School of Electical and Compute Engineeing, Conell Univesity ECE 303: Electomagnetic Fields and Waves Fall 007 Homewok 8 Due on Oct. 19, 007 by 5:00 PM Reading Assignments: i) Review the lectue notes. ii) Review sections 4.1-4.3, 5.1-5., 5.4, 6.1, 6.3-6.4, papeback book Electomagnetic Waves. These sections also include the mateial to be coveed in the next two weeks of the class. Special Note: Gades have been instucted to take off points (as much as 50%) if pope units ae not included in you numeical answes. You must specify the coect units with you numeical answes. Poblem 8.1: (Impedance tansfomations in micowave cicuits) Conside the following tansmission line cicuit: V s (t ) Z s 50 Ω V + V - 70 Ω 9.55 nh V s (t ) = 5 cos(ω t ) z = - l The cicuit is opeating at a fequency of 1.0 GHz. At the load end of the cicuit thee is a 70 Ω esisto in seies with a 9.55 nh = 9.55x10-9 H inducto. The length of the tansmission line is such that: 1 l = λ 16 whee λ is the wavelength of waves in the tansmission line at a fequency of 1.0 GHz. a) Find the load eflection coefficient Γ L (give a numeical value). b) Find the impedance Z(z = -l) looking into the tansmission line at z =-l so that the following equivalent cicuit can be used fo analysis: V s (t ) Z s 50 Ω Z(z =-l) Give a numeical value fo you answe. c) Find the time-aveage powe dissipated in the impedance Z(z = -l) in the above cicuit (give a numeical value). 1
d) Find the voltage V + of the fowad going voltage wave in the tansmission line cicuit (give a numeical value). e) Find the voltage V - of the backwad going voltage wave in the tansmission line cicuit (give a numeical value). f) Find the net time-aveage powe taveling in the +z-diection in the tansmission line cicuit (give a numeical value). Compae you answe with you answe in pat (c) and explain what you leaned by this compaison. g) Now find the Thevenin equivalent of the souce+tansmission line so that the following equivalent cicuit can be used fo analysis: V th Z th 70 Ω 9.55 nh You need to find the impedance Z th and the voltage phaso V th (give numeical answes fo each of these). h) Using you cicuit of pat (g) find the time-aveage powe dissipated in the 70 Ω load esisto (give a numeical value). Compae you answe to you answes in pats (c) and (f) and explain what you leaned by this compaison. Poblem 8.: (Enegy flow and powe in tansmission lines) Conside a co-axial tansmission line whose coss-section is shown below: y b a ε o µ o x Suppose a voltage-cuent wave given by: V ( z) = V+ e V+ I( z) = I+ e = e Zo is taveling in the co-axial line. In the lectue notes you wee told that the time-aveage powe is elated to the Poynting vecto though the elation: 1 1 * P z () t = S(, t ). zˆ dx dy = Re [ S( )]. zˆ dx dy = Re [ V+ I+ ]
In this poblem you ae going to pove the last equality of the above elation fo a co-axial line. a) Find the expession fo the impedance Z o of the co-axial line in tems of the dimensions specified in the figue above. You can wite down the answe using lectue notes no need to compute fom fist pinciples. b) If the voltage wave is given by ( ) V z = V+ e, find an expession fo the position dependent electic field vecto phaso associated with the wave in the annula egion between the two conductos. Hint: You can assume that a positive value of the voltage implies that the cente conducto is at a highe potential. E( ) =? c) If the cuent wave is given by ( ) I z = I+ e find an expession fo the magnetic field vecto phaso associated with the wave in the annula egion between the two conductos. Hint: You can assume that a positive value of the cuent implies that the cuent in the cente conducto is in the +z-diection. H( ) =? S d) Using you esults fom pats (b) and (c), find the complex Poynting vecto ( ) between the two conductos. Which way is the powe flowing? in the annula egion e) Integate the complex Poynting vecto S( ) obtained in pat (d) above ove the coss-sectional aea of the annula egion between the two conductos and show that: 1 Re 1 * [ S( )]. zˆ dx dy = Re [ V I ] Poblem 8.3: (Lossy Tansmission Lines) + + Pefect metals and dielectics ae geneally not available in this wold to make tansmission lines. Consequently, eal tansmission lines ae lossy (i.e. a wave popagating in a