Frequency characteristics of simplest circuits.

Frequency characteristics of first-order circuits

 

Consider the simplest rC-circuit (Fig.5.5.a).Let us define the complex voltage transfer ratio:

 

(5.26)

 

Fig.5.5

 

Пропущены формулы и текст  и рисунки (со стр. 130 )

 

Fig.5.6

 

Fig.5.7

 

        If τ is denoted as time constant: τ = l/r, than

                                             (5.27)

 

  The expression (5.84) coincides with (5.71) for the rC-circuit in Fig.5.5.a. Consequently, the frequency characteristics of the circuits in Fig.5.8.a and 5.5.a are identical.

  The analysis shows that the frequency characteristics on the circuits in Fig.5.8.b and 5.5.b are also identical.

 

  5.3.2. Frequency characteristics of second- and third-order circuits

Consider the second-order circuit in Fig.5.9.The matrix of node conductances for this circuit is as follows:

 

                                            (5.28)

 

Fig. 5.8                                                                                    Fig.5.9

 

 Hence, the algebraic adjuncts D₁₃ and  D₁₁ are:

Пропущена формула

Here M₁₃ and M₁₁ are the minor-determinants obtained from the matrix determinant (5.28) by deleting the first line and third column and the first line and first column, respectively, that is

 

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The voltage transfer ratio in (5.53) is:

                                                                                                                                                           (5.29)

Here

                                 (5.30)

                                       (5.31)

where τ = rC is  a time constant.

The diagrams of AFC and APFC are given in Fig.5.10.a and Fig. 5.10.b respectively.

 

Fig. 5.10

       Consider the second-order circuit in Fig. 5.11 (the Wien bridge).The matrix of node conductances for the circuit is as follows:

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                                 Fig.5.11                                                  Fig. 5.12

Hence, the algebraic adjuncts D₁₂ and D₁₁ are:

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The voltage transfer ratio in will be

                                   (5.32)

 

Here

                  (5.33)

                                   (5.34)

 

When comparing (5.33) and (5.34)with (5.30), (5.31), we see that these expressions are identical and, consequently, the circuit frequency characteristics in Fig.5.9 and 5.11 coincide

 Consider the third-order circuit in Fig. 5.2 (double T-form bridge).

The matrix of node conductances for this circuit is as follows:

Пропущена формула

 

Hence, the algebraic adjuncts D₁₄, D₁₁ are:

Пропущеныформулы

The voltage transfer ratio equals

 пропущена формула

 

  Here

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Phase angle equal

 

The diagrams of AFC and APHFC are represented in Fig. 5.13.a and Fig. 5.13.b.

 

Fig.5.13

 

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