Working through the analysis for a discharging

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Unformatted text preview: itor voltage will exponentially drop down to the source voltage. You can see this by going back to slides 2-4 and replacing the words “increasing” and “charging” with “decreasing” and “discharging”. You don’t need to change the direction of the current or any voltage polarities. Working through the analysis for a discharging capacitor leads to exactly the same equation as we obtained for the charging capacitor. EE 442 RC and RL transients – 12 transition (switching) times As noted, the transition time is governed by the RC time constant, and the “few time constants” rule works for order of magnitude estimates. However, in applications where RC circuits used as timing elements, it is necessary to have more quantitative results. In the example shown on slide 9, one might ask, “How long does it take to go from 2 V to 6 V?” The correct answer is “Forever”, due to the fact that the exponential asymptotically approaches the final value. Usually, we define a marker (or markers) that the voltage will cross during its transition from high-to-low or low-to-high. EE 442 RC and RL transients – 13 One commonly used measure is the 50% point, where the voltage is halfway through its swing. (VS − VCi ) V 0 .5 = V S − 2 ￿ ￿ (VS − VCi ) t0 . 5 VS − = VS − [VS − VCi ] exp − 2 RC t0.5 = ln (2) RC = 0.693RC Another is the 10%-to-90% rise (or fall) time. The 10% and 90% times can be calculated in a similar fashion. ￿￿ 10 t0.1 = ln RC t0.9 = ln (10) RC 9 ∆ t = t 0 . 9 − t0 . 1 = ln (10) RC − ln ￿ 10 9 ￿ RC = ln (9) RC = 2.20RC EE 442 RC and RL transients – 14 RL transient Now consider the RL circuit below, consisting of a current source, a resistor, a switched inductor, and an, all in parallel. IS R iL + L vL – The switch is open, so no current flows through the inductor. Consequently, there is no voltage across the inductor. All of the source current flows through the resistor. The circuit just sits here like this waiting for something to happen. iL (t < 0) = iLi = 0 vL (t < 0) = 0 EE 442 RC and RL transients – 15 Then the switch c...
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This document was uploaded on 04/05/2014.

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