EEE161 Applied Electromagnetics Practical Laboratory 1

Transcription

1 Dr. Milica Marković Applied Electromagnetics Laboratory page 1 EEE161 Applied Electromagnetics Practical Laboratory 1 Instructor: Dr. Milica Marković Office: Riverside Hall milica@csus.edu Web: milica

2 Chapter 1 Transmission Line Theory 1.1 Transmission Line Theory What is a transmission line? Any wire, cable or line that guides energy from one point to another is a transmission line. Whenever you make a circuit on a breadboard, every wire you attach is a transmission line. Wheather we see the propagation effects on a line depends on the line length. So, at lower frequencies we do not see any difference between the signal s phase at the generator and at the load, whereas at higher frequencies we do Types of transmission lines 1. Coaxial Cable, Figure Microstrip, Figure Stripline, Figure Coplanar Waveguide, Figure Two-wire line, Figure Parallel Plate Waveguide, Figure Rectangular Waveguide, Figure Optical fiber, Figure Wave types Types of waves include acoustic waves, mecahnical pressure waves, electromagnetic (EM) waves. Here we will focus on EM waves and transmission lines for EM waves. 2

3 Dr. Milica Marković Applied Electromagnetics Laboratory page 3 (a) Coaxial Cable (b) Microstrip (c) Stripline (d) Coplanar Waveguide (e) Two-wire Line (f) Parallel Waveguide Plate (g) Rectangular Waveguide (h) Optical Fiber Figure 1.1: Types of transmission lines What are Transmission Line Effects? Figure 1.3 shows a step voltage at the generator and the load of a circuit in Figure 1.4. The voltage needs T sec to appear at the load, once the switch closes. Figure?? shows the step signal as it traveles on the transmission line at different time steps t=0, t=t/4, t=t/2 and t=t. How much time it takes for this signal to go from AA end to BB end? t = l c, where c = If the signal at end AA is Then at the other end the transmission line the signal is v AA (t) = Acos(ωt) (1.1)

4 Dr. Milica Marković Applied Electromagnetics Laboratory page 4 Figure 1.2: Voltage as a function of time at the generator side z=0 (top) and the load side z=l (bottom) of the transmission line in Figure 1.4, if the switch closes at t=0 the voltage arrives at t=l/c=t at the load. These graphs can be obtained by observing the voltage on an oscilloscope at the load and at the generator side. Figure 1.3: Voltage along the transmission line in Figure 1.4, for four different time intervals t=0, switch closes, t=t/4, t=t/2 and t=t. It is assumed that the length of the transmission line is equal to l=t/c.

5 Dr. Milica Marković Applied Electromagnetics Laboratory page 5 Figure 1.4: Electronic Circuit with an emphasis on cables that connect the generator and the load. Since we know that angular frequency is ω = 2πf v BB (t) = v AA (t l c ) (1.2) v BB (t) = Acos(ω(t l )) (1.3) c v BB (t) = Acos(ωt ω l c ) (1.4) v BB (t) = Acos(ωt ω l) (1.5) c The quantity c f v BB (t) = Acos(ωt 2πf l) (1.6) c is the wavelength λ. We will talk more about wavelength in Section v BB (t) = A cos(ωt 2π λ l) (1.7) The quantity 2π is the propagation constant β λ Finally the expression for the voltage at BB end is v BB (t) = A cos(ωt βl) (1.8) v BB (t) = A cos(ωt Ψ) (1.9) We see that at BB the signal will experience a phase shift. We will derive this equation later again from the Telegrapher s equations Now let s see how the length of the line l affects the voltage at the end BB or a wire. Look at Equation 1.10.

6 Dr. Milica Marković Applied Electromagnetics Laboratory page 6 v BB (t) = A cos(ωt 2π l λ ) (1.10) The signal will experience a phase shift of 2π l. If this phase shift is small, there will not be much λ difference between the phase of the signal at the generator and at the load. This means that we don t have to use transmission line theory to account for the effects of the line. If the phase shift is significant, then we do have to use the transmission line theory. Let s look at some numbers in the following example. 1. If l < 0.01 then the angle 2π l is of the order of rad or about λ λ 20. In this case, the phase is obviosly something that we don t have to worry about. When the length of the transmission line is much smaller than λ, l << λ the wave propagation on the line can be ingnored If l λ > 0.01, say l λ = 0.1, then the phase is 200, which is a significant phase shift. In this case it may be necessary to account for reflected signals, power loss and dispersion on the transmission line. Problem 1. Using Matlab plot the sinusoidal signal y=sint as a function of time. Assume that the frequency of the signal is f=1ghz. Then repea Problem 2. Using Matlab plot the sinusoidal signal Assume that the frequency of the signal is f=1ghz. Then repeat the exercise by plotting the s Problem 3. Using Matlab plot the sinusoidal signal y=sin(t-40) Assume that the frequency of the signal is f=1ghz. Then repeat the exercise by plotting the s Problem 4. Plot the previous three plots on the same graph. Which signal is lagging the sinus Problem 5. Plot two periods of two sinusoidal signals y1(t) and y2(t) with different phases Problem 6. Now assume that the sinusoidal signal is generated by a source positioned at x=0 Problem 7. If the voltage on the generator side is y=sin(t) then the voltage on the load sid

7 Dr. Milica Marković Applied Electromagnetics Laboratory page 7 Solutions Problem 1 Using Matlab plot the sinusoidal signal y=sint as a function of time. Assume that the frequency of the signal is f=1ghz. Then repea Code: f=1e6; t=0:1/100/f:4/f; y=sin(2*pi*f*t); plot(t,y, Color, red ); hold on Graph of y(t)=sin(t) In this part of the lab, we were asked to display a simple sine wave with a frequency of 1GHz Problem 2 Using Matlab plot the sinusoidal signal Assume that the frequency of the signal is f=1ghz. Then repeat the exercise by plotting the s Code: fc=1e6; t=0:1/100/fc:4/fc; y=sin((2*pi*fc*t)+((2*pi)/9)); plot(t,y, Color, black ); Graph of y(t)=sin(t+40) With the phase change of +40 degrees, we see our graph shift to the left on the x-axis. This

8 Dr. Milica Marković Applied Electromagnetics Laboratory page 8 Problem 3 Using Matlab plot the sinusoidal signal y=sin(t-40) Assume that the frequency of the signal is f=1ghz. Then repeat the exercise by plotting the s Code: fc=1e6; t=0:1/100/fc:4/fc; y=sin((2*pi*fc*t)-((2*pi)/9)); plot(t,y); Diagram of y(t) = sin(t-40) With the same frequency for each plot, we notice this sin function shifts to the right. This Problem 4 Plot the previous three plots on the same graph. Which signal is lagging the sinusoidal signa Code: f=1e6; t=0:1/100/f:4/f; y=sin(2*pi*f*t); plot(t,y, Color, red ); hold on fc=1e6; t=0:1/100/fc:4/fc; y=sin((2*pi*fc*t)+((2*pi)/9)); plot(t,y, Color, black ); fc=1e6; t=0:1/100/fc:4/fc; y=sin((2*pi*fc*t)-((2*pi)/9)); plot(t,y);

9 Dr. Milica Marković Applied Electromagnetics Laboratory page 9 This Code allows us to plot all 3 graphs simultaneously on the same page. (different colors a Plot of : y(t) =sin(t) // RED y(t) =sin(t-40) // BLUE y(t) =sin(t+40) // BLACK This graph intricately shows all three sine waves together and their lagging/leading elements Problem 5 Plot two periods of two sinusoidal signals y1(t) and y2(t) with different phases and at the Code:

10 Dr. Milica Marković Applied Electromagnetics Laboratory page 10 fc=1e6; t=0:1/100/fc:4/fc; y=sin(2*pi*fc*t); plot(t,y); hold on fc=1e6; t=0:1/100/fc:4/fc; y=sin((2*pi*fc*t)+.025); plot(t,y); Plots both sine functions simultaneously with the minimal amount of phase change that display Diagram of stacked waves: y(t)=sin(t) y(t)=sin(t+.25) Problem 6 Now assume that the sinusoidal signal is generated by a source positioned at x=0 and the sou Code: c=3e8 t=y./c y=1:.001:100 plot(t,y); Diagram of signal generator(t) and cable length Graph shows the signal generated at specific lengths of cable with respet to time. As the cab Problem 7 If the voltage on the generator side is y=sin(t) then the voltage on the load side will lag Code: c=3e8; fc=1e11; %change this number for different frequencies t=0:1/100/fc:1/fc;

11 Dr. Milica Marković Applied Electromagnetics Laboratory page 11 y=sin(2*pi*fc*(t-(1/c))); plot(t,y); hold on F=1000Hz Dispersion is an effect where different frequencies travel with different speeds on the transmission line. Example Find what is the length of the cable at which we need to take into account transmission line effects if the frequency of operation is 10 GHz What is wavelength? HERE EXPLANATION OF WAVE ON TRANSLINE 1 FIGURE Propagation modes on a transmission line Coax, two wire line, microstrip etc can be approximated as TEM up to the GHz (unshielded), up to 140 GHz shielded. 1. TEM E, M field is entirely transverse to the direction of propagation 2. TE, TM E or M field is in the direction of propagation Derivation of the voltage on a transmission line In this section we will derive what is the expression for the signal along a wire as a function of space z. In previous classes, you learned about voltage or current as a function of time. In this section, we will see that the voltages and currents propagating on a cable may also be a function of distance from the generator. We want to derive the equations for the case when the transmission line is longer then the fraction of a wavelength. To make sure that we don t encounter the transmission line effects to start with, we can look at the piece of a transmission line that is much smaller then the fraction of a wavelenth. In other words we cut the transmission line into small pieces to make sure there are no transmission line effects, as the pieces are shorter then the fraction of a wavelength. Plan: Look at an infinitensimal length of a transmission line z. Model infinitesimal transmission line lenght with an equivalent circuit. Write KCL, KVL for the piece in the time domain (we get differential equations) Apply phasors (equations become linear)

