barcode reader code in asp.net R + C vC (t) _ RC circuit: 1 dvC 1 vC vS = 0 dt RC RC R in Software

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R + C vC (t) _ RC circuit: 1 dvC 1 vC vS = 0 dt RC RC R
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+ v (t) _ S
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R 1 diL iL vS = 0 dt L L
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Figure 59 Differential equations of rst-order circuits
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To evaluate the constant K, we need to know the initial condition The initial condition is related to the energy stored in the capacitor or inductor, as will be further explained shortly Knowing the value of the capacitor voltage or inductor current at t = 0 allows for the computation of the constant K, as follows: xN (t = 0) = Ke 0 = K = x0
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usually refer to the unforced solution as the homogeneous solution
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5
Transient Analysis
Thus, the natural solution, which depends on the initial condition of the circuit at t = 0, is given by the expression xN (t) = x0 e t/ (517)
where, once again, xN (t) represents either the capacitor voltage or the inductor current and x0 is the initial condition (ie, the value of the capacitor voltage or inductor current at t = 0) Energy Storage in Capacitors and Inductors
t=0 t=0 Switch VB vC Switch C i(t) R
Before delving into the complete solution of the differential equation describing the response of rst-order circuits, it will be helpful to review some basic results pertaining to the response of energy-storage elements to DC sources This knowledge will later greatly simplify the complete solution of the differential equation describing a circuit Consider, rst, a capacitor, which accumulates charge according to the relationship Q = CV The charge accumulated in the capacitor leads to the storage of energy according to the following equation: WC = 1 2 Cv (t) 2 C (518)
Exponential decay of capacitor current 1 09 08 07 06 05 04 03 02 01 0 0 05 1 15 2 25 3 35 4 45 5 Time, s
Figure 510 Decay through a resistor of energy stored in a capacitor
t=0 t=0 Switch IB vC Switch L iL(t) R
To understand the role of stored energy, consider, as an illustration, the simple circuit of Figure 510, where a capacitor is shown to have been connected to a battery, VB , for a long time The capacitor voltage is therefore equal to the battery voltage: vC (t) = VB The charge stored in the capacitor (and the corresponding energy) can be directly determined using equation 518 Suppose, next, that at t = 0 the capacitor is disconnected from the battery and connected to a resistor, as shown by the action of the switches in Figure 510 The resulting circuit would be governed by the RC differential equation described earlier, subject to the initial condition vC (t = 0) = VB Thus, according to the results of the preceding section, the capacitor voltage would decay exponentially according to the following equation: vC (t) = VB e t/RC (519)
Inductor current, A
Exponential decay of inductor current 1 09 08 07 06 05 04 03 02 01 0 0 05 1 15 2 25 3 35 4 45 5 Time, s
Physically, this exponential decay signi es that the energy stored in the capacitor at t = 0 is dissipated by the resistor at a rate determined by the time constant of the circuit, = RC Intuitively, the existence of a closed circuit path allows for the ow of a current, thus draining the capacitor of its charge All of the energy initially stored in the capacitor is eventually dissipated by the resistor A very analogous reasoning process explains the behavior of an inductor Recall that an inductor stores energy according to the expression WL = 1 2 Li (t) 2 L (520)
Inductor current, A
Figure 511 Decay through a resistor of energy stored in an inductor
Thus, in an inductor, energy storage is associated with the ow of a current (note the dual relationship between iL and vC ) Consider the circuit of Figure 511, which is similar to that of Figure 510 except that the battery has been replaced with a current source and the capacitor with an inductor For t < 0, the source current, IB , ows through the inductor, and energy is thus stored; at t = 0, the inductor current is equal to IB At this point, the current source is disconnected by means of the left-hand switch and a resistor is simultaneously connected to the inductor,
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