lectureslides11 - Initial conditions dvC Q = CvC iC = C dt...

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EE 442 RC and RL transients 1 Initial conditions i C = C dv C dt Because we cannot instantaneously change the charge on a capacitor we cannot instantaneously change the voltage across the capacitor. (Charge has to move, which takes a bit of time.) Q + = Cv C If we could change voltage instantaneously, then we would have current go to infinity for some period of time. By the same token, we cannot change the current through an inductor instantaneously. (Otherwise, we would create infinite voltages.) Inductor current must be continuous with time. v L = L i L dt There is no restriction on capacitor current or inductor voltage – these quantities can change instantaneously.
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EE 442 RC and RL transients RC transient Consider the circuit below, consisting of a voltage source, a resistor, a switch, and a capacitor, all in series. 2 The switch is open, so no current flows – essentially nothing is happening – and it has been this way for a while. The capacitor has some some voltage on it (i.e. it has stored charge), but since there is no path for current to flow, the charge stays in place and the capacitor voltage does not change. + + v C i C V S R v C ( t < 0) = v Ci i C ( t < 0) = 0
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EE 442 RC and RL transients 3 Then the switch closes at t = 0. Interesting things will happen. + + v C i C V S R t = 0 1. Because the voltage on a capacitor cannot change instantaneously, v C will remain at the voltage that it had before the switch closed. v C ( t = 0) = v Ci . 2. Capacitor current has no such restriction, and it’s value will instantaneously jump to a value determined by the resistor: i C ( t = 0) = i R = V S v Ci R
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EE 442 RC and RL transients 4 As we observe the progression for t > 0: 3. As current flows to the capacitor, the capacitor charges up and the capacitor voltage increases. v C ( t > 0) > v Ci , (and increasing). 4. As the capacitor charges up and its voltage increases, the capacitor current will decrease. i C ( t ) = V S v C ( t ) R + + v C i C V S R t > 0 decreasing increasing
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EE 442 RC and RL transients 5 5. Eventually, the capacitor voltage will increase until it is equal to V S . At that point, the current drops to zero. From that time nothing changes again. Now, it doesn’t matter if the switch is closed or open. + + i C = 0 V S R after a sufficiently long time v C = V S This is a transient effect. Initially, the capacitor voltage was at some fixed, steady value and the capacitor current was zero. When the switch closed, the current jumped up to a value determined by the difference between the source voltage, the initial capacitor voltage and the resistance connecting them. The current charged the capacitor to increasingly higher voltage. When the capacitor voltage reached the source voltage, the current dropped to zero, and the circuit to a static (quiescent) state with the capacitor at a new, higher voltage with no current flowing.
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EE 442 RC and RL transients
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