ANTENNAS and MICROWAVES ENGINEERING (650427)

Transcription

1 Philadelphia University Faculty of Engineering Communication and Electronics Engineering ANTENNAS and MICROWAVES ENGINEERING (65427) Part 2 Dr. Omar R Daoud 1

2 General Considerations It is a two-port network connecting a generator circuit to a load. The impact of a transmission line on the current and voltage in the circuit depends on the length of line, l and the frequency, f of the signal provided by generator. At low frequency, the impact is negligible At high frequency, the impact is very significant 2/27/218 Dr. Omar R Daoud 2

3 General Considerations It may be classified into two types: Transverse electromagnetic (TEM) transmission lines: waves propagating along these lines having electric and magnetic field that are entirely transverse to the direction of propagation Higher order transmission lines: waves propagating along these lines have at least one significant field component in the direction of propagation 2/27/218 Dr. Omar R Daoud 3

4 General Considerations It is represented by a parallel-wire configuration regardless of the specific shape of the line, i.e coaxial line, twowire line or any TEM line. It can be modelled as a umped element: It consists of four basic elements called the transmission line parameters : R : combined resistance of both conductors per unit length, in Ω/m : the combined inductance of both conductors per unit length, in H/m G : the conductance of the insulation medium per unit length, in S/m C : the capacitance of the two conductors per unit length, in F/m The following parameters µ, σ, ε refer to the insulating material between conductors 2/27/218 Dr. Omar R Daoud 4

5 General Considerations In the TEM transmission lines: The followings should be fulfilled 'C' The propagation constant is G' C' R' j' G' jc' j The line characteristics impedance is R' j' G' jc' The wave propagation phase velocity is u p f The following parameters µ, σ, ε refer to the insulating material between conductors 2/27/218 Dr. Omar R Daoud 5

6 Example For an air line with characteristic impedance of 5Ω and phase constant of 2 rad/m at 7MHz, find the Capacitance per meter and the inductance per meter of the line. j' jc' Im ' C' and C ' pf/m nh/m ' C' ' j' jc' An air line (a transmission line for which air is the dielectric material present between the two conductors, which renders G =. In addition, the conductors are made of a material with high conductivity so that R ). ' C' 2/27/218 Dr. Omar R Daoud 6

7 ossless Transmission ines In the lossless transmission lines, R and G are set to zero (very small values) : The propagation constant is (lossless line) ' C' (lossless line) The line characteristics impedance is since R' j' G' jc' R' and G', ' (lossless line) C' The following parameters µ, σ, ε refer to the insulating material between conductors 2/27/218 Dr. Omar R Daoud 7

8 ossless Transmission ines In the lossless transmission lines, R and G are set to zero (very small values) : The wave propagation phase velocity is (rad/m) 1 u p (m/s) The wavelength is u c f p f 1 r r The following parameters µ, σ, ε refer to the insulating material between conductors 2/27/218 Dr. Omar R Daoud 8

9 Reflection Coefficient Voltage Reflection Coefficient (Γ) : It is the ratio of the amplitude of the reflected voltage wave, V - to the amplitude of the incident voltage wave, V + at the load. V 1 (dimensionless) V 1 for lossless line is a real number while in general is a complex number. Hence, A load is matched to the line if = because there will be no reflection by the load (Γ = and V = ). When the load is an open circuit, ( = ), Γ = 1 and V - = V +. When the load is a short circuit ( =), Γ = -1 and V - = V +. j r e 2/27/218 Dr. Omar R Daoud 9

10 Example A 1-Ω transmission line is connected to a load consisting of a 5-Ω resistor in series with a 1pF capacitor. Find the reflection coefficient at the load for a 1-MHz signal. R 5, R C j / C 1 11 F, 1, f 1MHz j 5 j / 1.5 j / 1.5 j Hz 2/27/218 Dr. Omar R Daoud 1