12 Dr. Milica Marković Applied Electromagnetics Laboratory page 12 Figure 1.5: Section of a coaxial cable.

13 Dr. Milica Marković Applied Electromagnetics Laboratory page 13 Solve the linear system of equations to get the expression for the voltage and current on the transmission line as a function of z. Let s follow the plan now. Figure 1.6: Sinusoidal voltage propagation on a coaxial cable. Figure 1.7: Circuit Model for a section of a coaxial cable. KVL KCL v(z, t) + R z i(z, t) + L z i(z, t) t + v(z + z, t) = 0 i(z, t) = i(z + z) + i CG (z + z, t) v(z + z, t) i(z, t) = i(z + z) + G z v(z + z, t) + C z t Rearrange the KCL and KVL Equations 1.11, 1.14 divide them with z Equations 1.12, 1.15 let z 0 and recognize the expression for the derivative Equations, 1.13, 1.16.

14 Dr. Milica Marković Applied Electromagnetics Laboratory page 14 KVL KCL i(z, t) (v(z + z, t) v(z, t)) = R zi(z, t) + L z t v(z + z, t) v(z, t) i(z, t) = Ri(z, t) + L z v(z, t) z = Ri(z, t) + L t i(z, t) t (1.11) (1.12) (1.13) v(z + z, t) (i(z + z, t) i(z, t)) = G zv(z + z, t) + C z t i(z + z, t) i(z, t) v(z + z, t) = Gv(z + z, t) + C z i(z, t) = Gv(z + z, t) + C z We just derived Telegrapher s equations in time-domain: t v(z + z, t) t (1.14) (1.15) (1.16) i(z, t) z v(z, t) z = Gv(z + z, t) + C i(z, t) = Ri(z, t) + L t v(z + z, t) t These are two differential equations with two unknowns. It is not impossible to solve them, however we would prefer to have linear differential equations. So what do we do now? Express time-domain variables as phasors! v(z, t) = Re{V (z)e jωt } i(z, t) = Re{I(z)e jωt } And we get the Telegrapher s equations in phasor form z, V (z) = (R + jωl)i(z) z (1.17) I(z) = (G + jωc)v (z) z (1.18) Two equations, two unknowns. To solve these equations, we first integrate both equations over 2 V (z) z 2 2 I(z) z 2 = (R + jωl) I(z) z V (z) = (G + jωc) z

15 Dr. Milica Marković Applied Electromagnetics Laboratory page 15 Now rearange the previous equations 1 I(z) (R + jωl) z 1 V (z) (G + jωc) z now substitute Eq.1.19 into Eq.1.17 and Eq.1.20 into Eq.1.18 and we get = 2 V (z) z 2 (1.19) = 2 I(z) z 2 (1.20) Or if we rearrange 2 V (z) z 2 2 I(z) z 2 = (G + jωc)(r + jωl)v (z) = (G + jωc)(r + jωl)i(z) 2 V (z) (G + jωc)(r + jωl)v (z) = 0 z 2 (1.21) 2 I(z) (G + jωc)(r + jωl)i(z) = 0 z 2 (1.22) The above Eq.1.21 and Eq.1.22 are the equations of the current and voltage wave on a transmission line. γ = (G + jωc)(r + jωl) is the complex propagation constant. This constant has a real and an imaginary part. γ = α + jβ where α is the attenuation constant and β is the phase constant. α = Re{ (G + jωc)(r + jωl)} β = Im{ (G + jωc)(r + jωl)} What is the general solution of the differential equation of the type 1.21 or 1.22? V (z) = V + 0 e γz + V 0 e γz I(z) = I + 0 e γz + I 0 e γz In this equation V + 0 and V 0 are the phasors 1 of forward and backward going voltage waves, and I + 0 and I 0 are the phasors of forward and backward going current waves 1 complex numbers having an amplitude and the phase

16 Dr. Milica Marković Applied Electromagnetics Laboratory page 16 The time domain expression for the current and voltage on the transmission line we get v(t) = Re{(V 0 + e ( α+jβ)z + V0 e (α+jβ)z )e jωt } (1.23) v(t) = V 0 + e αz cos(ωt + βz + V 0 + ) + V0 e αz cos(ωt βz + V0 ) (1.24) Equation 1.24 can be separated into two Equations 1.27-reffws. (1.25) v f (t) = V 0 + e αz cos(ωt βz + V0 ) (1.26) v r (t) = V0 e αz cos(ωt + βz + V 0 + ) (1.27) Visualisation of Lossless Forward and Reflected Voltage Waves We will show next that if the signs of the ωt and βz are the same, the wave moves in the forward +z direction. If the signs of ωt and βz are opposite, the wave moves in the z direction. In order to see this, we will visualise Equations using Matlab code below. Figure 1.8 shows forward and reflected waves on a transmission line. On x-axis is the spatial coordinate z from the generator to the load, where the transmission line is connected, and on y-axis is the magnitude of the voltage on the line. Figure shows red and a dashed blue lines that represent the signal at two time points on the transmission line.top figure shows a forward, and the bottom figure shows the reflected signal. On the top graph, we will plot the Equation 1.26, and assume that the phase V 0 + = V0 =0, and the magnitude of both voltages is V 0 + = V0 =1 V, the frequency of this signal is 1 GHz, and there is no loss on the line, α = 0 Np. In Matlab code below, the top figure in Figure 1.8 shows equation cos(ωt βz), and the bottow figure shows equation cos(ωt + βz). The red line on both graphs is the voltage signal at a time.1 ns. We would obtain Figure 1.8 if we had a camera that can take picture of voltages, and took a picture of the signal at the time t 1 =.1 ns on the entire transmission line. The blue dotted line on both graphs is the same signal.1 ns later, at time t 2 =.2 ns. We see that the signal has moved to the right in the time of 1 ns, or from the generator to the load. On the bottom graph we see that at a time.1 ns, the red line represents the reflected signal. Dashed blue line shows the signal at a time.2 ns. We see that the signal has moved to the left, or from the load to the generator. clear all clc f = 10^9; w = 2*pi*f c=3*10^8; beta=2*pi*f/c; lambda=c/f; t1=0.1*10^(-9) t2=0.2*10^(-9) x=0:lambda/20:2*lambda;

17 Dr. Milica Marković Applied Electromagnetics Laboratory page 17 Figure 1.8: Forward (top) and reflected (bottom) waves on a transmission line. y1=sin(w*t1 - beta.*x); y2=sin(w*t2 - beta.*x); y3=sin(w*t1 + beta.*x); y4=sin(w*t2 + beta.*x); subplot (2,1,1), plot(x,y1, r ),... hold on plot(x,y2, --b ),... hold off subplot (2,1,2), plot(x,y3, r ) hold on plot(x,y4, --b ) hold off Visualisation of Lossless Forward and Reflected Voltage Waves Repeat the visualization pocedure in the previous section for a lossy transmission line. Assume that α = 0.1 Np, and all other variables are the same as in the previous section. How do the voltages compare in the lossy and lossless cases?

18 Dr. Milica Marković Applied Electromagnetics Laboratory page Relating forward and backward current and voltage waves on the transmission line -Transmissio Line Impedance Z 0 V (z) = V 0 + e γz + V0 e γz (1.28) I(z) = I 0 + e γz + I0 e γz (1.29) In this equation V + 0 and V 0 are the phasors of forward and backward going voltage waves, and I + 0 and I 0 are the phasors of forward and backward going current waves We will relate the phasors of forward and backward going voltage and current waves. When substitute the voltage wave equation into Telegrapher s Eq The equation is repeated here Eq We now rearrange Eq.1.31 Now we compare Eq.1.32 with the Eq V (z) = (R + jωl)i(z) z (1.30) γv 0 + e γz γv0 e γz = (R + jωl)i(z) (1.31) γ I(z) = R + jωl (V 0 + e γz + V0 e γz ) I(z) = γv 0 + R + jωl e γz γv 0 R + jωl eγz (1.32) I 0 + = γv 0 + R + jωl I0 = γv 0 R + jωl We can define the characteristic impedance of a transmission line as the ratio of the voltage to current amplitude of the forward going wave. Z 0 = V 0 + I 0 + Z 0 = R + jωl γ R + jωl Z 0 = G + jωc

19 Dr. Milica Marković Applied Electromagnetics Laboratory page Voltage Reflection Coefficient Γ, Lossless Case The equations for the voltage and current on the transmission line we derived so far are V (z) = V 0 + e jβz + V0 e jβz (1.33) I(z) = V 0 + e jβz V 0 e jβz Z 0 Z 0 (1.34) Figure 1.9: Transmission Line connects generator and the load. At z = 0 the impedance of the load has to be Z L = V (0) I0 Substitute the boundary condition in Eq.1.46 V V0 Z L = Z 0 V 0 + V0 We can now solve the above equation for V 0 (1.35) Z L (V 0 + V0 ) = V V0 Z 0 ( Z L 1)V 0 + = ( Z L + 1)V0 Z 0 Z 0 V 0 V + 0 V 0 V + 0 = Z L Z0 1 Z L Z0 + 1 = Z L Z 0 Z L + Z 0 (1.36) The quantity V 0 is called voltage reflection coefficient Γ. It relates the reflected to incident V + 0 voltage phasor. Voltage reflection coefficient isi n general a complex number, it has a magnitude and a phase. Examples

