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Zin s 2 (a) At s j2 rad/s, Zin j2 3:22 7:138

8:36

33. (a) 0; (b) 2; (c) 3; (d) 1 (b) 3; (c) (b) 1; (c) 6 4

V IZin 10 08 3:22 7:138 32:2 7:138 V and v 0:1 32:2 cos 18:598 30:5 V. (b) At s 1 j2 s 1 , Zin 1 j2 3:14 11:318

5 2 3 2

For typical values, the coe cient of s2 on the right side of (8.36) is several orders of magnitude smaller than the other terms; by approximating this coe cient as zero (i.e., neglecting the s2 term), we neglect a breakpoint at a frequency much greater than !H . Doing so and using (8.32), we obtain the desired high-frequency voltage-gain ratio: Av s RC kRL sC gm VL r VS r rx s 1 gm C RC kRL 1 8:37

Fig. 8-20

(b) 1; (c) 2; (d) 1

Fig. 8-21

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The small-signal high-frequency model for the FET is an extension of the midfrequency model of Fig. 7-1. Three capacitors are added: Cgs between gate and source, Cgd between gate and drain, and Cds between drain and source. They are all of the same order of magnitude typically 1 to 10 pF. Figure 8-10 shows the small-signal high-frequency model based on the current-source model of Fig. 7-1(a). Another model, based on the voltage-source model of Fig. 7-1(b), can also be drawn.

42. (a) line having y-intercept 5, slope 2; (b) line passing through (2, 3), slope 4; 5 (c) line passing through ( 2, 3), slope 4; (d) line passing through origin, slope 1; 2 1 (e) line having y-intercept 7, slope 1; (f) line passing through origin, slope 3 43. (a) y 44. (a) y (f) y 4x; (b) y 2x; (c) y

(b) s j4 rad/s,

Example 8.8. For the JFET ampli er of Fig. 4-5(b), (a) nd an expression for the high-frequency voltage-gain ratio Av s and (b) determine the high-frequency cuto point. (a) The high-frequency small-signal equivalent circuit is displayed in Fig. 8-11, which incorporates Fig. 8-10. We rst nd a Thevenin equivalent for the network to the left of terminal pair a; a 0 . Noting that vgs vi , we see that the open-circuit voltage is given by sCgd gm gm V Vi sCgd i sCgd

3 2x

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY 4 s2 3s 4 s Zin s 2 2 2 4 s s 2 2 s 1 s 2 s 1 (a) Zin 0 4

1 3x

8:38

, the impedance o ered to a constant (dc) source in the steady state. b Zin j4 2 j4 2 3 j4 4 2:33 29:058

2 5x

Express the impedance Z s of the parallel combination of L 4 H and C 1 F. frequencies s is this impedance zero or in nite

RG _

0 4x 3;

Z s 4s 1=s s 4s 1=s s2 0:25

+ LL _

By inspection, Z 0 0 and Z 1 0, which agrees with our earlier understanding of parallel LC circuits at frequencies of zero (dc) and in nity. For jZ s j 1, s2 0:25 0 or s j0:5 rad=s

4 1 7

A sinusoidal driving source, of frequency 0.5 rad/s, results in parallel resonance and an in nite impedance.

1 2x

Fig. 8-11

The circuit shown in Fig. 8-22 has a voltage source connected at terminals ab. The response to the excitation is the input current. Obtain the appropriate network function H s .

4 3;

CHAP. 8]

Fig. 12-45

51. (a) 10; (b) 10; (c) 20; (d) 20; (e) 7; (f) 25 52. (a) 16; (b) 12; (c) 20; (d) 24 53. (a) 10; (b) 12; (c) 22 54. (a) 5; (b) 13; (c) 72 55. (a) 6; (b) 10; (c) 1.2 56. (a) 15; (b) 49; (c) 53 57. (a) 30; (b) 49; (c) 88; (d) 24; (e) 16; (f) 18

If Vi is deactivated, Vi Vgs 0 and the dependent current source is zero (open-circuited). A driving-point source connected to a; a 0 sees only ZTh Vdp 1 Idp sCgd 8:39

Fig. 12-46

13

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