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EKT241 – ELECTROMAGNETICS THEORY Chapter 5 Transmission Lines Chapter Objectives Introduction to transmission lines Lump-element model that represent TEM lines Lossless line Smith Chart to analyze transmission line problem Chapter Outline 5-1) 5-2) 5-3) 5-4) 5-5) 5-6) 5-7) 5-8) 5-9) 5-10) 5-11) General Considerations Lumped-Element Model Transmission-Line Equations Wave Propagation on a Transmission Line The Lossless Transmission Line Input Impedance of the Lossless Line Special Cases of the Lossless Line Power Flow on a Lossless Transmission Line The Smith Chart Impedance Matching Transients on Transmission Lines 5-1 General Considerations • Transmission lines connect a generator circuit to a load circuit at the receiving end. • Transverse electromagnetic (TEM) lines have waves that propagate transversely. 5-2 Lumped-Element Model • Transmission lines can be represented by a lumped-element circuit model. 5-2 Lumped-Element Model • 1. 2. 3. 4. Lumped-element circuit model consists 4 transmission line parameters: R’ (Ω/m) L’ (H/m) G’ (S/m) C’ (F/m) 5-2 Lumped-Element Model • In summary, • All TEM transmission lines share the relations: L'C ' G' C' where µ, σ, ε = properties of conductor 5-3 Transmission-Line Equations • Transmission line equations in phasor form is given as ~ dV z ~ R' jL'I z dz ~ dI z ~ G ' jC 'V z dz 5-4 Wave Propagation on a Transmission Line • The wave equation is derived as ~ d V z 2 ~ V z 0 where 2 dz 2 R' jL'G' jC ' Complex propagation constant • γ has real part α (attenuation constant) and imaginary part β (phase constant). R' jL'G' jC ' (Np/m) Jm Jm R' jL'G' jC ' (rad/m) R e R e 5-4 Wave Propagation on a Transmission Line • Characteristic impedance Z0 of the line is Z0 • R' jL' R' jL' G ' jC ' Phase velocity for propagating wave is u p f where f = frequency (Hz) λ = wavelength (m) Example 5.1 Air Line An air line is a transmission line for which air is the dielectric material present between the two conductors, which renders G’ = 0. In addition, the conductors are made of a material with high conductivity so that R’ ≈0. For an air line with characteristic impedance of 50 and phase constant of 20 rad/m at 700 MHz, find the inductance per meter and the capacitance per meter of the line. Solution 5.1 Air Line The following quantities are given: Z 0 50, 20 rad/m, f 700 MHz 7 108 Hz With R’ = G’ = 0, Jm jL' jC ' L' C ' and Z 0 jL' L' jC ' C' The ratio is given by 20 C' 90.9 pF/m 8 Z 0 2 7 10 50 We get L’ from Z0 Z0 L' C' L' 50 90.9 1012 227 nH/m 2 5-5 The Lossless Transmission Line • Low R’ and G’ for transmission line is called lossless transmission line. 0 (lossless line) L' C ' (lossless line) Z0 • L' C' (lossless line) Using relation properties, (rad/m) p 1 (m/s) 5-5 The Lossless Transmission Line • Wavelength is given by p 0 c 1 f f r r • where εr = relative permittivity For the lossless line, there are 2 unknowns in the equations for the total voltage and current on the line. 5-5.1 Voltage Reflection Coefficient • The relations for lossless are V0 Z L Z 0 Z L Z 0 1 V0 Z L Z0 Z L Z0 1 I 0 V0 I0 V0 e jr • A load that is matched to the line when ZL = Z0, Γ = 0 and V0−= 0. Example 5.2 Reflection Coefficient of a Series RC Load A 100-Ω transmission line is connected to a load consisting of a 50-Ω resistor in series with a 10-pF capacitor. Find the reflection coefficient at the load for a 100-MHz signal. Solution 5.2 Reflection Coefficient of a Series RC Load The following quantities are given RL 50, CL 10 11 F, Z 0 100, f 100 MHz 108 Hz The load impedance is Z L RL j / CL 50 j 1 50 j159 8 11 2 10 10 Voltage reflection coefficient is Z L / Z 0 1 0.5 j1.59 1 0.67e j119.3 0.76 60.7 Z L / Z 0 1 0.5 j1.59 1 5-5.2 