Image Impedance - Transfer Function

Transfer Function

The transfer function of the half section, like the image impedance, is calculated for a network terminated in its image impedances (or equivalently, for a single section in an infinitely long chain of identical sections) and is given by,

where γ is called the transmission function, propagation function or transmission parameter and is given by,

The term represents the voltage ratio that would be observed if the maximum available power was transferred from the source to the load. It would be possible to absorb this term into the definition of γ, and in some treatments this approach is taken. In the case of a network with symmetrical image impedances, such as a chain of an even number of identical L sections, the expression reduces to,

In general, γ is a complex number such that,

The real part of γ, represents an attenuation parameter, α in nepers and the imaginary part represents a phase change parameter, β in radians. The transmission parameters for a chain of n half sections, provided that like impedance always faces like, is given by;

As with the image impedance, the transmission parameters approach those of a transmission line as the filter section become infinitesimally small so that,

with α, β, γ, Z and Y all now being measured per metre instead of per half section.

Image filter sections
Unbalanced
L Half section T Section Π Section
Ladder network
Balanced
C Half-section H Section Box Section
Ladder network
X Section (mid-T-Derived) X Section (mid-Π-Derived)
N.B. Textbooks and design drawings usually show the unbalanced implementations, but in telecoms it is often required to convert the design to the balanced implementation when used with balanced lines.

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