11 Standing Waves The Interference between the reflected wave and the incident wave along a transmission line creates a standing wave. n,1, 2,...if max Constructive interference gives maximum value, and occurs at r n lmax where n 4 2 n 1, 2, 3,...if l min l l max r r Destructive interference gives minimum value, and occurs at / 4 / 4 if if l l max max / 4 / 4 The repetition period for standing wave pattern is λ/2 (it is λ for incident and reflected wave individually). For a matched line, =, Γ = and V ~ z = V + for all values of z. For a short-circuited load, ( =), Γ = -1. For an open-circuited load, ( = ), Γ = 1. The wave is shifted by λ/4 from short-circuit case. 2/27/218 Dr. Omar R Daoud 11

12 Standing Waves Voltage Standing Wave Ratio (VSWR): It is the ratio of the maximum voltage amplitude to the minimum voltage amplitude and provides a measure of the mismatching between the load and the transmission line. ~ V 1 max VSWR ~ (dimensionless) V 1 min For a matched load, if Γ =, VSWR = 1 Γ - 1, VSWR = For a matched line, =, Γ = and V ~ z = V + for all values of z. For a short-circuited load, ( =), Γ = -1. For an open-circuited load, ( = ), Γ = 1. The wave is shifted by λ/4 from short-circuit case. 2/27/218 Dr. Omar R Daoud 12

13 Example A 5- transmission line is terminated in a load with = (1 + j5)ω. Find the voltage reflection coefficient and the voltage standing-wave ratio (VSWR).. 45e / 1 1 j5 5 / 1 1 j5 5 j VSWR /27/218 Dr. Omar R Daoud 13

14 ossless T: input impedance ( in ) It is the ratio of the total voltage (incident and reflected voltages) to the total current at any point z on the line. jl jl 2 jl V l V e e 1 e j tan l in jl jl 2 jl I l V e e 1 e j tan l For a line terminated in a short-circuit, = : ~ sc Vsc l in ~ j tan l I sc l For a line terminated in an open circuit, = : oc Voc l in ~ j cotl Ioc l Then: o sc in oc in tanl sc in oc in 2/27/218 Dr. Omar R Daoud 14

15 Example A source with 5 source impedance drives a 5 transmission line that is 1/8 of wavelength long, terminated in a load = 5 j25. Calculate: the voltage reflection coefficient the voltage standing-wave ratio (VSWR) The input impedance seen by the source. 2/27/218 Dr. Omar R Daoud 15

16 Example 2/27/218 Dr. Omar R Daoud j e j j VSWR tan tan tan j j j j j j in

17 ossless T: length of line At the line length of where n is an integer, then If the transmission line is a quarter wavelength, with then 2/27/218 Dr. Omar R Daoud 17 2 l n / tan 2 / / 2 tan tan n n l 2 / for in l n 2 / 4 / n l l 2 / 4 / for 2 in n l

18 Example A 5-Ω lossless transmission line is to be matched to a resistive load impedance with =1Ω via a quarter-wave section as shown, thereby eliminating reflections along the feedline. Find the characteristic impedance of the quarter-wave transformer. To eliminate reflections at terminal AA, the input impedance in looking into the quarter-wave line should be equal to 1 (the characteristic impedance of the feedline). Thus, in = 5Ω. 2 2 in Since the lines are lossless, all the incident power will end up getting transferred into the load. 2/27/218 Dr. Omar R Daoud 18

19 ossless T: Power Flow Average power for incident wave P i av V 2 2 (W) Average power for reflected wave 2 2 V r 2 i Pav Pav 2 The net average power delivered to the load: 2 V i r 2 Pav Pav Pav 1 (W) 2 2/27/218 Dr. Omar R Daoud 19

20 ossless T: Return oss (R) When the load is mismatched, not all the of the available power from the generator is delivered to the load. This loss is called return loss, and is defined (in db) as: R 2log At matching load case ( Γ =), R is db (no reflected power). At total reflection case ( Γ =1), R is db (all incident power is reflected). So return loss will have only values between < R < 2/27/218 Dr. Omar R Daoud 2

