Page 15 - Servo Motors and Industrial Control Theory

1.6 First Order Transfer Function 7

The above analysis shows that by taking Laplace Transform from the differential
equations, they become algebraic equation. After re-arrangement using partial frac-
tion method and referring to the Table 1.1 the solution can be obtained. Fortunately,
there is no need to solve complicated Laplace Transforms. There are certain pa-
rameters that the performance of a system for various input signal can be obtained.
The closed loop control of velocity and angular position and the effect of exter-
nal torque will be discussed in different chapters.

1.5 Transfer Function

Taking Laplace Transform from both sides of Eq. (1.3) and assuming zero initial
conditions yield

(as + as n−1 + + ay (b s + b s m−1 + + b ) ()xs (1.24)
) ()s =
n
m
n n−1 m m−1
which can be written in the form of
m m−1
ys() = bs + b m−1 s + + b (1.25)
m
n
xs() as + as n−1 + + a
n−1
n
The right hand side of Eq. (1.25) is called the transfer function. a …, b … are
m
m
constants and y( s), x( s) are called the output and input variables. Equation (1.25)
can be of any form but normally for real system n > m and n is called the order of
transfer function.
The principle of superposition may be used for simple multivariable systems.
Once the transfer function is obtained, the following performance must be
studied.
1. Stability
2. Transient response
3. Steady state error for various standard input
4. The above analysis should be carried out for various input functions
5. Frequency response
There are some standard transfer functions that can be solved and exact solution
may be obtained. In the following, some standard transfer function is studied.

1.6 First Order Transfer Function

First order transfer function in standard form may be written as
ys = A (1.26)
()
xs s τ + 1
()