After reading this topic First order control system, you will understand the theory, the open and close loop transfer function, Pole zero map, examples and block diagram.
In a system whose transfer function having the highest power of s equal to 1 in its denominator, is called the first order control system.
Closed-loop transfer function and block diagram
A block diagram of first order closed – loop control system with unity negative feedback is shown below in Figure 1,
Figure 1 First order control system block diagram. www.lightgreen-buffalo-211679.hostingersite.com
It is given unity negative feedback $H(s)$ so,
\[H(s) = 1\]
The open loop transfer function $G(s)H(s)$ of the first-order control system can be written as
\[G(s)H(s) = \frac{1}{{sT}}.1\]
So, the close loop transfer function of first order control system can be written as
\[\frac{{C(s)}}{{R(s)}} = \frac{{G(s)}}{{1 + G(s)H(s)}} = \frac{{1/sT}}{{1 + (1/sT)}} = \frac{1}{{1 + sT}}….(1)\]
Equation 1 describe the standard form of a first – order system.
Pole-zero map
Using Equation 1, the Pole-zero map of a first-order system is shown below in Figure 2.
Figure 2 First order Control System Pole zero map. www.lightgreen-buffalo-211679.hostingersite.com
Examples
Thermometer, thermal system, overhead tank, gas cylinder, series RL network, series RC network
Consider the series RC network which is an example of a first – order system, as shown below in Figure 3.
Using KVL gives,
\[{V_i}(s) = I(s)\left( {R + \frac{1}{{sC}}} \right)\]
or,
\[{V_i}(s) = I(s)\left( {\frac{{1 + sCR}}{{sC}}} \right)….(2)\]
Output voltage ${V_o}(s)$ can be written as,
\[{V_o}(s) = I(s)\left( {\frac{1}{{sC}}} \right)….(3)\]
\[{\text{Using Equation 2 and Equation 3 gives,}}\]
\[\frac{{{V_o}(s)}}{{{V_i}(s)}} = \frac{1}{{1 + sCR}}….(4)\]
Comparing Equation 1 and Equation 4 gives,
\[T = RC\]
where T is the time constant of series RC network.