21 Summary 2/27/218 Dr. Omar R Daoud 21

22 Smith Chart It is the T calculator and used to analyze and design T circuits. Smith Chart parameters are: The reflection coefficient: Γ = Γ e jθ, ( Γ 1), (-18 θ 18 ) The normalized impedance: z = / The normalized admittance: y = 1 / z SWR and R scale 1 Fundamentals of Electromagnetics With Engineering Applications by Stuart M. Wentworth Copyright 25 by John Wiley & Sons. All rights reserved. z y z z z 1 r jx 2/27/218 Dr. Omar R Daoud 22

23 Smith Chart z =.6 + j1.4 y =.25 - j.6 The families of circle for r and x. For = 5Ω, a = (S.C.) b = (O.C.) c = 1 + j1 Ω d = 1 - j1 Ω e = 5 Ω Fundamentals of Electromagnetics With Engineering Applications by Stuart M. Wentworth Copyright 25 by John Wiley & Sons. All rights reserved. The complex Γ plane. 2/28/218 Dr. Omar R Daoud 23

24 2/27/218 Dr. Omar R Daoud 24

25 Smith Chart The Smith Chart is a plot of normalized impedance. For example, if a = 5 Ω transmission line is terminated in a load = 5 + j1 Ω. To locate this point on Smith Chart, Normalize the load impedance, N = / N to obtain N = 1 + j2 Ω The normalized load impedance is located at the intersection of the r =1 circle and the x =+2 circle. A T-line terminated in a load (a) shown with v A location T-line terminated of the normalized in a load load (a) impedance shown with is va location of the normalized load impedance is f Fundamentals of Electroma Fundamentals of Electromagn (c) 2/28/218 Dr. Omar R Daoud 25

26 Smith Chart: Input Impedance The input impedance, in : j2l 1 e in j2l 1 e Γ is the voltage reflection coefficient at the load. The phase angle of Γ is shifted by 2βl to get Γ ( from z in to z ; the same Γ, but the phase is changed by 2βl = 2(2π/λ)l= 2π). On the Smith chart, this means rotating in a clockwise direction (WTG). For one complete rotation corresponds to l = λ/2. The objective of shifting Γ to Γ is to find in at an any distance l on the transmission line. A 5-Ω transmission line is terminated with =(1-j5)Ω. Find in at a distance l =.1λ from the load. 2/28/218 Dr. Omar R Daoud 26

27 Smith Chart: Reflection Coefficient The reflection coefficient has a magnitude j and an angle : e The magnitude can be measured using a scale for magnitude of reflection coefficient provided below the Smith Chart, The angle is indicated on the angle of reflection coefficient scale shown outside the circle 1 on chart. Fundamentals of Electromagnetics With Engineering Applications by Stuart M. Wentworth Copyright 25 by John Wiley & Sons. All rights reserved. e.7 e j j45 Scale for magnitude of reflection coefficient 2/28/218 Dr. Omar R Daoud 27

28 Smith Chart: VSWR Point A is the normalized load impedance with z =2+j1. VSWR = 2.6 (at P max ). The distance between the load and the first voltage maximum is max l =( )λ. The distance between the load and the first voltage minimum is min l =( )λ. j Move along the constant e circle is akin to moving along the transmission line. 2/28/218 Dr. Omar R Daoud 28

29 Example Given that the voltage standing-wave ratio, VSWR = 3. On a 5-Ω line, the first voltage minimum occurs at 5 cm from the load, and that the distance between successive minima is 2 cm, find the load impedance. The distance between successive minima is equal to λ/2. Hence, λ = 4 cm. First voltage minimum (in wavelength unit) is at 5 l min on the WT scale from point B. Intersect the line with constant SWR circle = 3. The normalized load impedance at point C is: z.6 j.8 De-normalize (multiplying by ) to get : j.8 3 j /28/218 Dr. Omar R Daoud 29

