Problem 4-70

Find the Thevenin equivalent with respect to terminals a and b.

NOTE: items in blue are not part of original diagram.

Since there are no independent sources, we cannot determine the Thevenin equivalent by finding v_oc and i_sc. Also, the lack of an independent source implies that the Thevenin equivalent will be a resistor (v_Th will equal 0, thus can be replaced by a wire.)

Since there is a dependent source,w e cannot fine the Thevenin equivalent by circuit reduction techniques.

The only remaining option to find the Thevenin equivalent resistance is thus to drive the circuit with an external source and see what happens. I have chosen to drive it with a 1 amp source and determine the voltage developed using mesh analysis.

Dependent source parameter:

iD = i_b - i_c

Current source:

i_c = - 1 A

Mesh a:

5 (i_a - i_b) - 6.5 iD + 50 i_a = 0

Substitute iD

5 (i_a - i_b) - 6.5 (i_b - i_c ) + 50 i_a = 0

55 i_a - 11.5 i_b + 6.5 i_c = 0

Substitute i_c

55 i_a - 11.5 i_b = 6.5 (Equation 1)

Mesh b:

10 i_b + 25 (i_b - i_c) + 6.5 iD + 5 (i_b - i_a) = 0

Substitute iD

10 i_b + 25 (i_b - i_c) + 6.5 (i_b - i_c ) + 5 (i_b - i_a) = 0

- 5 i_a + 46.5 i_b - 31.5 i_c = 0

Substitute i_c

- 5 i_a + 46.5 i_b = - 31.5 (Equation 2)

No KVL needed around mesh c due to current source.

Solving Equations 1 and 2 yields

i_a = -23.07 mA

i_b = -679.9 mA

To determine V_T, write KVL around the rightmost loop.

12 i_c + V_T + 25 (i_c - i_b) = 0

V_T = - 12 i_c + 25 (i_b - i_c)

V_T = - 12 (- 1) + 25 (-0.6799 + 1)

V_T = 20.00 V

Thus

R_Th = V_T / 1

R_Th = 20.00 W

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