Standing waves and resonance

Chapter 14.6 Standing waves and resonance

Lessons In Electric Circuits Volume II – AC Book
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Lessons In Electric Circuits Volume II – AC Book

  • 492CHAPTER 14. TRANSMISSION LINESpropagation time is a large fraction or multiple of the signal’s period, the line is consideredelectrically long.• A signal’s wavelength is the physical distance it will propagate in the timespan of one period.Wavelength is calculated by the formula λ=v/f, where “λ” is the wavelength, “v” is thepropagation velocity, and “f” is the signal frequency.• A rule-of-thumb for transmission line “shortness” is that the line must be at least 1/4 wave-length before it is considered “long.”• In a circuit with a “short” line, the terminating (load) impedance dominates circuit behavior.The source effectively sees nothing but the load’s impedance, barring any resistive losses inthe transmission line.• In a circuit with a “long” line, the line’s own characteristic impedance dominates circuitbehavior. The ultimate example of this is a transmission line of infinite length: since thesignal will never reach the load impedance, the source only “sees” the cable’s characteristicimpedance.• When a transmission line is terminated by a load precisely matching its impedance, there areno reflected waves and thus no problems with line length.14.6Standing waves and resonanceWhenever there is a mismatch of impedance between transmission line and load, reflections willoccur. If the incident signal is a continuous AC waveform, these reflections will mix with more ofthe oncoming incident waveform to produce stationary waveforms called standing waves.The following illustration shows how a triangle-shaped incident waveform turns into a mirror-image reflection upon reaching the line’s unterminated end. The transmission line in this illustrativesequence is shown as a single, thick line rather than a pair of wires, for simplicity’s sake. Theincident wave is shown traveling from left to right, while the reflected wave travels from right to left:(Figure 502,14.21)If we add the two waveforms together, we find that a third, stationary waveform is created alongthe line’s length: (Figure 503,14.22)This third, “standing” wave, in fact, represents the only voltage along the line, being the repre-sentative sum of incident and reflected voltage waves. It oscillates in instantaneous magnitude, butdoes not propagate down the cable’s length like the incident or reflected waveforms causing it. Notethe dots along the line length marking the “zero” points of the standing wave (where the incidentand reflected waves cancel each other), and how those points never change position: (Figure 504,14.23)Standing waves are quite abundant in the physical world. Consider a string or rope, shaken atone end, and tied down at the other (only one half-cycle of hand motion shown, moving downward):(Figure 505,14.24)Both the nodes (points of little or no vibration) and the antinodes (points of maximum vibration)remain fixed along the length of the string or rope. The effect is most pronounced when the free endis shaken at just the right frequency. Plucked strings exhibit the same “standing wave” behavior,with “nodes” of maximum and minimum vibration along their length. The major difference between