20 Dr. Milica Marković Applied Electromagnetics Laboratory page Ω transmission line is terminated in a series connection of a 50 Ω resistor and 10 pf capacitor. The frequency of operation is 100 MHz. Find the voltage reflection coefficient. 2. For purely reacive load Z L = jx L find the reflection coefficient. The end of this lecture is spent in the lab making a Matlab program to make a movie of a wave moving left and right Lossless transmission line In many applications it is sufficient to assume that the transmission line is lossless, R 0 2 and G 0 3. This is a lossless transmission line. In this case the transmission line parameters are Propagation constant Transmission line impedance Phase velocity γ = (R + jωl)(g + jωc) γ = jωl jωc γ = jω LC = jβ R + jωl Z 0 = G + jωc jωl Z 0 = jωc L Z 0 = C v = v = ω ω β ω LC v = 1 LC 2 metal resistance is low 3 dielectric conductance is low

21 Dr. Milica Marković Applied Electromagnetics Laboratory page 21 Group velocity Wavelength Low-Loss Transmission Line λ = 2π β λ = 2π ω LC 2π λ = ɛ0 µ 0 ɛ r λ = c f ɛ r λ = λ 0 ɛr In some practical applications, losses are small, but not negligible. R << ωl 4 and G << ωc 5. In this case the transmission line parameters are Propagation constant We can re-write the propagation constant as shown below. In somel applications, losses are small, but not negligible. R << ωl and G << ωc, then in Equation 1.38, RG << ω 2 LC. 1 j ( R ωl + G ωc ) in Equation 1.39 is shown in Equa- Taylor s series for function 1 + x = tions γ = (R + jωl)(g + jωc) (1.37) γ = jω ( R L C 1 j ωl + G ) RG (1.38) ωc ω 2 LC γ jω ( R L C 1 j ωl + G ) (1.39) ωc γ jω L C 1 j x 1 + x = x2 8 + x3... for x < 1 (1.40) 16 ( R ωl + G ) = jω ( L C 1 j ( R ωc 2 ωl + G )) (1.41) ωc 4 metal resistance is lower than the inductive impedance 5 dielectric conductance is lower than the capacitive impedance

22 Dr. Milica Marković Applied Electromagnetics Laboratory page 22 The real and imaginary part of the propagation constant γ are: α = ω L C 2 ( R ωl + G ) ωc (1.42) β = ω L C (1.43) We see that the phase constant β is the same as in the lossless case, and the attenuation constant α is frequency independent. This means that all frequencies of a modulated signal are attenuated the same amount, and there is no dispersion on the line. When the phase constant is a linear function of frequency, β = const ω, then the phase velocity is a constant v p = ω = 1, and the group velocity is also a constant, and equal to the phase velocity. β const In this case, all frequencies of the modulated signal are propagated with the same speed, and there is no distortion of the signal. This is the case only when the losses in the transmission line are small. We usually represent phase and group velocity on ω β diagrams, shown in Figure (a). At a frequency ω 1, the ratio of ω gives the phase velocity (graphically, this is the slope of the β red line on the graph), whereas the slope of the ω β curve (blue line on the graph) gives the group velocity at this freqency. These diagrams are usefull, as they show how phase constant β varies with frequency, and it also shows how phase and group velocities vary with frequency. We can see that in this case, both group and phase velocity for this line are positive quantities, which is a representation of what is called a forward wave. In a forward wave, both signal and the energy propagate in the forward direction. Backward waves are waves where the signal propagates forward, however energy propagates backwards. In backward waves, phase and group velocities have opposite signs. When the phase constant is a linear function of frequency, then, phase velocity and group velocity are equal, and do not depend on frequency. Both velocities are equal to the slope of the line in Figure (b). (a) Phase constant β is a nonlinear function of frequency. (b) Phase constant β is a linear function of frequency. Figure 1.10: Omega-Beta diagrams.

23 Dr. Milica Marković Applied Electromagnetics Laboratory page 23 Transmission line impedance Transmission line impedance is the impedance that the forward voltage and current see as they propagate down the line from the generator to the load. R + jωl Z 0 = G + jωc jωl Z 0 = jωc L Z 0 = C Phase velocity Phase velocity is the velocity of the carrier frequency. v = v = ω ω β ω LC v = 1 LC Group velocity Group velocity is the velocity of the modulated signal impressed on the carrier frequency. v g = 1 β ω (1.44) v g = 1 LC (1.45) Wavelength Wavelenght is how far the signal travels on the line while the carrier signal goes through one whole period in time, see Section??. λ = 2π β λ = 2π ω LC 2π λ = ɛ0 µ 0 ɛ r λ = c f ɛ r λ = λ 0 ɛr

