ssrs barcodelib PARALLEL RLC CIRCUIT in Software

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PARALLEL RLC CIRCUIT
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The response of the parallel RLC circuit shown in Fig. 8-7 will be similar to that of the series RLC circuit, since a second-order di erential equation can be expected. The node voltage method gives v 1 t dv 0 1 v dt C R L 0 dt Di erentiating and dividing by C yields d 2v 1 dv v 0 dt2 RC dt LC A solution is of the form
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CHAP. 8]
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HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
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Fig. 8-6
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v A 1 es 1 t A 2 es 2 t 2 s   q 1 1 2 1 2 !2 where s1 0 2RC 2RC LC s   q 1 1 2 1 2 !2 s2 0 2RC 2RC LC p where 1=2RC and !0 1= LC . Note that , the damping factor of the transient, di ers from in the series RLC circuit.
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Fig. 8-7
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The transient response is easiest to visualize by assuming an initial charge Q0 on the capacitor and a switch that closes at t 0. However, a step function voltage applied to the circuit will initiate the same transient response. Overdamped Case 2 > !2 0 In this case, the solution (2) applies.
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EXAMPLE 8.4 A parallel RLC circuit, with R 1000
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, C 0:167 mF, and L 1:0 H, has an initial voltage V0 50:0 V on the capacitor. Obtain the voltage v t when the switch is closed at t 0. We have 1 2994 2RC 2 8:96 106 !2 0 1 5:99 106 LC
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HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
[CHAP. 8
Since 2 > !2 , the circuit is overdamped and from (2) we have 0 q q s1 2 !2 1271 and s2 2 !2 4717 0 0 dv At t 0; and s A s2 A 2 V0 A1 A2 dt t 0 1 1 From the nodal equation (1), at t 0 and with no initial current in the inductance L, V0 dv dv V0 0 or C dt dt t 0 R RC Solving for A1 , A1 Substituting into (2) v 155:3e 1271t 105:3e 4717t See Fig. 8-8. V V0 s2 1=RC 155:3 s2 s1 and A1 V0 A1 50:0 155:3 105:3
Fig. 8-8
Underdamped (Oscillatory) Case !2 > 2 0 The oscillatory case for the parallel RLC circuit results in an equation of the same form as that of the underdamped series RLC circuit. Thus, v e t A1 cos !d t A2 sin !d t 3
q where 1=2RC and !d !2 2 . !d is a radian frequency just as was the case with sinusoidal 0 circuit analysis. Here it is the frequency of the damped oscillation. It is referred to as the damped radian frequency.
EXAMPLE 8.5 A parallel RLC circuit, with R 200
, L 0:28 H, and C 3:57 mF, has an initial voltage V0 50:0 V on the capacitor. Obtain the voltage function when the switch is closed at t 0. 1 1 700 2RC 2 200 3:57 10 6 2 4:9 105 !2 0 1 1 106 LC 0:28 3:57 10 6
Since !2 > 2 , the circuit parameters result in an oscillatory response. 0
CHAP. 8]
HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY q q !2 2 106 4:9 105 714 0
!d
At t 0, V0 50:0; hence in (3) A1 V0 50:0. From the nodal equation V0 1 t dv 0 v dt C dt R L 0 dv V 0 dt t 0 RC
at t 0,
Di erentiating the expression for v and setting t 0 yields dv !d A2 A1 or dt t 0 Since A1 50:0, A2 and so V0 =RC V0 49:0 !d V
!d A2 A1
V0 RC
v e 700t 50:0 cos 714t 49:0 sin 714t
The critically damped case will not be examined for the parallel RLC circuit, since it has little or no real value in circuit design. In fact, it is merely a curiosity, since it is a set of circuit constants whose response, while damped, is on the verge of oscillation.
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