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Transmission Lines The problem of plane waves propagating in air represents an example of ungu ided wav e propag ation. Tra nsmi ssion lines and waveg uides offer an alternative way of transmitting signals in the form of guided wave propagation. Tra nsmi ssion lines ar e typically electrically large (sever al wavelengths) such that we cannot accurately describe the voltages and currents along the t ransmi ssion line u sing a s imple lumped-element equivalent circuit. We must use a distributed-element equivalent circuit which describes each short segment of the transmission line by a lumped- element equivalent circuit. Consider a simple uniform two-wire transmission line with its conductors parallel to the z-axis as shown below. Uniform transmission l ine - condu ctors and i nsulating m edium maintain the same cross-sectional geometry a long the entire transmi ssion line. The equivalent circuit of a short segment Äz of the two-wire transmission line may be represented by simple lumped-element equivalent circuit. R =series resistance per unit length (Ù/m) of the transmission line conductors. L =series induc tance per unit length (H/m) of the transmission line conductors (internal plus external inductance). G =shunt conductance per unit length (S/m) of the med ia bet ween the transmi ssion line con ductors. C =shunt capacitance per unit length (F/m) of the transmission line conductors. We ma y relate the values of voltage and current at z and z+Äz by writing KVL and KCL equations for the equivalent circuit. KVL KCL Grouping the voltage and current terms and d ividing by Äz gives Taking the limit as Äz 6 0, the terms on the right hand side of the equations above become partial derivatives with respect to z which gives Time-domain transmission line equations (coupled PDE’s) For time-harmonic signals, the instantaneous voltage and current may be defined in terms of phasors such that The derivatives of the voltage and c urrent with respect to time yield jù times t he respect ive phasor w hich gives Frequency-domain (phasor) transmission line equations (coupled DE’s) Note the si milarity in the fu nctional form of t he time- and frequency- domain transmi ssion line equations to the respective source-free Maxwell’s equations (cu rl equations). Ev en though these equ ations were der ived without any consideration of the electromagnetic fields associated with the transmissio n line, remember th at circu it theory is based on Maxwell’s equations. Given the similarity of the phasor tra nsmission line equations to Maxwell’s equations, we find that the voltage and current on a transmission line sat isfy wave equat ions. These voltage and current wave equations are derived using the same techniques as the electric and m agnetic field wave equations. Beginning wi th the phasor transmission line equations, we take derivatives of both sides with respect to z. We then insert the first derivatives of the voltage and current found in the original phasor transmi ssion line equ ations. The voltage and current wave equ ations may be w ritten as where ã is the complex propagation constant of the wa ve on the transmission line given by Just as with unguided waves, the real part of the propagation constant (á) is the attenuation constant while the imaginary part (â) is the ph ase constant. The general equations for á and â in terms of the per-unit-length transmissio n line parameters are The general solutions to the voltage and current wave equations are ~~~~~ ~~~~~ + z-directed w aves !z-directed w aves _ a The current equation may be written in terms of the voltage coefficients throug h the original phasor transmi ssion line equ ations. o The complex constant Z is defined as the transmission line characteristic impedance and is given by The transmission line equations written in terms of voltage coefficients only are The complex voltage coefficients may be written in terms of magnitude and phase as The instantaneo us voltage beco mes The wavelength and p hase velocity of the waves on t he transmission line may be found using the p oints of constant pha se as was don e for plane waves. Lossless Transmission Li ne If the transmission line loss is neglected (R = G = 0) , the equivalent circuit reduces to Note that for a true lossless transmission line, the insulating m edium between the conductors is characterized by a zer o conductivity (ó = 0) , and real-valued per mittivity å and per meabi lity ì (åO = ìO= 0) . The propagation constant on the lossless transmission line is Given the purely imaginary propagation c onstant, the transmission line equations for the lossless line are The char acteristic impedan ce of t he lossless transmi ssion line is purely real and g iven by The phase velocity a nd wavelength on the lossless line are Examp le (Lossless coaxia l transmi ssion line) The dom inant mode on a coaxial transmission line is the TEM zz (transverse electromagnetic) mode defined by E = H = 0. Due to the symmetry of the coaxial transmission line, the transverse fields ar e indepen dent of ö. Th us, we may write Between the conductors, the fields must satisfy the sou rce-free M axwell’s equations: ñö The field