24 Dr. Milica Marković Applied Electromagnetics Laboratory page Voltage Reflection Coefficient, Lossless Case The equations for the voltage and current on the transmission line we derived so far are Standing Waves V (z) = V 0 + e jβz + V0 e jβz (1.46) I(z) = V 0 + e jβz V 0 e jβz Z 0 Z 0 (1.47) In the previous section we introduced the voltage reflection coefficient that relates the forward to reflected voltage phasor. Γ = V 0 V 0 + = Z L Z 0 Z L + Z 0 (1.48) If we substitute this expression to Eq we get for the voltage wave V (z) = V 0 + e jβz + V0 e jβz (1.49) V (z) = V 0 + e jβz + ΓV 0 + e jβz V (z) = V 0 + (e jβz Γe jβz ) (1.50) since Γ = Γ e jθr Eq.1.51 becomes V (z) = V + 0 (e jβz Γ e jβz+θr ) (1.51) and for the current wave I(z) = V 0 + e jβz V 0 e jβz (1.52) Z 0 Z 0 I(z) = V 0 + e jβz + Γ V 0 + e jβz Z 0 Z 0 I(z) = V 0 + (e jβz Γe jβz ) (1.53) Z 0 The voltage and the current waveform on a transmission line are therefore given by Eqns.1.51, Now we have two equations and one unknown V 0 +! We will solve these two equations in Lecture 7. Now let s look at the physical meaning of these equations. 6 repeated here as 1.49

25 Dr. Milica Marković Applied Electromagnetics Laboratory page 25 In Eq.1.51, Γ is the voltage reflection coefficient, V + 0 is the phasor of the forward going wave, z is the axis in the direction of wave propagation, β is the phase constant 7, Z 0 is the impedance of the transmission line 8. V (z) is a complex number, phasor. We will find the magnitude and phase of the voltage on the transmission line. The magnitude of a complex number can be found as z = zz 9. V (z) = V (z)v (z) V (z) = V 0 + (e jβz Γ e jβz+θr )V 0 + (e jβz Γ e (jβz+θr) ) V (z) = V 0 + (e jβz Γ e jβz+ Θr )(e jβz Γ e (jβz+θr) ) V (z) = V Γ e (2jβz+ Θr) + Γ e j2βz+θr + Γ 2 ) V (z) = V Γ 2 + Γ (e (2jβz+Θr) + e (j2βz+θr) ) V (z) = V Γ Γ cos(2βz + Θ r ) (1.54) The magnitude of the total voltage on the transmission line is given by Eq It seems like a complicated function. Let s start from a simple case when the voltage reflection coefficient on the tranmission line is Γ = 0 and draw the magnitude of the total voltage. HERE PICTURE OF THE FLAT LINE Let s look at another case, Γ = 0.5 and Θ r = 0. The equation for the magnitude V (z) = V cos2βz (1.55) 4 The function 1.55 is at it s maximum when cos(2βz) = 1 or z = k λ, and the function value is 2 V (z) = 1.5V 0 +. It is at it s minimum when cos(2βz) = 1 or z = 2k+1 λ and the function value 4 is V (z) = 0.5V 0 + It is important to mention here that the function that we see looks like a cosine with an average value of V 0 +, but it is not. The minimums of the function are sharper then the maximums, so when the reflection coefficient is at it s maximum of Γ = 1 the function looks like this: PICTURE OF STANDING WAVE PATTERN WITH SHORT 7 imaginary part of the complex propagation constant 8 defined as the ratio of forward going voltage and current 9 Prove this in Carthesian and Polar coordinate system

26 Dr. Milica Marković Applied Electromagnetics Laboratory page 26 General Case. In general the voltage maximums will occur when cos(2βz) = 1 V (z) max = V Gamma Γ V (z) max = V 0 + (1 + Gamma ) 2 V (z) max = V 0 + (1 + Γ ) (1.56) In general the voltage minimums will occur when cos(2βz) = 1, V (z) min = V Gamma 2 2 Γ V (z) min = V 0 + (1 Gamma ) 2 V (z) min = V 0 + (1 Γ ) (1.57) The ratio of voltage minimum on the line over the voltage maximum is called the Voltage Standing Wave Ratio (VSWR) or just Standing Wave Ratio (SWR). SW R = V (z) max V (z) min SW R = 1 + Γ 1 Γ (1.58) The voltage maximum position on the line is where cos(2βz) = 1 2βz + Θ r = 2nπ z = 2nπ Θ r 2β z = 2nπ Θ r 4π (1.59)

27 Chapter 2 Artificial Transmission Line Lab 2.1 Instrumentation Needed 1. One Artificial Transmission Line 2. One Low-Frequency Signal Generator 3. One Multimeter and/or Oscilloscope Connect to the experimental setup as shown in Figure 2.2(a). The artificial transmission line box is shown in Figure 2.2(b). 2.2 Transmission Line Impedance and Voltage on a Low- Loss matched line. Understanding the equivalent model Equivalent model of the transmission line consists off 50 sections of an RLC filter. The number of sections will affect the bandwidth off the artificial transmission line and how close it simulates the actual cable. The bandwidth of the model can be calculated from the following equation BW N 1 10 τ This means that a 50 section artificial transmissio line with a delay of Where τ = C total L total is the time delay of the transmission line. (2.1) Question 1 How many wavelengths long is the transmission line at f = 8MHz frequency if L = 3.55H/mile, C =.1µF/mile, and the length of the whole transmission line is miles? Each capacitor in the 27

