Lecture 7: Linear Dependent Sources
Today we will look at special voltage and current sources called
dependent sources.
A dependent source has a voltage or current that depends on a
different voltage or current somewhere in the circuit (or even in
a different, detached circuit).
R1
+
VS
R2
VX
_
G VX
R3
Four Types of Linear Dependent Sources
Constant of proportionality
Parameter being sensed
(controlling voltage/current)
Output
Voltage-controlled voltage source … V = Av Vc
Current-controlled voltage source … V = Rm Ic
Current-controlled current source … I = Ai Ic
Voltage-controlled current source … I = Gm Vc
+
_
Av Vc
+
_
Rm Ic
Ai Ic
Gm Vc
Analyzing Circuits with Linear Dependent
Sources




Circuits with linear dependent sources can be
analyzed using the tools we have learned so far.
A dependent voltage source acts like an
independent ideal voltage source: it tells us
what its voltage is, but the current is unknown.
Similarly, a dependent current source tells us
what its current is, but the voltage is unknown.
We just need to write an extra equation that
specifies what the controlling voltage or
current is.
Example
Find VR.
R1
+
VS
R2
VX
G VX
_


By Ohm’s law, VR = (R3)(G VX)
Extra equation specifying VX:
R3
+
VR
_
R1
VX  VS
R1  R 2
Nodal Analysis with Dependent Sources
V
R
a
IS
V
1
I
2
R
b
R
3
2
Va  Vb
 IS  0
R1
R
4
+
_
R
I
m 2
Vb  Va Vb Vb  Rm I2


0
R1
R2
R3
Vb
I2 
R2
Floating Dependent Voltage Sources
R1
Rm I2
Va
Vb
| +
V1
+

R3
R2
R4
I2
Va  V1 Va Vb


0
R1
R2 R 4
Va
I2 
R2
Vb  Va  Rm I2
Linearity of I-V Relationship

Whenever a circuit is composed only of the elements
we have studied so far,
 Ideal
Independent Voltage and Current Sources
 Linear Dependent Voltage and Current Sources
 Resistors
the I-V relationship is always a line. Simple examples:
i
i
i
+
_
v
v
v
Equivalent Circuits
Consider the simple circuit composed of a
voltage source and resistor.
I
R
I

+
VS
VS
v
V
VS/R
_


This circuit has a linear I-V relationship:
I = (V – VS) / R
I = (1/R) V – VS / R
With proper choice of VS and R, this circuit can
mimic any other circuit we have studied so far.