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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

Fig. 2-26

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.

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where the rst two terms on the right-hand side are forced-response terms (called the frequency response), and the balance of the terms constitute the transient response. The transient response diminishes to zero with time, provided the roots of D s 0 are located in the left half plane of complex numbers (the condition for a stable system). The coe cients k1 and k2 are evaluated by the method of residues, and the results are used in an inverse transformation to the time-domain steady-state sinusoidal response given by v2 t V1m jAv j! j sin !t V2m sin !t (see Problem 8.23). The network phase angle is de ned as tan 1 ImfAv j! g RefAv j! g 8:4 8:3

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, 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

From (8.4), it is apparent that a sinusoidal input to a stable, linear, two-port network results in a steadystate output that is also sinusoidal; the input and output waveforms di er only in amplitude and phase angle. For convenience, we make the following de nitions: 1. 2. 3. Call A j! the frequency transfer function. De ne M jA j! j, the gain ratio. De ne Mdb 20 log M 20 log jA j! j, the amplitude ratio, measured in decibels (db).

5. (a) Two sides are congruent and i. (b) Opposite sides are congruent. (c) Opposite angles are congruent. (d) AD and BC are congruent and parallel (AD > EF > BC). 6. (a) x 7. (a) x 10. (a) x (f) x 13. (a) x 14. (a) x (c) x 15. (a) m 16. (a) x 17. (a)

CHAP. 8]

1 222;

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The subscript v or i may be added to any of these quantities to speci cally denote reference to voltage or current, respectively. The graph of Mdb (simultaneously with if desired) versus the logarithm of the input signal frequency (positive values only) is called a Bode plot.

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY q q !2 2 106 4:9 105 714 0

1 42;

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

Example 8.1. A simple rst-order network has Laplace-domain transfer function and frequency transfer function A s 1 s 1 and A j! 1 1 j!

1 172;

at t 0,

where is the system time constant. (a) Determine the network phase angle and the amplitude ratio Mdb and (b) construct the Bode plot for the network.

45 4; (e) x 25, y 25;

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

1 232

For the circuit and values of Problem 1.7, nd the Norton equivalent for the network to the left of terminals a; b.

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.

12 (since y does not join midpoints, Pr. 3 does not apply);

Fig. 2-2

Then, by superposition,

18. (a) 21; (b) 30; (c) 14; (d) 26

Fig. 2-3

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