Standing Waves • 3 types of voltage standing-wave patterns: (a) Matched load (b) Short-circuited line (c) Open-circuited line 5-5.2 Standing Waves • To find maximum and minimum values of voltage magnitude, we have r 2n r n 2 4 2 n 1,2... if r 0 n 0,1,2... if r 0 z lmax 5-5.2 Standing Waves • First voltage maximum occurs at lmax • First voltage minimum occurs at lmin • r where n 0 4 lmax / 4 if lmax / 4 lmax / 4 if lmax / 4 Voltage standing-wave ratio S is defined as ~ V 1 max S ~ (dimension less) 1 V min Example 5.4 Standing-wave Ratio A 50- transmission line is terminated in a load with ZL = (100 + j50)Ω . Find the voltage reflection coefficient and the voltage standing-wave ratio (SWR). Solution We have, Z L / Z 0 1 100 j50 50 0.45e j 26.6 Z L / Z 0 1 100 j50 50 S is given by 1 1 0.45 S 2 .6 1 1 0.45 5-6 Input Impedance of the Lossless Line • • Voltage to current ratio is called input impedance Zin. The input impedance at z = −l is given as Z L cos l jZ0 sin l Z L jZ0 tan l Z 0 Z in l Z 0 Z 0 cos l jZL sin l Z 0 jZL tan l and ~ V Z in 1 g V0 Z g Z in e jl e jl Example 5.6 Complete Solution for v(z, t) and i(z, t) A 1.05-GHz generator circuit with series impedance Zg = 10Ω and voltage source given by vg t 10 sin t 30 V is connected to a load ZL = (100 + j50) through a 50-Ω, 67-cm-long lossless transmission line. The phase velocity of the line is 0.7c, where c is the velocity of light in a vacuum. Find v(z, t) and i(z, t) on the line. Solution 5.6 Complete Solution for v(z, t) and i(z, t) We find the wavelength from 0.7 3 108 0.2m 9 f 1.05 10 up and 2l 2 0.67 tan l tan tan tan 126 0.2 The voltage reflection coefficient at the load is Z L Z 0 100 j50 50 0.45e j 26.6 Z L Z 0 100 j50 50 The input impedance of the line Z jZ0 tan l Z in Z 0 L 21.9 j17.4 Z 0 jZ0 tan l Solution 5.6 Complete Solution for v(z, t) and i(z, t) Rewriting the expression for the generator voltage, vg t 10 sin t 30 10 cost 60 e 10e 60e jwt Thus the phasor voltage is V ~ Vg 10e j 60 10 60 V The voltage on the line is ~ V 1 g Z in j159 V0 10 . 2 e 10.2159 V jl Z g Z in e e jl and phasor voltage on the line is ~ V z V0 e jz e jz 10.2e j159 e jz 0.45e j 26.6e jz Solution 5.6 Complete Solution for v(z, t) and i(z, t) The instantaneous voltage and current is ~ vz, t e V z e jt 10.2 cost z 159 4.55 cost z 185.6 V i z , t 0.20 cost z 159 0.091cost z 5.6 A 5-7 Special Cases of the Lossless Line • Special cases has useful properties. 5-7 .1 Short-Circuited Line • For short-circuited line at z = −l, Z insc ~ Vsc l ~ jZ0 tan l I sc l Example 5.7 Equivalent Reactive Elements Choose the length of a shorted 50- lossless transmission line (Fig. 5-16) such that its input impedance at 2.25 GHz is equivalent to the reactance of a capacitor with capacitance Ceq = 4 pF. The wave velocity on the line is 0.75c. Solution 5.7 Equivalent Reactive Elements We are given u p 0.75c 2.25 10 8 m/s Z 0 50 f 2.25 10 9 Hz Ceq 4 10 12 F The phase constant is 2nd quadrant is 2 l1 2.8 rad or l1 4th quadrant is l2 5.94 rad 2f 1 62.8 rad/m, tan l 0.354 up Z 0Ceq 2.8 or l2 2.8 4.46 cm 62.8 5.94 9.46 cm 62.8 Any length l = 4.46 cm + nλ/2, where n is a positive integer, is also a solution. 5-7.2 Open-Circuited Line • With ZL = ∞, it forms an open-circuited line. V l Z inoc ~oc jZ0 cot l I oc l 5-7.3 Application of Short-Circuit and Open-Circuit Measurements • Product and ratio of SC and OC equations give the following results: Z o Z insc Z inoc tan l Z insc Z inoc • Radio-frequency (RF) instruments measure the impedance of any load. Example 5.8 Measuring Z0 and β Find Z0 and β of a 57-cm-long lossless transmission line whose input impedance was measured as Zscin = j40.42Ω when terminated in a short circuit and as Zocin = −j121.24Ω when terminated in an open