28 Dr. Milica Marković Applied Electromagnetics Laboratory page 28 (a) Experimental Setup (b) Artificial Tranmission Line Box Figure 2.1: Artificial Transmission Line Laboratory Question 2 What is the characteristic impedance of the transmission line? To calculate the characteristic impedance of the line, use the equations for the lossy transmission line impedance given in Equation??. Comment on the value of the imaginary part of the transmission line impedance. Is this a low-loss line? Check out the section about low-loss lines, and see if you can 2.3 Matched Line Question 3 Terminate the line with the matched load. What is the value of the matched load Z L? Since the characteristic impedance of the transmission line is a complex number, you will need to use both resistors and capacitors for Z L Measure the voltage on the line. Using a multimeter measure the line voltage and plot it as a function of distance along the line. Collect data, then use Matlab to plot it.

29 Dr. Milica Marković Applied Electromagnetics Laboratory page Question 4 Are there any losses in the transmission line? Was your line well matched? Question 5 Calculate the attenuation coefficients α from your measurements, and from the theory. How do they compare? How will you find the current from the measured voltage along the line? 2.4 Unmatched Line We will now look at what happens when the line is not well matched. 1. Terminate the line with an impedance equal to 2Z 0, where Z 0 in the characteristic impedance of the lossless line. 2. Choose a frequency so that the transmission line is λ/2 long. 3. What frequency is this? 4. Measure the voltage along the line in the same manner you did before, and make a new graph with Matlab. 5. Was the output voltage smaller than the input voltage? 6. Now select the frequency so that the line is a quarter of a wavelength long and plot the line voltage on the same graph as in the previous part. As the last part, terminate the lossless line in the short and open circuit and plot the line voltage for both cases. 7. What would happen to the output voltage if λ/4 transmission line was a perfect one? How do the currents and voltages compare the case of half and quarter long transmission lines? What standard circuit element is the quarter wavelength long line remind you off? Explain. 2.5 Transmission Line Impedance Measurement Short/Open etc.

30 Chapter 3 The Slotted Line Laboratory There are 12 different measurements in this lab: 6 the coaxial slotted line and six for the waveguide slotted line Coaxial Slotted Line 1. Explain a block diagram of the slotted line. What are the variables you re trying to measure? SWR meter is a narrow band hi again audio amplifier with the arson Val in parenthesis rectified, metered out for display. SWR scale calibration assumes a perfect square load detector (diode detector having a fee out proportional to input power). Coaxial line may be connected to the 1000 Hz output of the linear SWR meter amplifier,, to monitored the date demodulated signal coming from the diode is the out. SWR meter amplifier is tuned to their maximum gain at about 1000 Hz with approximately 25 Hz bandwidth for noise reduction purposes, when the signal coming from the diode detector is quite small. 2. The UHF generator frequency has to be set at 500 MHz. With a short-circuit at the and off slotted line, produces 100% reflection. This reflection produces knowledge that are spaced lambda by 2. SWR in that case is infinite. To locate nulls the SWR meter should be set to cry again 50 or 60 DB s. Then mouse can be located with good accuracy by locating to equal voltage points to either side of the no. 3. measured the impedance of an unknown loads by use of slotted line to measure the SWR produced by an unknown loads,, first move the probe carriage along the slotted line to the observable Vmax location along the slot, adjusting the SWR meter again (preferably under 40 DB gain position) to produce a full-scale deflection on the SWR scale of the outputs meter. Then the probe carriage is moved to the the minimum location a quarter wave away at which location the SWR scale gives a direct reading of the design SWR value. Check this measurement a couple of times to confirm. 4. To find the proxy load position and that the minimum location along the slotted line, use the location reference of the short-circuit at the load plane. The difference between the minimum of the location of the short and the minimum when the unknown load is connected corresponds to the rotation needed on the Smith chart to find unknown loan-to-value. this business is in a fraction of the wavelength.. Wavelength can be found from the measurement of halfway decent is between denials whenever the slotted line length permits locating two adjacent dolls. Otherwise Landa may 30

