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Reasons 1. A straight line may be drawn between two points. 2. Radii of a circle are congruent. 3. Given 4. Perpendiculars form right angles. 5. Reflexive property 6. hy-leg 7. Corresponding parts of congruent triangles are congruent. 8. In a circle, congruent central angles have congruent arcs. 9. A diameter bisects a circle. 10. In a circle, congruent arcs are equal arcs; Subtraction Postulate

_ _ D (b)

A more complex circuit containing resistors, sources, and a single energy storage element may be converted to a Thevenin or Norton equivalent as seen from the two terminals of the inductor or capacitor. This reduces the complex circuit to a simple RC or RL circuit which may be solved according to the methods described in the previous sections. If a source in the circuit is suddently switched to a dc value, the resulting currents and voltages are exponentials, sharing the same time constant with possibly di erent initial and nal values. The time constant of the circuit is either RC or L=R, where R is the resistance in the Thevenin equivalent of the circuit as seen by the capacitor or inductor.

6. An angle inscribed in a circle is measured by one-half its intercepted arc. Case I: The center of the circle is on one side of the angle.

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EXAMPLE 7.5 Find i, v, and i1 in Fig. 7-11(a).

Fig. 7-12

Given: /A is inscribed in circle O. O is on side AC. To Prove: A/5 1BC 2 Plan: When radius OB is drawn, isosceles ^AOB is formed. /A is proved to be equal in measure to one-half central /1, which is measured by BC.

Fig. 7-11 The Thevenin equivalent of the circuit to the left of the inductor is shown in Fig. 7-11(b) with RTh 4

more easily determined than vgs , has been labeled. With terminals a; b opened in Fig. 7-12(a), KVL around the S; G; D loop yields vgd vgs 1 Then the Thevenin voltage at the open-circuited terminals a; b is v vTh vgs 1 gd 1

PROOF:

and vTh 3u t (V). The time constant of the circuit is L=RTh 5 10 3 =4 s 1:25 ms. The initial value of the inductor current is zero. Its nal value is i 1 Therefore, i 0:75 1 e 800t u t A v L di 3e 800t u t dt V i1 9 v 1 3 e 800t u t A 12 4 vTh 3V 0:75 A RTh 4

Statements 1. Draw OB. 2. AO > OB 3. /A > /B 4. m/A m/B m/1

The Thevenin impedance is found as the driving-point impedance to the left through a; b (with vi deactivated or shorted), as seen by a source vab driving current ia into terminal a. Since vgs vab , KVL around the output loop of Fig. 7-12(a) gives vab vgs ia rds vab ia rds v r RTh ab ds ia 1

v can also be derived directly from its initial value v 0 9 6 = 12 6 3 V, its nal value v 1 0 and the circuit s time constant.

Reasons 1. A straight line can be drawn between two points. 2. Radii of a circle are congruent. 3. If two sides of a triangle are congruent, the angles opposite these sides are congruent. 4. In a triangle the measure of an exterior angle equals the sum of the measures of the two adjacent interior angles. 5. Substitution Postulate 6. Halves of equals are equal. 7. A central angle is measured by its intercepted arc. 8. Substitution Postulate

CHAP. 7]

Expressions (1) and (2) lead directly to the circuit of Fig. 7-12(b).

Given: /BAC is inscribed in circle O. O is inside /BAC. 1 To Prove: /BAC 2 BC Plan: When a diameter is drawn, /BAC is divided into two angles which can be measured by applying Case I.

EXAMPLE 7.6 In Fig. 7-12 the 9-mF capacitor is connected to the circuit at t 0. is v0 17 V. Find vA , vB , vC , iAB , iAC , and iBC for t > 0:

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