a. Find the mathematical expression for the voltage across the capacitor once the switch is closed. b. Find the mathematical expression for the current during the transient period. c. Sketch the waveform for each from initial value to final value. 60
11 Example 3-Cont’d 61 • Solution: a. Substituting the open-circuit equivalent for the capacitor will result in a final or steady-state voltage v C of 24 V . The time constant is determined by: And: Example 3-Cont’d 62 b. Since the voltage across the capacitor is constant at 4 V prior to the closing of the switch, the current (whose level is sensitive only to changes in voltage across the capacitor) must have an initial value of 0 mA. At the instant the switch is closed, the voltage across the capacitor cannot change instantaneously, so the voltage across the resistive elements at this instant is the applied voltage less the initial voltage across the capacitor. The resulting peak current is: The current will then decay (with the same time constant as the voltage v C ) to zero because the capacitor is approaching its open circuit equivalence. The equation for i C is therefore : Example 3-Cont’d 63 c. Example 4 • For the circuit shown below: a. Find the mathematical expression for the transient behaviour of the voltage v C and the current i C following the closing of the switch (position 1 at t=0 s). b. Find the mathematical expression for the voltage v C and current i C as a function of time if the switch is thrown into position 2 at t= 9 ms. c. Draw the resultant waveforms of parts (a) and (b) on the same time axis. 64 Example 4-Cont’d 65 a. Applying Thévenin’s theorem to the 0.2-mF capacitor, we obtain: Example 4-Cont’d 66
12 Example 4-Cont’d 67 The resultant Thévenin equivalent circuit with the capacitor replaced is shown in Fig. For the current: Example 4-Cont’d 68 b. Ans: c. Prove Interpretation of τ • In general, you should note that an equation of the form: means exponential growth (!) . The time constant, τ, is the amount of time necessary for an exponential to grow to 0.63 of its final value. / (1 ) t y Y e τ - = - t y 0 Y t= τ 0.63Y 69 Charging Interpretation of τ- Cont’d • An equation of the form: means exponential decay . The time constant, τ, is the amount of time necessary for an exponential to decay to 36.7% of its initial value. / t y Ye τ - = t y 0 Y t= τ 0.37Y 70 Discharging Example 1 • Find the mathematical expressions for the transient behaviour of i L and v L for the circuit below after the closing of the switch. Sketch the resulting curves. 71 Example 1-Cont’d • Solution: 72
13 Example 2 The inductor of the figure below has an initial current level of 4 mA in the direction shown. a. Find the mathematical expression for the current through the coil once the switch is closed. b. Find the mathematical expression for the voltage across the coil during the same transient period.