1250
HOMEWORK #3-5 prob 3 solution
F14
EX:
Find the Thevenin equivalent circuit at terminals a-b. vx must not appear in your
solution. The expression must not contain more than circuit parameters α, R1, R2, R3,
and is. Note: 0 < α < 1.
SOL'N:
vTh is the voltage across the a, b terminals with nothing attached at a, b.
If we use the node-voltage method to find vTh, then if the bottom node is
reference, the top node is vTh.
We must define the dependent source variable, vx, in terms of the node
voltage, but we discover that vx is actually the node voltage!
vx = vTh
Now we can write the current summation equation for the vTh node:
vTh − 0V
v − α vTh
+ −is + Th
= 0A
R1 + R2
R3
or
⎛ 1
1− α ⎞
vTh ⎜
+ −is +
= is
R3 ⎟⎠
⎝ R1 + R2
or
vTh =
is
1
1− α
+ −is +
R1 + R2
R3
or
vTh =
(R1 + R2 )R3
is (R1 + R2 )R3
=
1
1− α (R1 + R2 )R3 R3 + (1− α )(R1 + R2 )
+
R1 + R2
R3
is
⋅
To find RTh, we may short out the a, b terminals and measure the current
in the short circuit and use RTh = vTh/isc.
If we examine the circuit, we discover that vx = 0, so the dependent source
turns into a wire (0V), and we have R3 in parallel with the wire from a to
b. Current is will all flow through the wire, since it has no resistance.
isc = is
Now we can compute RTh.
v
(R1 + R2 )R3
RTh = Th =
isc
R3 + (1− α )(R1 + R2 )