30 Smith Chart The Smith Chart is a plot of normalized impedance. For example, if a = 5 Ω transmission line is terminated in a load = 5 + j1 Ω and has a line length of.3λ. Normalize the load impedance, N = / N to obtain N = 1 + j2 Ω The normalized load impedance is located at the intersection of the r =1 circle and the x =+2 circle. Moving away from the load (towards generator) corresponds to moving in the clockwise direction on the Smith Chart. Moving towards the load corresponds to moving in the anti-clockwise direction on the Smith Chart. To find IN, move towards the generator by: Drawing a line from the center of chart to outside Wavelengths Toward Generator (WTG) scale, to get starting point a at.188λ j Adding.3λ moves along the constant e circle to.488λ on the WTG scale. Read the corresponding normalized input impedance point c, NIN = j.8ω Denormalizing, to find an input impedance, IN IN VSWR is at point b; NIN 8.75 j4 VSWR 5.9 ( ( A T-line terminated in a load (a) shown with va (c) A location T-line terminated of the normalized in a load load (a) impedance shown with is vafo location of the normalized load impedance is fo A T-line terminated in a load (a) shown with values normalized to in (b). (c) The location of the normalized load impedance is found on the Smith Chart. Fundamentals of Electromagne Fundamentals of Electromagnetics With Engineering Applications by Stuart M. Wentworth C Copyright 25 by John Wiley & Sons. All rights reserved. Fundamentals of Electromagne C 2/28/218 Dr. Omar R Daoud 3

31 Smith Chart = 5 - j25 and =5 Ohm. Find in, VSWR and Γ using the Smith Chart. ocate the normalized load, and label it as point a, where it corresponds to N 1 j.5 Draw constant j circle. e It can be seen that.245e Move from point a (at.356λ) on the WTG scale, clockwise toward generator a distance λ/8 or.125λ to point b, which is at.481λ. j76 VSWR 1.66 We could find that at this point, it corresponds to NIN IN.62 j.7 31 j3. 5 2/28/218 Dr. Omar R Daoud 31

32 Smith Chart The input impedance for a 1 Ω lossless transmission line of length λ is measured as 12 + j42ω. Determine the load impedance. ocate th Normalize the input impedance: z in in ocate the normalized input impedance and label it as point a Take note the value of wavelength for point a at WT scale. At point a, WT =.436λ Move a distance 1.162λ towards the load to point b (WT =.436λ λ = 1.598λ ). But, to plot point b, 1.598λ 1.5λ =.98λ. Read the point b: N Denormalized it: 12 j j.7 N 15 j7 j.42 2/28/218 Dr. Omar R Daoud 32

33 Smith Chart On a 5 lossless transmission line, the VSWR is measured as 3.4. A voltage maximum is located.79λ away from the load (towards generator). Determine the load. Use the given VSWR to draw a constant circle. Then move from maximum voltage at WTG =.25λ (towards the load) to point a at WTG =.25λ -.79λ =.171λ. At this point we have N = 1 + j1.3 Ω, or = 5 + j65 Ω. j e 2/28/218 Dr. Omar R Daoud 33

34 Smith Chart: Impedance Matching Transmission line is matched to the load when = ; which no reflection occurs. This is usually not possible since is used to serve other application. Alternatively, we can place an impedancematching network between load and transmission line (The load impedance will be transformed to be equal to of the transmission line. The important of impedance matching or tuning: Maximum power is delivered when the load is matched to the line. The power loss in the feed line is minimized. (increase power handling capability by optimizing VSWR) Improved the signal-to-noise ratio (SNR). (e.g with controlled mismatch, an amplifier can operate with minimum noise generation) Reduced the amplitude and phase errors. Matching Network oad Techniques Adding an impedance-matching of network impedance ensures that all power matching will make it to the : load. Quarter-wave transformer Single / double stub tuner umped element tuner Fundamentals of Electromagnetics With Engineering Applications by Stuart M. Wentworth Copyright 25 by John Wiley & Sons. All rights reserved. Multi-section transformer 2/28/218 Dr. Omar R Daoud 34

