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An angle formed by two secants intersecting outside a circle is measured by one-half the difference of its intercepted arcs 8b An angle formed by a tangent and a secant intersecting outside a circle is measured by one-half the difference of its intercepted arcs 8c An angle formed by two tangents intersecting outside a circle is measured by one-half the difference of its intercepted arcs 9 If three angles of one triangle are congruent to three angles of another triangle, the triangles are similar 10 If the altitude is drawn to the hypotenuse of a right triangle, then (a) the two triangles thus formed are similar to the given triangle and to each other, and (b) each leg of the given triangle is the mean proportional between the hypotenuse and the projection of that leg upon the hypotenuse 11.

appearing at the output terminals is vTh vds gm vgs rds vgs , where we have de ned a new equivalent-circuit constant, Amplification factor gm rds Proper series arrangement of vTh and RTh leads to Fig. 7-1(b).

Fig. 7-9

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The square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides 12 The area of a parallelogram equals the product of the length of one side and the length of the altitude to that side 13 The area of a triangle is equal to one-half the product of the length of one side and the length of the altitude to that side 14 The area of a trapezoid is equal to one-half the product of the length of the altitude and the sum of the lengths of the bases 15 The area of a regular polygon is equal to one-half the product of its perimeter and the length of its apothem..

In the drain-feedback-biased ampli er of Fig. 4-9(a), RF 5 M ; RL 14 k ; rds 40 k ; and gm 1 mS. Find (a) Av vds =vi ; b Zin ; c Zo looking back through the drain-source terminals, and (d) Ai ii =iL .

At times a transient is only partially displayed (on chart paper or on the face of an oscilloscope), and the simultaneous values of function and slope needed in the preceding method are not available. In that case, any pair of data points, perhaps read from instruments, may be used to nd the equation of the transient. Thus, referring to Fig. 7-10, f1 Ae t1 = which may be solved simultaneously to give and then A in terms of and either f1 or f2 . t2 t1 ln f1 ln f2 f2 Ae t2 =

1. If two sides of a triangle are congruent, the angles opposite these sides are congruent. (Base angles of an isosceles triangle are congruent.)

Fig. 12-43

(a) The voltage-source small-signal equivalent circuit is given in Fig. 7-10. With vds as a node voltage, vi vds vds vds vi RF RL rds

Given: ^ABC, AB > BC To Prove: /A > /C Plan: When the bisector of the vertex angle is drawn, the angles to be proved congruent become corresponding angles of congruent triangles.

CHAP. 12]

PROOF:

Fig. 7-10 Substituting for gm rds and rearranging yield Av vds RL rds 1 RF gm RF rds RL rds RL RF vi 14 103 40 103 1 5 106 1 10 3 10:35 5 10 40 103 14 103 40 103 14 103 5 106

Clearly, as R ! 0, this expression reduces to that given for the pure LC tank. Substituting the numerical values produces a value of 0.791 for the radical; hence, !a 408:2 0:791 322:9 rad=s or fa 51:4 Hz

Statements 1. 2. 3. 4. 5. 6. Draw BD bisecting /B. /l > /2 AB > BC BD > BD ^ADB > ^CDB /A > /C 1. 2. 3. 4. 5. 6.

(b) KVL around the outer loop of Fig. 7-10 gives vi ii RF vds ii RF Av vi , from which Zin (c) vi RF 5 106 440 k ii 1 Av 1 10:35 With vi 0,

12.17 Measurements on a practical inductor at 10 MHz give L 8:0 mH and Qind 40. (a) Find the ideal capacitance C for parallel resonance at 10 MHz and calculate the corresponding bandwidth . (b) Repeat if a practical capacitor, with a dissipation factor D Q 1 0:005 at 10 MHz, is cap used instead of an ideal capacitance.

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