components E and H are related to the transmission line voltage and current by 1 If we integrate the Faraday’s law equation along the path C and integrate 2 the Ampere’s law equation along the path C, which ar e the lossless transmi ssion line equ ations. Terminated Los sless Transmission Li ne If we choose our reference point (z = 0) at the load termination, then the lossless transmission line equations evaluated at z = 0 give the load voltage and current. The ratio of voltage to current at z = 0 must equal the load impedan ce. Solving this equation for the voltage coeffi cient of the !z traveling wav e gives where à is the reflection coeff icient which defines t he ratio of the reflected wave t o the incident wave. Note that the reflection coefficient is in general complex with 0 # *Ã*# 1. Lo If the reflection coefficient is zero (Z = Z), there is no r eflected wave and Lo the load is said to be matched to the transmi ssion line. If Z … Z, the magn itude of t he reflection coeff icient is no n-zero (there is a reflected wave). The presence of forward and r everse traveling wav es on t he transmi ssion line produces standing waves . We may rewrite the transmission line equations in terms of the reflection coeff icient as or The mag nitude of t he transmi ssion line voltage may be w ritten as The maximum and minimum voltage magnitudes are The ratio of maximum to m inimum vol tage magnitudes defines the standing wave ratio (s). Note that the standing wav e ratio is real with 1 # s # 4. The time-average power at any point on t he transmission line is given by ~~~~~~~~~~~~~~ P urely ima ginary A ! A = 2j Im{A}* iP = incident power rP = reflected power The return loss (RL) is defi ned as the ratio of incident power to reflected power. The return loss in dB is Matched load *Ã* = 0 s = 1 RL = 4 (dB) Total reflection *Ã* = 1 s = 4 RL = 0 (dB) Transmi ssion Line I mpedan ce The impedance at any point on t he transmission line is given by The impedance at the input of a transmission line of length l terminated L with an impedan ce Z is Lo Lossless Transmission Li ne with Matched Loa d (Z = Z) Note that the input impedan ce of the lossless transmi ssion line terminated with a mat ched impedan ce is indepen dent of the line length. Any mi smat ch in the transmission line system will cause standing wav es and m ake the input impedan ce depen dent on the length of the line. L Short-Circuited L ossless Transmi ssion Line ( Z = 0) [See Figure 2.6 (p.61)] The input impedance of a short-circuited lossless transmission line is purely reactive and can t ake on any val ue o f capaci tive o r inductive reactance depending on the line length. The incident wave is to tally reflected (with min inversion) from the load setting up s tanding wav es with *V* = 0 and max o *V* = 2*V*.+ L Open-Circuited L ossless Transmi ssion Line ( Z = 4) [See Figure 2.8 (p.62)] The input impedance of an open-circuited lossless transmission line is purely reactive and can take on any value of capacitive or inductive reactance depending on the line length. The incident wave is totally reflected (without inversion) from the load setting up standing waves with min max o *V* = 0 and *V* = 2*V*.+ Transmission Li ne Conne ctions The anal ysis of a connection between t wo distinct transmi ssion lines can be performed us ing t he same techniques used for plane wave transmi ssion/refle ction at a mat erial interface. Consider two lossless transmi ssion lines of dif ferent char acteristic impedan ces connected as shown below. Assume that a source is conne cted to transmission line #1 a nd transmission L2 o2 line #2 is terminated w ith a mat ched impedan ce (Z = Z). Transmission o2 line #1 is then effectively terminated w ith a load impedan ce of Z which constitutes a mismatch. At the transmission line connection, a portion o f the incident wave on transmission line #1 is transmitted onto transmission line #2 while the remai nder is reflected back on transmi ssion line #1. Thus, we may write the voltages on the two transmi ssion lines as where à is the reflection coefficient on t ransmission line #1 a nd T is the transmi ssion coeffi cient on transmi ssion line #2. Equating the voltages at the transmission line connection point (z = 0) gives Note the similarity between the equations for p lane wave (unguided wave) transmission/reflection a t a material interface and t he gui ded wave transmission/reflection at a transmission line conne ction. Insertion Loss Strictly speaki ng, insertion loss i s the ratio of power absorbed b y a load before and after a network is inserted into the line. For the previous example conne ction b etween two transmission lines, we co nsider the matched case for transmission line #1 a nd the case with transmission line #2 inserted into the system. The insert ion loss (IL) is then Smith Chart The Smith chart is a useful graphical tool used to ca lculate the reflection c oefficient and impedance at various points on a transmission line syst em. Th e Smi th ch art is actually a pol ar plot of t he com plex reflection coefficient Ã(z) overlaid with the corr espon ding impedan ce Z(z). The voltage at any point on the transmission line is The reflection coeff icient