2. DC ANALYSIS 16
2.10 DEPENDENT (CONTROLLED) ELEMENTS
Arbitrary dependent element - description
Two-terminal elements are normally characterized by an analytic function, linear or nonlinear, of one argument (2.1.4). Another category of elements can be distinguished, namely dependent or controlled elements. A controlled element is described by the following relationship:
(2.10.1) Then, such element is described by a family of characteristics, with X as the second parameter, so called control variable, which can be: temperature t, lightning flux , other voltage or other current . The two latter elements are called the controlled sources and they will be discussed in details.
Controlled sources - description
Controlled source is a source that provides a current or voltage that is dependent on other current or voltage elsewhere in the circuit.
Four types of controlled sources can be distinguished:
a) Voltage Controlled Voltage Source (VCVS), b) Current Controlled Voltage Source (CCVS), c) Voltage Controlled Current Source (VCCS), d) Current Controlled Current Source (CCCS).
Then, two branches are assigned to each controlled source: source branch and control variable branch, as depicted in Fig. 2.7.1a, b, c, d, for ideal (resistiveless) sources. These elements
where, are control coefficients, constants
characterizing corresponding sources.
Fig. 2.10.1 a) VCVS, b) CCVS, c) VCCS, d) CCCS
Families of relationships characterizing VCVS and CCCS are presented in Fig. 2.10.2a and d.
Fig. 2.8a) Family of relationships characterizing VCVS, b) CCCS
Fig. 2.10.2 Family of I−U relationships characterizing a) VCVS, d) CCCS
Use of controlled sources to element modeling
Controlled sources are used in modeling of circuit elements, such as transistor, operational amplifier or any other multi-terminal element.
Transistor
A transistor circuit symbol and the simplified model, for the common emitter mode of operation, are presented in Fig. 2.10.3. As can be seen, the CCCS is used. Then, after linearization of a diode characteristic (2.12.1c), i.e. its replacement by the voltage source , the transistor equations are:
(2.10.3)
J
G
CE BE C
B
U U I
I (2.10.3a)
U I
U I
Uf
B BE
B
B (1/R )U U /R
I f
B BE
B
C ( /R )U U /R
I f
a)
E
b)
E
c)
J
d)
J
a) I
U
d) I
U
Fig. 2.10.3 Transistor a) circuit symbol, b) simplified model
For the assumed model, the transistor is characterized by the following conductance matrix and short-circuit current vector:
(2.10.3a)
and then, practically by two parameters ( for a silicon transistor V):
current gain ,
Base resistance .
It should be observed, that resistance matrix R does not exist.
Operational amplifier
An operational amplifier (op-amp) circuit symbol and the model are presented in Fig. 2.10.4.
As can be seen, the VCVS is used.
Fig. 2.10.4 Op-amp a) circuit symbol, b) model
For the assumed model, an op-amp is characterized by equations (2.10.4), i.e. by two resistances and one control coefficient.
in
Arbitrary three-terminal or two-port element
An arbitrary linear active three-terminal element (Fig. 2.9.2) or two-port (Fig. 2.9.5) can be described by equations (2.9.20). These equations are KVL equations and equivalent circuit built of two-terminal elements can be constructed, as presented in Fig. 2.10.5. For a two-port element, connection denoted by the bold line should be removed.
Fig. 2.10.5 Model of three-terminal or two-port element Analysis of circuits containing controlled sources
If a circuit contains controlled source(s), then such circuit nodal equations should be supplemented by equation(s) of controlled source(s), with controlling variable(s) expressed by nodal voltages.
Example 2.10.1
Find nodal equations of the circuit presented in Fig. 2.10.6.
Fig. 2.10.6 Circuit for Example 2.10.1
For the assumed , the circuit nodal equations are:
(2.10.5)
Equations of the controlled sources, with controlling variables expressed by nodal voltages are:
(2.10.6)
After setting equations (2.10.6) into (2.10.5) and reordering, the following system is obtained:
(2.10.7)
It should be observed, that the conductance matrix is not symmetrical, . This is due to the presence of the controlled sources.
The next example illustrates a strategy of determination of the Thevenin equivalent when two-terminal circuit contains dependent sources.
Example 2.10.2
Find the circuit relationship and then the Thevenin equivalent.
Fig. 2.10.7 Circuit for Example 2.10.2
By attaching a fictitious external current I, KVL can be formulated
(2.10.8a) From KCL
(2.10.8b) and the circuit relationship is
(2.10.9) Then, the Thevenin equivalent parameters are:
(2.10.10)
The next example illustrates the proper use of superposition when there are dependent sources present in a circuit.
B
Example 2.10.3
Two subcircuits are separated by an ideal voltage source as shown. Find . The circuit
parameters are: .
Fig. 2.10.8 Circuit for Example 2.10.3 and its superposition components
÷
An ideal voltage source isolates two subcircuits, however the substitution theorem can not be applied as the controlling variable and the dependent source are located in different subcircuits.
When superposition is applied, then only independent sources give the superposition components. Thus, the example circuit can be divided into two subcircuits, as shown. The computed components of the current are:
, . (2.10.11)
Then, the total current is:
Drill problems 2.10
1. Find model of a three-terminal/two-port element, if the element is described by the conductance matrix and short-current vector.
2. Draw characteristic or a practical controlled source: a) VCCS, b) VCVS, c) CCCS, d) CCVS, characterized by the following parameters: [X/X], for the controlling variable of 5 [X], where X means V or A.
IE
3. Model a passive two-port characterized by the following resistances:
, . Use controlled sources and resistors.
4. Find the power absorbed by the load resistance .
Fig. P.2.10.4 5. Find the mesh current.
Fig. P.2.10.5
6. Find the collector resistance that gives . Assume a diode voltage drop of . The supply voltages are and the Base resistance is
.
Fig. P.2.10.6
7. A practical source of is connected across the input terminals of an op-amp and the load resistance of is connected between the output and ground.
0.5 , 12 21 2
11 R R
R R22 4
1 Rl
RC UCE 5V
V 7 .
0 EB 1.7V,EC 10V
100k RB
1mV, t 5
o R
E
1k Rl
I
4 V 2I
C
B
E
Determine the load voltage . Use the idealized op-amp model, Rin ,Rout 0, with
open-loop gain of .
Fig. P.2.10.7 Ul
V/V 105
k
1
2
3 4
+