35 Smith Chart: Impedance Matching Quarter Wave Transformer It can only be constructed if the load impedance is all real (no reactive component) Designed to achieve matching at a single frequency/narrow bandwidth Easy to build and use The input impedance, in (looking into the quarter wave long section of lossless S impedance line terminated in a resistive load R ): in S R S j jr S tan l tan l Quarter-wave transformer. 2 l, 4 2 tan l Fundamentals of Electromagnetics With Engineering Applications by Stuart M. Wentworth Copyright 25 by John Wiley & Sons. All rights reserved. S R 3/1/218 Dr. Omar R Daoud 35

36 Smith Chart: Impedance Matching Quarter Wave Transformer Example Calculate the position and characteristic impedance of a quarter wave transformer that will match a load impedance, R = 15Ω; to a 5 Ω line. The transformer s impedance S R (5)(15) Quarter-wave transformer. the position of quarter-wave transformer from the load: d =.25 λ Fundamentals of Electromagnetics With Engineering Applications by Stuart M. Wentworth Copyright 25 by John Wiley & Sons. All rights reserved. 3/1/218 Dr. Omar R Daoud 36

37 Smith Chart: Impedance Matching Quarter Wave Transformer Example A transistor has an input impedance of = 25 Ω, at an operating frequency of 5 MHz. Assume: Thickness of the dielectric is d = 1 mm, The relative dielectric constant ε r is 4. Find: The surface resistance, R and shunt conductance, G, are be neglected. The length, l The width, w The characteristic impedance of the quarter-wave parallel plate line transformer for which matching is achieved. 3/1/218 Dr. Omar R Daoud 37

38 Smith Chart: Impedance Matching Quarter Wave Transformer Example = 25 Ω, f = 5 MHz, d= 1mm, ε r = 4, and R=G= S The characteristic of the line is ine From the table, R (5)(25) C d p w d w w The line length l follows from the condition p 5.329mm The input impedance of the combined transmission line and the load is: where d = l = λ/4 and the reflection coefficient is d p line 235.8nH / m 1 l mm 4 4 f C in line ( d) e line j j line tan tan r w C pf / m d d 1 ( d) line d 1 ( d) exp j2 2 jd line 2 line p f d v p o = 8.85 x 1-12 F/m µ o = 4 x 1-7 H/m 3/1/218 Dr. Omar R Daoud 38

39 d Transmission ines Y Y Y Smith Chart: Impedance Matching Stub Matching Shunt stub matching network The matching network has to transform the real part of load impedance, R to and reactive part, X to zero Use two adjustable parameters e.g. shunt-stub. Thus, The main idea of shunt stub matching network is to: Find length d and l in order to get y d and y l. Ensure total admittance y tot = y d + y l = 1 for complete matching network. Open or shorted stub l l Open or shorted stub Y=1/ (a) =1/Y Single-stub tuning circuits. (a) Shunt stub. (b) Series stub. (b) d Y 3/1/218 Dr. Omar R Daoud 39

40 d Transmission ines Y Y Y Smith Chart: Impedance Matching Stub Matching Shunt stub matching network using Smith Chart ocate the normalized load impedance N. Draw constant SWR circle and locate Y N. Move clockwise (WTG) along e intersect with 1 ± jb value of y d. circle to The length moved from Y N towards y d is the through line length, d. ocate y l at the point jb. Depends on the shorted/open stub, move along the periphery of the chart towards y l (WTG). The distance traveled is the length of stub, l. j Open or shorted stub l l Open or shorted stub Y=1/ (a) (b) =1/Y Single-stub tuning circuits. (a) Shunt stub. (b) Series stub. d Y 3/1/218 Dr. Omar R Daoud 4