at any po int on the transmi ssion line is defi ned as where à is the reflection coefficient at the load (z = 0). Smith chart ce nter Y *Ã* = 0 (no reflection - matched) Outer circle Y *Ã* = 1 (total reflection) The reflection coefficient for a lossless transmission line of characteristic oL impedan ce Z, terminated w ith an impedan ce Z is given by LL where z is the normalized load impedance. If we solve (1) for z, we find LL where r and x are the normalized l oad resistance and reactance, respectively. Equa tion (2) shows that the reflection c oefficient on t he Smith chart correspond s to a specific normalized load impedance for the given transmi ssion line / load comb ination. Solving (2) for the resistance and reactance gives equ ations for the “resistance” and “ reactance” ci rcles: (2)(1) In a similar fashion, the general impedance at any point along the length of the transmi ssion line may be w ritten as n The norma lized val ue of the impedan ce z(z) is Note the similarity between Equa tions (2) and (3). The magnitude of the reflection coefficient is constant on a lossless line. Thus, Equation (3) shows that once we locate the normalized load impedance on the Smith chart, we simply rotate throug h an angle of 2 âz on the *Ã* circle to find the impedance at a given po int on t he transmission line. Eva luating the reflection coefficient at the transmission line input (z = !l) gives which defines a negative phase shift moving toward the generator from the load. Thus, in general CW rotationY toward the generator CCW rotationY toward the load One complete revolution o n the Smith chart occurs for Thus, each revolution on the Smith chart represents a movement of one-half wavelength along the transmi ssion line (ë = 720).o Once an impedance point is located on the Smith chart, the equivalent admittance point is found by rotating 180 from the impedance point on theo constant reflection coeff icient circle.(3) Example Problem 2.19 (using equations and Smith chart) (b.) (a.) (c.) (d.) (e.) (f) 54m -6 "(180 -0.3Zug=OE8____\ 180),_n-03= (evenn) 2B4n 4 _ _03, _ _ m=OEgg SEA=-007SR Igy=0.0752 n=2>Fac=AEA=0.425% Lossy T ransmission Lines The g eneral transmi ssion line equ ations (defined in terms of a comp lex propagation constant) must be u sed w hen deal ing with lossy l ines. The general transmissio n line equations are The reflection coeffi cient for a lossy t ransmi ssion line may be d etermined by replacing each jâ term with ã which gives In a similar fashion, the input impedance of a terminated lossy transmission line is Most transmission lines are designed with materials which produce small losses (low loss l ines). The general expressions for the characteristic impedance and p ropagation constant of a lossy transmission line may be simplified somew hat for a low-loss l ine. Lossless lineY R = G = 0 Low l oss lineY R << ùL, G << ùC Thus, for the equ ivalent circuit of a low loss line, the reactance terms must be much larger than the resistance terms at all frequencies of operation. The g eneral equ ation for the propagation con stant on a l ossy transmi ssion line may be app roximated as f ollows for a low loss l ine. Note th at we still include the dominant loss terms (even though they are small fo r a lo w loss line). Using a Taylor series expansion for the square root term above yields Using the same approximations for the characteristic impedance, we find The power level delivered to the load is lo wer than that delivered to the transmission line input due to the line losses. The line losses can be determined by calculating the difference in these power levels. The voltage and current at the load are The power delivered to the load is The voltage and current at t he input to the tran smissio n line are The power delivered to the input of the transmission line The power lost in the transmission line is Distortionless Tran smission Li ne On a lossless transmission line, the propagation constant is p urely imaginary and g iven by The phase velocity on the lossless line is Note that the ph ase constant which varies linearly with frequency pr oduces a const ant phase velocity (indepen dent of fr equency) so that all frequencies propagate along the lossless transmission line at the same velocity. From Fourier theory, we know that any t ime-domain signal may be r epresented as a weighted sum o f sinusoids. Thus, signals transmi tted along a lossless transmi ssion line will suffer no distortion since al l of t he fr equency comp onents propagate at the same v elocity. When the phase velocity of a transmi ssion line is a function of fr equency, signals will becom e distorted as different comp onents of the signal arrive at different times. This effect is called dispersion. For the low-loss line, using the appropriate approximations, we found which implies that the phase velocity on a low loss line is near constant. However, the smal l variations in the phase vel ocity on a low loss l ine may produc e significant distortion if the line is very long. There is a speci al case of lossy line w ith the linear phase constant that produces a distortionless line. A t ransmission line is a distortionless line if the per-unit-length paramet ers