31 Dr. Milica Marković Applied Electromagnetics Laboratory page 31 be calculated using CRF assuming ideally lossless airline in assuming that the operating frequencies known.. Draw the SWR circle on the Smit chart.. This is the circle whose radius is equal to the appropriate SWR ratio on the chart. The point of the minimum for short-circuit on the chart is at the intersection of SWR circle in the negative GR axis. 6. To measure the antenna input impedance versus frequency,, first obtain proxy load positions beta over the desired frequency range by putting their reference short at the desired antenna load plane (which is at the end of the cable used to connect the slotted line and the antenna).. Take data on the proxy load plane locations versus the frequency.. Then with a monopole antenna replacing the short, take SWR in the minimum location data in each of the same Dial set frequencies.. obtained the input impedance for each of those frequencies and label them on the Smith charts. Plot also real and imaginary parts of the antennas impedance versus frequency on using Matlab. 3.2 Waveguide Slotted Line Newer Labs the last time we looked at how micropower is measured. Now we will look at another important measurement column data voltages, impedances and reflection coefficients. One way of doing this is by using the slotted line conflagration, which enables direct sampling of the electric field amplitude of the standing wave. It is made of a cortex are waveguide section that has a longitudinal slot in which a probe with a diode detectors inserted. There is a generator and one end of the line, and the unknown load terminates the line at the other end. The probe is a need like small pose that acts as a receiving antenna and samples the electric field. (You ll understand better how this works when we study antennas later.) If Saudi line measurements, we want to find the unknown load. We measured the SWR on the line and the distance from the low to the first voltage minimum aliment. We need to measure to quantities, says the load impedance is a complex number with both in amplitude and phase. From the SWR, the magnitude of the reflection coefficient is SWR formula we know that the volume minimum occurs for formula from here, we can find the unknown complex load impedance impedance formula the slotted line is now a story construing, except that high millimeter wave frequencies. The modern instrument used for measurement impedances is a vector network analyzer, which we will study next time experiment for the slotted line click the slotted line setup is shown in figure 41. The HP 864B RF signal generator goes up to 500 MHz of frequency. It has the possibility of being amplitude modulated with a 1 khz signal. The output power you will uses 20 DVM. The output from the generator goes to a 500 MHz low pass filter. The purpose of this filter is to get rid of all the higher harmonics, in order to maintain is pure for sine wave as possible. The output of the future is the input coaxial slotted line, which is about 60 cm long. The other end of the slotted line is terminated with the unknown load we wish

32 Dr. Milica Marković Applied Electromagnetics Laboratory page 32 to measure. All connect tutors in this system are GR (general radio) connectors. The slotted line probe has the diode detector built-in one of the two lands. Connect the send to the HP 415 ESW art meter. The diode detects the amp to the of the 1 khz low-frequency signal that modulates the art signal picked up by the probe. The SWR meter is just a narrow band high gain amplifier for the 1 khz that the 500 MHz is modulated with. It it s high gain assures good measurement sensitivity. The other GR connectors on the probe is used for optimizing the match of the probe to the line. Connect the adjustable stop tuna to this end. This unit is just a section of quarks with a short bed the other and so that this impedance can change by physically moving this short. Number one in this first part of the experiment you will use a short circuit termination. Set the frequency at 500 MHz.find and also the standing wave pattern. Question one: why are the nulls of shorted circuit standing wave pattern sharp? the sharp nulls will be buried in noise. The best way to find their position is to move towards the zero from both sides, locate points a and B from figure 42, and then take them zero to be between them. measure and record the position of several successive zeros along an arbitrary position scale. question two: what is the wavelength in the line equal to? Number two: what is the wavelength in the line equal to? Increase the generator power and repeat the measurement. Question four: why is this measurement not as good as the previous one? What gets worse: the minimum or the maximum of the voltage? Number three in this part of the experiment, you will measure two loads at both 500 and 250 MHz. The first load is the unknown load from part two, and the second one is along to sister of the same value, mounted in the metal jig from experiment to. First determine the position of the proxy plane by connecting a short termination. Then connect the load, and measure the distance between the first the next of the proxy plane in the proxy plane. You can either move towards or away from the load, but make sure you are consistent on the Smith chart. Repeat this measurement for the two loads at F1 equals 250 MHz and after equals 500 MHz. Feel a table as the one shown in table 41. The distance fell in the fifth column is the electrical vistas between columns 3 and four. Use this this this and this may chart to find the normalized load impedance Z from the measures SWR. Then find the load impedance Z by multiplying by the characteristic impedance of the Coax. include your Smith chart. Four at 500 MHz measured the impedance of a car X 50 ohm termination, repeating the procedure from part three. Include the Smith chart. Question 5Y this the harder measurement than for badly matched loads? In the past part of this experiment you will measure the impedance of the monopole antenna above a metal plane at 500 MHz. The antennae 15 cm long, which is about a quarter wavelength of this frequency. This is called the quarter wave monopole. They impedance of the monopole with or without the ground plane is very different. Question six: this antenna can be viewed as a current flowing through a Short St., Finn Y which is located about an incident perfectly conducting plane. What basic fear massage among medics would you use to find electric and magnetic field of such a system? Illustrate your answer with a simple drawing you do not have time to find the fields. Question seven: the impedance of an antenna is defined as the voltage across its terminal by

33 Dr. Milica Marković Applied Electromagnetics Laboratory page 33 current. Conclusion Solve all of the above examples by hand and compare the results with the ones you found during the lab. In addition, for each of the problems, write what you have learned. Do not write an essay, just in a few sentences write what you have done in each problem, and specifically what you have learned from the work done. Due Date Submit the printout of the lab work and the conclusion by NEXT FRIDAY. LAB IS DUE BY noon (stamped) next Friday in the main office. No late labs. Do not bring previous lab writeup for the next Lab session.