gains

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gains

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Steady State Gain

Given the transfer function:

gain1

the Steady State Gain is defined as:

gain4

Note that the steady state gain can be zero or infinite depending on the element values of the numerator and denominator!

Root Locus Gain

Given an system described by the transfer function:

gain1

This transfer function can be rewritten in pole zero notation with

gain2

where pi .. p1 are the poles and zi .. z1 are the zeros of the system. The gain KRL is known as the Root Locus Gain. Note that it can easily be derived from the transfer function as:

gain3

System Gain

Given the transfer function:

gain1

If n0 and d0 are unequal to zero, this transfer function can be rewritten in pole zero notation with:

gain5

The gain KS is known as the System Gain. If n0 is zero and n1 is nonzero an equivalent notation can be found with an extra s multiplied:

gain5a

If more numerator element are zero, extra multiplications with s are added. The same goes for denominator elements equal to zero. In general the System Gain can be derived from the transfer function as:

gain6

Relating the Gains

The elements of the System Gain are related to the poles and zeros of the Root Locus Gain as:

gain7

If z0 and p0 are unequal to zero, the following equation holds:

gain8

The Root Locus Gain and the Steady State Gain are related as:

gain9