tansmission line looses powe as it popagates). One contibuting facto towads this loss is the I R dissipation in the impefect metals of the tansmission line. Any eal metal will have some finite esistance and cuent flow will esult in powe dissipation. In this poblem you will conside a moe ealistic model of a tansmission line. Conside the paallel plate tansmission line shown below. y z ε o µ o d The width of the metal plates is W (in the x-diection). The capacitance and inductance pe unit length of the tansmission line ae C and L, espectively. Suppose the total combined esistance pe unit length of the top and bottom plates is R. a) Show that when esistance is pesent the telegaphe s equations become, 3
V z ( z, t ) I( z, t ) = L t ( z, t ) V ( z t ) I = C, z t R I ( z, t ) Hint: Use the same methods as discussed in the lectue notes to deive the above equations. b) Convet the above equations into phaso notation, and then deive the complex wave equation fo the voltage phaso V(z). c) Assume a popagating solution of the fom V ( z) V e = +, plug it into the wave equation you deived in pat (b), and find the k-vs-ω dispesion elation fo the lossy tansmission line. Check that you esult gives the coect k-vs-ω dispesion elation in the case when R=0. If you did eveything coect to this point you will discove that the k-vecto has eal and imaginay pats and the imaginay pat descibes wave decay due to powe loss in the signal popagating in the tansmission line as a esult of the I R dissipation in the esistance associated with the metal plates. d) Find the chaacteistic impedance Z o of the tansmission line. Hint: it will be complex. Poblem 8.4: (Stub tuning in micowave cicuits) Conside the following tansmission line cicuit shown in the figue below. All tansmission lines have an impedance of 50 Ω. On the left is a tansmission line caying an input signal specified by the amplitude V + of the fowad going voltage wave on that tansmission line. On the ight is a load impedance of 5 Ω. The goal is to tansfe all the input powe to the 5 Ω load impedance. Since the load is not matched to the tansmission line impedance of 50 Ω, if the load wee to be diectly connected to the input tansmission line then some input powe will get eflected back. So we use the cicuit shown below. V + 5 Ω z = - l 1 z = - l An open-cicuit stub tune is used to match the total impedance of the stuctue on the ight (of the dashed line) to the 50 Ω impedance of the tansmission line on the left caying the input signal. You have at you disposal two design paametes - you can choose the lengths l 1 and l of both the tansmission lines. 4
Assuming that the wavelength of the waves at the fequency of opeation is λ in all the tansmission lines, you need to specify the lengths l 1 and l in tems of the wavelength λ such that the impedance of the stuctue to the ight of the dashed line is exactly 50 Ω. You need to design using Smith Chats. Matlab wok: Go to the couse website and download the matlab file fo the function smith303.m. The function smith303 is called as follows: >> smith303(z L, Z o ) whee Z L and Z o ae the load and tansmission line impedances, espectively. In esponse, smith303 does the following: i) daws a smith chat ii) points out the stating point (i.e. Γ(z=0) and Z n (z=0)) on the smith chat iii) daws the cicle that shows the values of Γ(z) and Z n (z) on the smith chat as ones moves back fom the load on the tansmission line. Solution stategy: i) Fist choose the smallest length l 1 such that the nomalized admittance Y n (z=-l 1 ) has a eal pat of unity ii) Then choose the smallest length l such that the nomalized admittance Y n (z=-l ) has an imaginay pat that exactly cancels the imaginay pat of Y n (z=-l 1 ). a) Find the smallest length l 1 in tems of the wavelength λ such that Y n (z=-l 1 ) has a eal pat of unity. Use smith chat to calculate l 1 and include a pintout of the smith chat showing you wok with you answe sheet. Note that the nomalized admittance is always diagonally opposite to the nomalized impedance on a smith chat. b) Find the smallest length l in tems of the wavelength λ such that Y n (z=-l ) has an imaginay pat that exactly cancels the imaginay pat of Y n (z=-l 1 ). Use smith chat to calculate l and include a pintout of the smith chat showing you wok with you answe sheet. Note that the nomalized admittance is always diagonally opposite to the nomalized impedance on a smith chat. c) Now suppose that instead of using an open-cicuit stub you use a shot cicuit stub as shown in the figue below. V + 5 Ω z = - l z = - l 1 5
