1 4Z 2 Z1 0 4Z 2 Stop band. Z1 1 0 4Z 2 pass band Z1 1 4Z2 stop band Cut-off frequency The frequency at which the network changes from pass band to stop band, or vice versa, are called cut-off frequencies. These frequencies occur when Z1 0 or Z1 0 4Z 2 Z1 1 or Z1 4 Z 2 4Z 2 (48) where Z1 & Z2 are opposite types of reactances . Since Z1 and Z2 may have number of combinations, as L and C elements, or as parallel and series combinations, a variety of types of performance are possible. Constant k- type low pass filter (a) Low pass filter section; (b) reactance curves demonstrating that (a) is a low pass section or has pass band between Z1 = 0 and Z1 = - 4 Z2 If Z1 and Z2 of a reactance network are unlike reactance arms, then Z1Z 2 k 2 where k is a constant independent of frequency. Networks or filter circuits for which this relation holds good are called constant-k filters. j Z1 jL and Z2 Z1Z 2 L C R 2 k C (51) (b) reactance curves demonstrating that (a) is a low pass section or has pass band between Z1 = 0 and Z1 = - 4 Z2 Low pass filter Pass band : Z1 0 to Z1 - 4Z2 f 0 to f f c f f c stopband fc 1 LC f sinh j 2 fC Variation of and with frequency for the low pass filter f Z1 For 1 so that - 1 0 , then fc 4Z2 f 1, fc 0, f 2sin ( ) fc -1 Phase shift is zero at zero frequency and increases gradually through the pass band, reaching at cut-off frequency and remaining same at at higher frequencies. Characteristic Impedance of T filter Z OT 2 f L 1 C fC Z OT RK f 2 1 f C ZOT varies throughout the pass band, reaching a value of zero at cut-off, then becomes imaginary in the attenuation band, rising to infinity reactance at infinite frequency Z OT Variation of R k filter. with frequency for low pass Constant k high pass filter (a) High pass filter; (b) reactance curves demonstrating that (a) is a high pass filter or pass band between Z1 = 0 and Z1 = - 4Z2 m-derived T section (a) Derivation of a low pass section having a sharp cut-off section (b) reactance curves for (a) m-derived low pass filter Variation of attenuation for the prototype amd m-derived sections and the composite result of two in series. Variation of phase shift for mderived filter Variation of Z 0 over the pass band for T and networks (a) m-derived T section; (b) section formed by rearranging of (a); © circuit of (b) split into L sections. Variation of Z1 of the L section over the pass band plotted for various m valus Cascaded T sections = Transmission Line Circuit Model/Lumped constant Model Approach • Normal circuit consists of Lumped elements such as R, L, C and devices. • The interconnecting links are treated as good conductors maintaining same potential over the interconnecting links. Effectively links behaves as short between components and devices. • Circuits obey voltage loop equation and current node equation. Lumped constants in a circuit Transmission Line Theory Transmission Line = N sections symmetrical T networks with matched termination If the final section is terminated in its characteristic impedance, the input impedance at the first section is Z0. Since each section is terminated by the input impedance of the following section and the last section is terminated by its Z0. , all sections are so terminated. Characteristic impedance of T section is known Z1 ) asZ OT Z1Z 2 (1 4Z 2 There are n such terminated section. I s , I r = sending and receiving end currents Is n e Ir the n = Propagation constant for one section Z1 Z1 2 Z1 e 1 2 ( ) 2Z 2 2Z 2 Z2 Z1 Z1 2 Z1 ln 1 ( ) 2Z 2 Z2 2Z 2 A uniform transmission can be viewed as an infinite section symmetrical T networks. Each section will contributes proportionate to its share ,R, L, G, C per unit length. Thus lumped method analysis can