node voltages or mesh currents

Example
The voltages v1, v2, v3 and v4 are the node voltages corresponding to nodes 1, 2, 3 and 4. The
values of these voltages are
v1 = 10 V, v2 = 75 V, v3 = −15 V and v4 = 22.5 V
Determine the values of the gains of the dependent sources, A and B, and of the resistance R1.
Solution:
Express the controlling voltage and current of the
dependent sources in terms of the node voltages:
and
i
b
va = v4
= 22.5 V
v3−v4 −15 −22.5
= = =− 0.75
R 50
2
Express the dependent voltage source voltage in
terms of the node voltages:
so
v2 − v3 = Ava = Av4 4
( )
v2 − v3
75 − −15
A = = = 4 V/V
v 22.5
Apply KCL to the supernode corresponding to the dependent voltage source
Apply KCL at node 4:
v −v v −v 75 −10 −15 −22.5
+ = ⇒ + = 2.5 ⇒ = 20 Ω
2 1 3 4
Is R1
R1 R2 R1
50

v3−v4 v4 v3−v4 −15 −22.5 22.5 −15−22.5 = + B ⇒ = + B
⇒ B = 2.5 A/A
R R R
50 20 50
2 3 2
Example
The value of the node voltage at node b in the circuit is v b = 18 V.
(a) Determine the value of A, the gain of the dependent source.
(b) Determine the power supplied by the dependent source.
Solution
(a) Express the controlling voltage of the dependent source in terms of the node voltages:
Apply KCL at node b to get
v = 9 − v
a b
9−vb vb 18−3vb
= A( 9 − vb) + ⇒ A=
= 0.02
100 200 200 9
(b) The power supplied by the dependent source is
( Av ) v ( )
( )( )
( − v b )
− a b =− 0.02 9 − 18 18 =
3.24 W

Example
The currents i1, i2 and i3 are the mesh currents corresponding to meshes 1, 2 and 3. The values of
these currents are
i1 = −1.375 A, i2 = −2.5 A and i3 = −3.25 A
Determine the values of the gains of the dependent sources, A and B.
Solution:
Express the controlling voltage and current of the dependent sources in terms of the mesh
currents:
v = 20 i − i = 20 −1.375 − − 2.5 = 22.5
and
( ) ( ( ) )
a 1 2
( )
ib = i3− i2
=−3.25 − − 2.5 =− 0.75 A
Express the current source currents in terms of the mesh currents:
and
i = − 2.5 A
2
( ) ( )
i3− i1 = Bib ⇒ −1.375 − − 2.5 = B −0.75 ⇒ B=
2.5 A/A
Apply KVL to the supermesh corresponding to the dependent current source
3 a b a
( ) ( ) ( )
0 = 20 i + Av + 50 i + v − 10 = 20 − 3.25 + A 22.5 + 50 − 0.75 + 22.5−10⇒ A=
4 V/V