Let's study the last set of parameters that are called
Transmission Parameters. In the Figure 35-01 we see the various variables
involved in determining of the Transmission Parameters.
Figure 35-01
The Transmission Parameters are also called ABCD Parameters and their
equations are given by:
V1 = A V2 - B I2
eq. 35-01
I1 = C V2 - D I2
eq. 35-02
These parameters are widely used in the analysis of circuits connected in
cascade.
We can define the ABCD parameters as:
a)A ⇒ Open circuit voltage ratio.
b)B ⇒ Negative short-circuit transfer impedance.
c)C ⇒ Admittance of open circuit transfer.
d)D ⇒ Negative ratio of the short-circuit current.
2. Calculation of ABCD Parameters
Let's determine the ABCD Parameters of the circuit shown in the Figure 35-02. At the
problem 31-3 we analyze this circuit for the calculation of the Z parameters.
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Figure 35-02
To determine the parameters A and C we must do I2 = 0,
that is, let the output port open. In the input port we can place
a voltage source or a current source. We chose to place a current source
of value I1, as shown in the Figure 35-03.
Figure 35-03
From the circuit, as we use I2 = 0, then there will be no voltage drop on the resistor
of 10 ohms. It is also possible to notice that in the 20 ohms resistor the current that circulate
for it is a I1. Then, by making the mesh in the direction indicated by the arrow red,
in the figure above, V2 is:
V2 = 4 I1 + 20 I1 = 24 I1
On the other hand, by making the mesh in the direction of the violet arrow, in the figure above,
the current I1 will circulate through the 5 and 20 ohms resistors.
Therefore, the input voltage V1 will be equal to:
V1 = 5 I1 + 20 I1 = 25 I1
With this data, it is possible to calculate the values of A and C.
A = V1 / V2 = 25 / 24
C = I1 / V2 = 1 /24
To calculate the values of B and D, we must do V2 = 0,
that is, the output port must be short-circuited. See in the Figure 35-04 the new configuration of the circuit.
Figure 35-04
Notice that if we make the right mesh, in the direction indicated by the green arrow,
we can find a relationship between I1 and I2. Then:
4 I1 + 10 I2 + 20 I1 + 20 I2 = 0
Working algebraically this equation, we find:
I1 = - (5/4) I2
But, from the equation of the quadripole we know that:
D = - (I1/ I2)
Comparing the last two equations, we conclude that the value of D is:
D = 5/4
On the other hand, making the mesh in the left of the circuit, in the direction of the arrow
brown, we can find the value of B. Then:
- V1 + 5 I1 + 20 I1 + 20 I2 = 0
Remember that I1 = - (5/4) I2. So, replacing in the equation above
and working algebrically the expression, we find:
V1 = - (45/4) I2
But, from the equation of the quadripole we know that:
B = - (V1/ I2)
In this way, we conclude that:
B = 45/4
Therefore, we determine all the ABCD parameters of the quadripole. To conclude,
let's write the equations of this quadripole.
V1 = (25/24) V2 - (45/4) I2
I1 = (1/24) V2 - (5/4) I2
Alternative Way to Calculate Parameters
It is possible to use an alternative way to calculate the quadrupole parameters knowing that there is a relationship between the different parameters.
So, let's start by writing the quadrupole mesh equations. Starting with port 1.
V1 = 25 I1 + 20 I2
And for port 2, we have:
V2 = 4 I1 + 10 I2 + 20 I1 = 24 I1 + 30 I2
Note that these equations are exactly the equations of the Z parameters of the quadrupole, where we have Z11 = 25, Z12 = 20,
Z21 = 24 and Z22 = 30. In possession of these values and using the
Table 30-01 we can transform the Z parameters into
ABCD parameters. Therefore, let's calculate the value of Δ Z.
Δ Z = h11 h22 - h12 h21 = 25 x 30 - 20 x 24 = 270
Now let's calculate the ABCD parameters in accordance with Table 30-01.