be extended to Transmission line Certain the analysis developed for lumped constants can be extended to distributed components well. The constants of an incremental length x of a line are indicated. Series constants: R + j L ohms/unit length Shunt constants: Y + jC mhos/unit length Thus one T section, representing an incremental length x of the line has a series impedance Zx ohms and a shunt admittance Yx mhos. The characteristic impedance of all the incremental sections are alike since the section are alike. Thus the characteristic impedance of any small section is that of the line as a whole. Thus eqn. (1) gives the characteristic of the line with distributed constant for one section is given as Zx ZxYx Z0 (1 ) Yx 4 Z ZYx Z0 (1 ) Y 4 2 (4) Allowing x to approach zero in the limit the value of Z0 for the line Z of distributed constant is obtained Ohms as Z 0 Y (5) Z and Y are in terms unit length of the line. The ration Z/Y in independent of the length units chosen. Propagation Constant Under Z0 termination I1/ I2 = eγ γ = Propagation constant α + jβ I1/ I2 = ( Z1/2 + Z2 + Z0 ) / Z2 = eγ = 1 + Z1/ 2Z2 + Z0/ Z2 I1/ I2 = 1 + Z1/ 2Z2 + √ Z1/Z2 ( 1 + Z1 / 4Z2 ) Propagation Constant Z1 / Z2 ( 1 + Z1 / 4 Z2 ) = Z / Z2 [ 1 + ½ (Z1 /4Z2) – 1/8 (Z1 / 4Z2)2 + ……..] e = 1+ Z1 / Z2 + ½ (Z1 / Z2 )2 + 1/8 (Z1 / Z2 )3 – 1/128 (Z1 / Z2 )5 + …… Applying to incremental length x e x = 1 + ZY x + ½ (ZY)2 x 2 + 1/8 ((ZY)23 x3 – 1/128 (ZY)5 x5 + … 6.6) Series expansion is done e x e x = 1 + x + x 2 x2 / 2! + 3 x3 / 3! + … (6.7) Equating the expansions and canceling unity terms x + 2 x2 / 2 + 3 x3 / 6 + … = ZYx + ( ZY)2 x2 )/2+ ( ZY)3 x3) / 8 + … Divide x + 2 x2 / 2 + 3 x3 / 6 … = ZY + (ZY)2 x / 2 + (ZY)3 x2 / 8 + … as x 0 γ = ZY (8) Characteristic Impedance γ = ZY Propagation Constant as x Z0 = Z / Y Ohms 0 Characteristic or surge impedance Since there no energy is coming back to the source , there is no reactive effect. Consequently the impedance of the line is pure resistance. This inherent line impedance is called the characteristic impedance or surge impedance of the line. The characteristic impedance is determined by the inductance and capacitance per unit length . These quantities are in turn depending upon the size of the line conductors and spacing Dimension of line decides line impedance The closer the two conductors of the line and greater their diameter, the higher the capacitance and lower the inductance. A line with large conductors closely spaced will have low impedance. A line with small conductors and widely spaced will have relative large impedance. The characteristic impedance of typical lines ranges from a low of about 50 ohms in the coaxial line type to a high of somewhat more than 600 ohms for a open wire type. Z0 jL L jC C Thus at high frequencies the characteristic impedance Z0 of the transmission line approaches a constant and is independent of frequency. Z0 depends only on L and C Z0 is purely resistive in nature and absorb all the power incident on it. jL L (5.5x106 ) Z0 2500 50 12 jC C (2200 x10 ) Characteristic line impedance Z1 R R1 10 100 110 RS Z1 100 x110 Z2 R 10 10 52.38 62.38 RS Z1 100 110 RS Z 2 100 x62.38 Z3 R 10 10 38.42 48.32 RS Z 2 100 62.38 100 x 48.32 Z 4 10 10 32.63 42.62 100 48.32 With additional section added the input impedance is decreasing further till it reaches its characteristic impedance of 37. For a single section with termination of 37 RS xZL 100 X 37 3700 Z 0 Z1 R 10 10 37 RS Z L 100 37 137 Transmission Line Transmission line is a critical link in any communication system. Transmission lines behaves as follows: a)Connecting link b) R – L – C components c)Resonant circuit d)Reactance impedance e) Impedance Transformer