Assuming that the length l 1 is the same as that calculated in pat (a) above, find the smallest length l in tems of the wavelength λ such that Y n (z=-l ) has an imaginay pat that exactly cancels the imaginay pat of Y n (z=-l 1 ). Use smith chat to calculate l and include a pintout of the smith chat showing you wok with you answe sheet. Note that the nomalized admittance is always diagonally opposite to the nomalized impedance on a smith chat. Poblem 8.5: (Powe splitting in micowave cicuits) Powe splittes ae commonly used in integated micowave cicuits on a chip to split micowave powe into two o moe output diections. A schematic of a 1x micowave splitte is shown below. You need to figue out what faction of the input powe is eflected, and what faction of the input powe is tansmitted into each of the output tansmission lines. Befoe you can do that you need to find the amplitudes V -1, V +, and V +3 of the voltage waves in tems of the input wave amplitude V +1. Z o1 =10 Ω V +1 V -1 V + Z o = 10 Ω V +3 Z o3 =10 Ω a) Looking to the ight of the dashed line, the two output tansmission lines can be epesented as lumped impedances so the equivalent cicuit becomes as shown below: V +1 Z o1 V -1 + V T - Z o Z o3 Find the amplitude V -1 of the eflected wave in tems of the input wave amplitude V +1 and find the faction of the input powe that is eflected (give a numeical answe). b) Find the total voltage V T at the point z=0 in the figue above in tems of the input wave amplitude V +1. 6
c) The total voltage V T found in pat (b) must also equal V + and V +3 since they ae in paallel. Knowing this, find the faction of the input powe tansmitted in the each of the two output tansmission lines (give numeical answes). Do all you factions (eflected and tansmitted) add up to unity? They should. d) Suppose you could choose the impedances Z 0 and Z 03 of the output tansmission lines to be whateve you wanted. Choose these values such that you simultaneously satisfy the following two conditions: i) No faction of the input powe is eflected ii) The output tansmission line with impedance Z 0 has twice as much powe going into it as the tansmission line with impedance Z 03. Poblem 8.6: (Plasma cut-off fequency) In the lectues you wee told that if the fequency ω of an electomagnetic wave is less than the plasma fequency ω p then the wave is completely eflected at the suface of a plasma. In this poblem, you will exploe this futhe and see if the above statement holds in all cases. Conside an electomagnetic wave given by: jk z xˆ E i i e incident nomally fom fee-space on a plasma whose pemittivity is given by the elation: ( ) ωp ε ω = ε o 1 ω In the pevious homewok you showed that when the fequency ω of the electomagnetic wave is less E than the plasma fequency ω p, the magnitude of the eflection coefficient Γ = is unity. Ei x E i ε o µ o H i v k = ki zˆ ε ( ω) µ o v k = ki zˆ E H plasma z a) The fequency below which a wave is completely eflected (i.e. Γ = 1) at the suface of a plasma is called the cut-off fequency ω c. In the pevious homewok you essentially showed that the cut-ff fequency ω c is just the plasma fequency ω p povided the wave is incident nomally on the plasma. Now suppose a TE-wave is incident at an angle of incidence θ i as shown in the figue below. 7
x ε o µ o k E H θ θ i ε ( ω) µ o E i k i plasma H i z ω c as a function of the angle of incidence θ i fo the TE-Wave. Is the cut-off Find the cut-off fequency fequency ω c now lage o smalle than the plasma fequency ω p? Hint: think what is equied to get Γ = 1. Although this poblem is not about total intenal eflection, you may want to study caefully how Γ becomes unity in the case of total intenal eflection. b) Same as pat (a) but now suppose a TM-wave is incident at an angle of incidence θ i. Find the cut-off fequency ω c as a function of the angle of incidence θ i fo the TM-wave. Is the cut-off fequency ω c now lage o smalle than the plasma fequency ω p? 8