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para_Z.png
Figure 31-01
    eq.   31-02

    Since the parameters Z are obtained by opening the input or output port circuitry, they are known as Open Circuit Impedance Parameters and its unit of measure is OHM.

    We can define the parameters as:

    a) Z11 ⇒ Open circuit input impedance.

    b) Z12 ⇒ Open circuit transfer impedance from Port 1 to Port 2.

    c) Z21 ⇒ Open circuit transfer impedance from Port 2 to Port 1.

    d) Z22 ⇒ Open circuit output impedance.


    2.   Calculation of the Parameters Z

    We calculate Z11 and Z21 connecting a voltage source V1 or a current source I1to Port 1 while we left Port 2 as an open circuit, that is,I2 = 0.Thus, knowing V1 , I1 and V2 we can determine these parameters.

    Z11 = V1 / I1        and       Z21 = V2 / I1
    eq.   31-03

    In the same way, we calculate Z12 and Z22 connecting a voltage source V2 or a current source I2 to Port 2     while we left Port 1 as an open circuit,     that is, I1 = 0.

    Z22 = V2 / I2        and       Z12 = V1 / I2
    eq.   31-04

    When Z11 = Z22 we say that the circuit is SYMMETRIC. This it means that we can divide it into two similar halves.

    When Z12 = Z21, we say that the circuit is PASSIVE or RECIPROCAL. This means that the quadripole is linear and has no voltage sources DEPENDENTS.

    See the Figure 31-02 a model of an equivalent circuit T, only valid for circuits Reciprocal or Passive.

    Note that if we want to calculate Z11 we have to put a source of voltage on Port 1, in this case, represented by V1 and leave open the Port 2, that is, I2 = 0. This means that we do not have the impedance Z22 - Z12 (represented in the color "pink") in the circuit. Then we have to add the impedances represented by the "blue" and "yellow" part and this neutralizes the impedance Z12 left alone Z11.

    The same reasoning can be applied to calculate the other impedances. Thus, our model achieves the objectives.

quad31-1K.jpg
Figure 31-02

    If the circuit is not Reciprocal or Passive, that is, have dependent sources, then we have to modify the model. See the Figure 31-03 for an equivalent circuit model for general cases.

quad31-1M.jpg
Figure 31-03

    Notice that this circuit is derived directly from the equations of the Parameters Z given at the top of the page.

    It should be noted that not always the Parameters Z can be described by the equations. As an example, we have the ideal transformers.



    3.   Equivalence between Y and Z Parameters

    There is a relation between the parameters Z and Y given by the equations:


    Δ Y = Y11 Y22 - Y12 Y21
    eq.   31-05
    Z11 =  Y22 / Δ Y
    eq.   31-06
    Z12 = - Y12 / Δ Y
    eq.   31-07
    Z21 = - Y21 / Δ Y
    eq.   31-08
    Z22 =  Y11 / Δ Y
    eq.   31-09