Transmission Lines: Coaxial Cables and Waveguides

Transcription

1 Transmission Lines: Coaxial Cables and Waveguides Jason ClIItis 1 April 8, Introdution This pap er investigates the wave veloities in oaxial ables and retangular waveguides. Coaxial ables and waveguides are important omponents in our radio reeiver systems. For example, oaxial ables are used to transmit the signal from the antenna on the roof down to our reeiver in the UG Astro Lab. The pyramidal horn a ntenna we used to observe the HI line is a waveguide that gradually tapers down, guiding the radio waves from the free spae air into a oax transmission line, while minimizing refletions. In this lab, we are fortunate to have slotted oaxial abl es and waveguides, with probes attahed that an slide along the length, allowing us to take measurements of the wave patterns and find nulls in t he standing waves. With these data, I will alulate the wavelengths a nd veloities in a slotted oaxial able, a flexible oaxial able, and a retangular waveguide.. The Slotted Coax This setion looks at a slotted oaxial able, and alulates the wave veloity in the line. The able has a probe that an slide along its length, reading the Eletri Field, whih is proportional t o the voltage. The osillosope shows the wave pat tern as the probe slides along the line, with periodi nulls every half wavelength. The slotted oax is approximately half a meter in length. The position of the probe an be thought of in terms of the number of wavelengths t hat are in the line up to that point. We an express this mathematially as, (1) where X m is the position along the oax, m is the number of the null (zero point), whih is divided by two, representing half a wavelength, and is multiplied by A, the wavelength inside the able. I want to alulate t he wave veloity, v = fa, where f is the freqneny. I ontrol the frequeny in this experiment, with the Kruse-Storke wave generator. What I need to find is the wavelength in the able. I do this by measuring the position of the nulls along the able, at a frequeny, f=97 MHz, and then Least Squares Fitting the data to equation 1 above. tjeur tis@ugastro. berkeley.edu

2 - I Null Point I Position on Coax Line (m) Table 1: Data for Slotted Coax Wave Veloity Calulation. Frequeny is 97 MHz My fit yields a wavelength, A = lo.09m, and when plugged into the veloity equation yields a wave veloity, v =.0006xl0 1 0 r; = l G~ ~VR?. Slotted Coax With A Flexible Coax Piee Attahed Next, I attah a flexible piee of oax able to the end of the slotted oax. This flexible piee has been measuled at a length, I = 8m. I intend to derive the wave veloity, v, in this able. Unlike the slotted oax, this piee does not have an E-Field probe attahed, whih a llowed me to measure the position of the nulls previously. I thought about what I had observed on the osillosope. I slid the probe along the oax, and disovered nulls in the wave pattern. T hese nulls are ideal spots for measuring the wavelength, with the distane between adjaent nulls orresponding to half a wavelength. With t his in mind, I then thought about what had happened when I hanged frequenies before. At a higher frequeny, the wavelength shortened, allowing for more waves to fit inside the oax. Consider the flexible able attahed to the slotted oax. I positioned the probe at the boundary between the slotted oax and the flexi able. If I inrease the frequeny, eventually t he probe will read a null, meaning that there is now an extra half a wavelength in t he flexi able. W it h this insight, I an write an equation whih rela tes the number of wavelengths in the flexi able to the total length of the able: I=A(n + m) ( ) where I is the length of the flexi able, A is the wavelength of the wave inside the able, n is the number of wavelengths in the able at'the initial frequeny, and m is the number of wavelengths

3 - introdued into the able as the frequeny is hanged. The equation an be rewritten in terms of the wave veloity, v, whih is onst.ant, and t he frequeny I am operating at using v = >.f. v l = (n + m) y () The initial number of wavelengths, n, is also onstant. I Data Point I Frequeny (MHz) m / / / 7 18 Table : Data for Flexibe Coax Wave Veloity Calulation I solve for n and v in equation using a simple linear Least Squares Fit. This yields a value for the veloity, v =. X ~' =.78. The X-band Retangular Waveguide Next, I will investigate the retangular waveguide. I want to find the veloity, Vguide, of the transverse eletromagneti waves (TE waves) in the waveguide, and with this information, solve for the width, a, of the waveguide. Note in the diagram below that while the Eletri Field is traveling at the speed of light, it is traveling along the waveguide at an angle, e, whih gives t.he wave veloi ty, Vguide = s i~o.

4 - ~/\g u jde~ r a I a 1 Vguide = sin 0 Fig. 1.- Left: Our retangular waveguide, with mode T E IO. Right: TE waves traveling down waveguide. Equation gives the "guide wavelength," ).g, for the T ElO mode. () Solving for v p, () First, I use a aliper to measure the width, a, of our waveguide: I measure a =.6 m. Next, I ondut an experiment to test the above equation, by taking measurements of the null points in the waveguide at various frequenies and doing a non-linear least squares fit of equation for a. The table below presents my null measurements at four different frequenies. Before I fit for a, I need to alulate t he wave veloity. I an do this with eah independent freq ueny data set, yielding four different values..1. Non-linea r Least Squares Fit I p resent a method for transforming a non-linear equation into a linear equation allowing foj" a simple linear Least Squares Fit to be onduted. First, reognize that I have an array, V p, with four elements, orresponding to the four frequenies I used in this experiment. Next, I will define a veloity, V guess, whih is equivalent to V p, exept that a has b een replaed by a guessed value for a = a g "",. I then define the differene between Vp and Vguess>

5 - (6) Calulating t he derivative, with a = a g yields, = 1 ( 1 _ ( ) ) - ~ (7) J ag f a g Inserting equation 7 into equation 8 yields an equation of known variables, with one unknown onstant to fit for, D.a. One D.a has been found, orresponding to the intia l guess, a g, I redefine ag,new = ag + D.a, and refit for a new D.a. I ontinue this proess until D.a drops toward zero and a g onverges on an aurate value for a. I worked through this method with two separate data sets (only one is shown below), and eah time I find a value a = 1.66 m. This is muh smaller than the value I measmed with the aliper. I believe I have gone wrong somewhere, but I do not yet understand where. I will ontinue investigating this, hopefuljy hange this, and end this paper on a happy note with a good answer. I Frequeny (GHz) I Point I Position of Null (em) II Frequeny (GHz) I Point I Position of Null (em) l l Table : Data for Waveguide Veloity and Dimension Calulation