circuit. From other measurements, we know that the line is between 3 and 3.25 wavelengths long. Solution 5.8 Measuring Z0 and β We have, Z 0 Z insc Z inoc j 40.42 j121 .24 70 Z insc 1 tan l oc 3 Z in True value of βl is l 6 6 19.4 rad and 19.4 34 rad/m 0.57 Example 5.9 Quarter-Wave Transformer A 50-Ω lossless transmission line is to be matched to a resistive load impedance with ZL = 100Ω via a quarterwave section as shown, thereby eliminating reflections along the feedline. Find the characteristic impedance of the quarter-wave transformer. Solution 5.9 Quarter-Wave Transformer To eliminate reflections at terminal AA’, the input impedance Zin looking into the quarter-wave line should be equal to Z01, the characteristic impedance of the feedline. Thus, Zin = 50 . 2 Z 02 Z in ZL Z 02 50 100 70.7 Since the lines are lossless, all the incident power will end up getting transferred into the load ZL. 5-8 Power Flow on a Lossless Transmission Line • We shall examine the flow of power carried by incident and reflected waves. 5-8.1 Instantaneous Power • Instantaneous power is the product of instantaneous voltage and current. 5-8.2 Time-Average Power • More interested in time-averaged power flow. 5-8.2 Time-Average Power • There are 2 types of approach: 1) Time-Domain Approach • Incident power and reflected wave power are Pavi • V0 2 2Z 0 (W) Pavr 2 V0 2 2Z 0 2 Pavi For net average power delivered to the load, Pav Pavi Pavr V0 2 2Z 0 1 2 (W) 5-8.2 Time-Average Power 2) Phasor-Domain Approach • Time-average power for any propagating wave is 1 ~ ~* Pav R e V I 2 5-9 Smith Chart • The Smith Chart is used for analyzing and designing transmission-line circuits. 5-9 Smith Chart • • Impedances represented by normalized values, Z0. Reflection coefficient is 1 zL 1 • Normalized load admittance is 1 1 yL (dimension less) zL 1 Example 5.11 Determining ZL using the Smith Chart Given that the voltage standing-wave ratio is S = 3 on a 50-Ω line, that the first voltage minimum occurs at 5 cm from the load, and that the distance between successive minima is 20 cm, find the load impedance. Solution The first voltage minimum is at 5 l min 0.125 40 Solution 5.11 Determining ZL using the Smith Chart From Smith Chart, rL S 3 The normalized load impedance at point C is z L 0.6 j 0.8 Multiplying by Z0 = 50Ω , we obtain Z L 500.6 j 0.8 30 j 40 5-10 Impedance Matching • • Transmission line is matched to the load when Z0 = ZL. Alternatively, place an impedance-matching network between load and transmission line. Example 5.12 Single-Stub Matching 50-Ω transmission line is connected to an antenna with load impedance ZL = (25 − j50). Find the position and length of the short-circuited stub required to match the line. Solution The normalized load impedance is zL Z L 25 j50 0.5 j Z0 50 Located at point A. Solution 5.12 Single-Stub Matching Value of yL at B is yL 0.4 0.115λ on the WTG scale. At C, yd 1 j1.58 j 0.8 which locates at position located at 0.178λ on the WTG scale. Distant B and C is d 0.178 0.155 0.063 Normalized input admittance at the juncture is yin ys yd 1 j 0 ys 1 j1.58 ys j1.58 Solution 5.12 Single-Stub Matching Normalized admittance of −j 1.58 at F and position 0.34λ on the WTG scale gives l1 0.34 0.25 0.09 At point D, yd 1 j1.58 Distant B and C is d2 0.322 0.115 0.207 Normalized input admittance ys j1.58 at G. Rotating from point E to point G, we get l2 0.25 0.16 0.41 Solution 5.12 Single-Stub Matching 5-11 Transients on Transmission Lines • • Transient response is a time record of voltage pulse. An example of step function is shown below. 5-11.1 Transient Response • Steady-state voltage V∞ for d-c analysis of the circuit is V Vg Z L Rg Z L where Vg = DC voltage source • Steady-state current is Vg V I Z L Rg Z L