Problema 54-5
Source: Adapted from Example 8-25 - page 421 - THOMAS, Roland E. ,
ROSA, Albert J. , THOUSSAINT, Gregory J. - Book: The Analysis & Design of Linear Circuits
- 6ª Edição - Ed. John Willey & Sons, Inc. - 2009.
Use the current loop method and find the output voltage V2 and the
input impedance Zin of the circuit shown in the Figure 54-5.1.
Figure 54-5.1
Solution of the Problem 54-5
In this problem there is a need to find three variables, that is, Vx , IA and IB.
Directly from the circuit, we can see the following relation:
Vx = 400 (IA - IB )
eq. 54-05.1
Note that this is the voltage over the 400 ohm resistor. On the other hand, we can make the two loops that
appear on the circuit indicated by the blue arrows. Starting with the mesh B.
-400 IA + (450 + j500) IB = - 2 Vx
eq. 54-05.2
By substituting the value of Vx given by eq. 54-05.1 in eq. 54-05.2, it is possible to obtain
a new relationship
which reveals the interdependence between the variables in question. This algebraic process is fundamental to simplify the
problem solving process. See the result in eq. 54-05.3.
400 IA + (- 350 + j500) IB = 0
eq. 54-05.3
And for the mesh A:
(600 + j500) IA - 400 IB = 10 ∠ 0
Soon, a system of two equations was built to two unknowns that can be solved with a scientific calculator or a specific program
like "Octave". Using Octave, the following values are found:
IA = 12.90 - j9.37 = 15.94 ∠-36° mA
And the value of IB:
IB = - 2.00 + j7.92 = 8.17 ∠104.17° mA
Now, we can calculate the value of Vx given in eq. 54-05.1:
Vx = 400 (14,9 - j17,29) = 5.960 - j 6.916 mV
We will express this voltage in volts, or:
Vx = 5,96 - j6,916 = 9,13 ∠-49,24° V
To calculate the value of V2, we must make the outer mesh on the right, and get
to the following expression:
V2 = 50 IB + 2 Vx
By making the numerical substitution and making the calculation, we will find V2 the value of:
V2 = 11,82 - j13,43 = 17,89 ∠-48,65° V
Now, lets to calculate the value of Zin which is the impedance that the circuit offers to the voltage source.
This impedance can be expressed by the quotient between the value of the
voltage source and current value IA. Thus, performing the calculation we have:
Zin = V / IA = 627.35 ∠36° = 507 + j368 Ω
What does means the value of Zin ?
For the voltage source, it means that you can have the entire circuit shown in the figure above connected
to its terminals, or simply replace the circuit with a resistance equal to 507 ohms in series with
an inductor with an impedance of 368 ohms. For the voltage source this is exactly the same.
