4-Lossless Lines (Sinusoids).pdf

4-Lossless Lines (Sinusoids).pdf - ECE 303 Sum2017 Notes...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
ECE 303 - Sum2017 Notes Set 4: Lossless Lines - Sinusoids 1 INTRODUCTION In an earlier set of Notes we studied the behavior of pulses transmitted on lossless lines. We learned that for short pulses we could actually “see” the propagation and reflection processes using the bounce diagram. In the current set of Notes we will extend these concepts to the case of transmitting sinusoids onto lossless lines. The propagation and reflection processes are still present but we will see that they now take a di ff erent form. Many information systems can be analyzed and designed by considering the input to be a single frequency sinusoid. For example, an analog cable television system might transmit a television signal by modulating a single high frequency sinusoid called a carrier. The time-domain variation of the video information signal is slower (i.e., at a lower frequency) than the rapid time-domain variations of the sinusoidal carrier. Therefore, the physical cable tends to propagate the modulated signal very similarly to how it would propagate the carrier by itself. Even “digital” communication systems, like digital cable TV or digital cable modem, fre- quently use a carrier modulation method to transmit the “digital” data. For example, a logical 1 might be 100 cycles of a positive cosine at the carrier frequency and a logical 0 might be 100 cycles of a negative cosine at the carrier frequency. The transmitted signal looks very much like a single frequency sinusoid except at the these “phase transition” times. Consequently, design and analysis of the physical system can often be accomplished by considering the input as a single frequency sinusoid. In this set of Notes we will see how the sinusoidal nature of the propagating signal requires a di ff erent analysis than did pulse propagation. Many of the tools needed in this analysis are complex-valued and therefore we will be using quite a bit of complex functions in this set of Notes. You can see the Supplement Notes on Complex Numbers and Functions (accessible from the Home Page) for a review of complex numbers.
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
ECE 303 - Sum2017 Notes Set 4: Lossless Lines - Sinusoids 2 PROPERTIES OF SINUSOIDS WAVELENGTH: The velocity of propagation v p (in m/sec), the Hertz frequency f (in Hz), and the wavelength λ (in m) of a wave on a transmission line are related by v p = f λ (1) For example, suppose a sinusoid with frequency f = 400 MHz has velocity v p = 2(10) 8 m/sec on a transmission line. The wavelength on the line is thus λ = v p f = 2(10) 8 4(10) 8 = 0 . 5 meters PHASE CONSTANT: The phase constant β (in rad/m), is given by the equivalent expressions β = ω v p = 2 π λ (2) The parameter β is sometimes called the Wavenumber, since it computes the number of radians in one meter of line. COMPLEX-EXPONENTIAL (PHASOR) NOTATION: It is often simpler to solve a problem in the complex domain, and then translate the final complex-valued math result back into the real domain to find the actual physical quantity.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern