Linear Network Theory by K. F. Sander and P. Hammond (Auth.)

Show description

5. Cascade connection order, n, have elements utj, vij9 the product is defined as the matrix having the element w^ where w,ij — ZJ U ik V (25) kj When n = 2 this may be explicitly written Γ^ιι w 1 2 l [_w21 w 22 J = ΓΜ11Ι)11 + Μ12Ι;21 [u2ivli + u22v2i ullv12 + u21vi2 + ul2v22l u22v22] Using this multiplication rule it is easily shown that [A, Β,Ι \Α2 B2~][AiA2+BiC2 Α,Β^Β,Ώ^ [C, D J \C2D2\ [C1A2 + DXC2C1B2 + D1D2\ o ^ v " V A^ - yL,) z2 FIG. 6. T-network This method enables the cascade matrix for a circuit to be evaluated in terms of the cascade matrices of simpler circuits.

The reason for the negative signs in equations (10) is merely that, as will be seen, in a cascade R -ΛΛΛΛ- (a) (b) FIG. 3. Simple 2-ports connection —12 is the output current, and is equal to the input current for the next unit. The matrix formed by the coefficients of equations (11) is used in connection with transistor equivalent circuits, where the co­ efficients happen to be easily measured quantities. CLASSIFICATION AND DESCRIPTION OF NETWORKS 41 The elements of any one of these matrices can be expressed in terms of those of any other, providing the equations are soluble.

This yields the algebraic equation Ur=jLr is known as the impedance of Lr at frequency ù/2ð. The equations expressing the constraints imposed by the net­ work are typified by such equations as equation (3) of Chapter 1. E 4 i . ,n (6) If /s has a sinusoidal waveform and is represented by the complex number Is these equations become i^rv(/se>f + /*e-^) = 0 s=l If this is true for all time ÓÁ„É,= s= 1 ÓÁ ç É? =0 s= 1 The two sums are complex conjugates since the coefficients Ars are real, and hence equations (6) may be replaced by ÓË,/, = 0 FOURIER T R A N S F O R M (7) METHOD A method which gives the same result but which introduces the idea of transformation rather than replacement is to use a simple case of the Fourier transform.