41 Smith Chart: Impedance Matching Shunt stub matching network using Smith Chart Example Construct the shorted shunt stub matching network for a 5Ω line terminated in a load = 2 j55ω ocate the normalized load impedance, N = / =.4 j1.1ω j Draw constant e circle. ocate Y N. (.112λ at WTG) Figure 6-28bc (p. 32) Moving to the (b) first Adding intersection shunt admittances. with the ± jb (c) circle, Using which the Smith Chart to find through line and Thus, the values are: is at 1 + j2. stub y d lengths. Values on the chart apply to Example 6.7. d =.75 λ Get the value of through line length, d Fundamentals from.112λ of Electromagnetics to With Engineering Applications by Stuart M. Wentworth Copyright 25 by John l Wiley =.74 & Sons. λ All rights reserved..187λ on the WTG scale, so d =.75λ y d = 1 + j2. Ω ocate the location of short on the Smith Chart (note: when y short circuit, =, hence Y = ) on the right side of the l = -j2. Ω chart with WTG=.25λ. Move clockwise (WTG) until point jb, which is at - j2., Where located at WTG=.324λ y l Y TOT = y d + y l = (1 + j2.) + (-j2.) = 1 Determine the stub length, l.324λ.25λ =.74 λ 3/1/218 Dr. Omar R Daoud 41

42 Smith Chart: Impedance Matching Shunt stub matching network using Smith Chart Example Construct the open ended shunt stub matching network for a 5Ω line terminated in a load = 15 + j1ω ocate the normalized load impedance, N = / = 3. + j2.ω j Draw constant e circle. ocate Y N. (.474λ at WTG) Moving to the first intersection with the ± jb circle, which is at 1 + j1.6 y d Get the value of through line length, d from.474λ to.178λ on the WTG scale, so d =.24λ ocate the location of open end on the Smith Chart (note: when open circuit, =, hence Y = ) on the left side of the chart with WTG=.λ. Move clockwise (WTG) until point jb, which is at j1.6, Thus, the values are: d =.24 λ l =.339 λ y d = 1 + j1.6 Ω y l = -j1.6 Ω Where Y TOT = y d + y l = (1 + j1.6) + (-j1.6) = 1 located at WTG=.339λ y l Determine the stub length, l.339λ.λ =.339 λ 3/1/218 Dr. Omar R Daoud 42

43 Smith Chart: Impedance Matching Shunt stub matching network using Smith Chart Example 5-Ω transmission line is connected to an antenna with load impedance = (25 j5)ω. Find the position and length of the short-circuited stub required to match the line. j The normalized load impedance is 25 5 z. 5 j (located at A). 5 Value of y at B is y.4 j.8 which locates at position.115λ on the WTG scale. Draw constant SWR circle that goes through points A and B. There are two possible matching points, C and D where the constant SWR circle intersects with circle r =1 (now g =1 circle). First matching points, C. At C, yd 1 j1.58 is at.178λ on WTG scale. Distance B and C is d= (.178λ-.115λ)=.63λ Normalized input admittance yin ys yd at the juncture is: 1 j y 1 j1.58 ys j1.58 E is the admittance of short-circuit stub, y =-j. Normalized admittance of j 1.58 at F and position.34λ on the WTG scale gives: l 1 = (.34λ-.25λ)=.9λ s 3/1/218 Dr. Omar R Daoud 43

44 Smith Chart: Impedance Matching Shunt stub matching network using Smith Chart Example 5-Ω transmission line is connected to an antenna with load impedance = (25 j5)ω. Find the position and length of the short-circuited stub required to match the line. j The normalized load impedance is 25 5 z. 5 j 5 (located at A). Value of y at B is y.4 j.8 which locates at position.115λ on the WTG scale. Draw constant SWR circle that goes through points A and B. There are two possible matching points, C and D where the constant SWR circle intersects with circle r =1 (now g =1 circle). Second matching points, D. At D, y d 1 j1.58 is at.322λ on WTG scale. Distance B and D is d= ( λ)=.27λ Normalized input admittance at the juncture is: ys j1.58 at G Rotating from point E to point G, we get l /1/218 Dr. Omar R Daoud 44