satisfy Inserting the p er-unit-length paramet er relationship into the g eneral equation for the propagation constant on a lossy l ine gives Although the shape of t he si gnal is not distorted, the si gnal will suffe r attenuation as the w ave prop agates along the line si nce t he distortionless line is a lossy tran smission line. N ote th at th e attenuation constant for a distortionless transmission line is also independent of frequency. If this were not true, the si gnal would suffer distortion due to different frequencies being attenuated by different amou nts. In the previous derivation, we have assumed that the per-unit-length parameters of the transmission line are independent of frequency. This is also an approximation t hat depends on t he spectral content of the propag ating signal. For very wideband signals, the at tenuation and phase constants will, in general, be fun ctions of f requency. For most practical transmission lines, we find that RC > GL. In order to satisfy the d istortionless line r equirement , series loading coils are typically placed periodically along the line to increase L. Perturbation M ethod f or Determining At tenuat ion Given only a forward wave propagating along a low loss transmission line, the voltage and current of the w ave may be w ritten as The power flow as a function of position along the transmission line is given by l The power loss per unit length along the transmi ssion line [P(z)] may be w ritten as Solving for the at tenuation constant gives Since t he fi elds of a low loss t ransmi ssion line ar e ver y close to those of a lossless line, we may use t he lossless line fi elds to cal culate the power loss per unit length (perturbation met hod). Note that power P(z) and the power l loss per unit length P(z) may be evaluated at any point on the transmission line. The perturbation method allows for the calculation of the attenuation constant using the transmission line fields rather than using the per-unit- length paramet ers in the general propagation constant formula. Example (Perturbation method - coaxial line attenuation constant) The fields within a lossless coaxial line are The attenuation constant, according to the perturbation method, is The power flow at any point on t he transmission line may be found by integrating the P oynting vect or over the surf ace S where the fields are located [S is defi ned by ( a#ñ#b) and (0#ö#2ð)]. Assuming that the dielectric and mag netic losses are negli gible, the power loss per unit length in the condu ctors of the coaxial line is given by si so where J and J are the surface currents on the inner and outer co nductors si so while R and R are the surf ace resistances of the inner and outer i conductors. Th e surface S on the inner con ductor is defi ned by ñ=a, o 0#ö#2ð, and 0#z#l while the surf ace S on the outer conductor is defi ned by ñ=b, 0#ö#2ð, and 0#z#l. Using the surface impedance approx-imation, the surface current on a goo d condu ctor is approximately that on a PEC which may be w ritten as Thus, we may replace the surface currents in the power loss per unit len gth equation by the surf ace magn etic fields ass ociated w ith a lossless coaxia l line. The power loss per unit length is then ti to where H and H are the tangential magnetic fields on t he surface of the inner and out er condu ctors. If the inner and ou ter condu ctors are the same material, then Evaluation o f the integrals in the power loss per unit length expression yields c The at tenuation con stant due to co nductor loss ( á) on t he coaxial line beco mes ss The surf ace resistance ( R) of the condu ctors is related to the skin depth (ä) by o For RG-59 coaxia l cabl e (Z = 75 Ù, copper conductors, ó = 5.8 × 1 0 É/m,7 o a = 0.292 mm, b = 1.854 mm, ì = ì ) at 500 MHz, scR = 5. 83 × 10 Ù, á = 0.0245 Np/m!3 Manufacturers normally specify the transmission line attenuation factor in units of dB/m as opposed to Np/m. The conversion factor between the two units is determined below. nep With á defined i n Np/m (á), the transmission line voltage attenuation is dB With á defined i n dB /m (á), the transmission line voltage attenuation is Equa ting the two expressions yields For the R G-59 coaxia l line, Wheeler Incremental Inductance Rule By noting that the con ductor loss in a transmi ssion line can b e related to the small change in the transmission l ine induc tance due to the penetration of the fields into the condu ctors, Wheeler derived an equation for the transmission line condu ctor loss in terms of the change in the characteristic impedance. The result of the W heeler increment al inductance rule may be w ritten as so where is R the surface resistance of the conductor, Z is th e characteristic impedance of the transmission line assuming perfect conductors, ç is the intrinsic impedance of the dielectric between the condu ctors, and l defi nes the direction into the condu ctors. The characteristic impedance of the lossless coaxia l transmi ssion line is given by The derivative term in the attenuation factor expression is The attenuation factor due to condu ctor loss in the coaxial transmission line beco mes which is identical to the result found using the perturbation m ethod.