The second order leading element
The general form of the differential equation is given by
(3.119)
and can be represented in the operational form as
(3.120)
or
, (3.121)
where T=T2, 
, 
.
We can obtain the equation for the time response from Eq.(3.139) as follows [7]
, (3.122)
where 
is the 
-function ( the unit impulse function), that is the derivative from the unit step function.
The transfer function can be represented as follows
(3.123)
Then the transfer function in the frequency domain is
(3.124)
The following equations describe the frequency response
, (3.125)
, (3.126)
(3.127)
, 
. (3.128)
The transient response to a step unit input is shown in Fig. 3.34(a). The log-magnitude and phase angle diagram is drawn in Fig. 3.34(b).
Fig. 3.34. Transient response to a step unit input (a) and log-magnitude and phase diagram (b) for the second order leading element.
The second order leading element is represented by the block diagram as shown in Fig. 3.35.
Fig.3.35. Block diagram of the second order leading element.
For 
we can rewrite Eq. (3.143) as
.
The transfer function in the frequency domain is
.
The equations for the frequency response can be written as follows
, 
,
(3.129)
, 
. (3.130)
The transient response for this case is shown in Fig.3.36(a). The graphs of log-magnitude and phase versus 
for k<1 are shown in Fig.3.36(b).
Fig. 3.36. Transient response to a step unit input (a) and log-magnitude and phase diagram (b) of the second order leading element for .
Terms and Concepts
Bode plot (logarithmic plot).The logarithm of the magnitude of the transfer function is plotted versus the logarithm of , the frequency. The phase, 
, of the transfer function is separately plotted versus the logarithm of the frequency.
Break frequency.The frequency at which the asymptotic approximation frequency response for a pole (or zero) changes of the slope.
Decibel, dB. The units of the logarithmic gain.
Fourier transform.The transformation of a function of time, f(t), into the frequency domain.
Frequency response.The steady-state response of a system to a sinusoidal input signal.
Logarithmic magnitude. The logarithm of the magnitude of the transfer function, 
.
Polar plot.A plot of the real part of 
versus the imaginary part of .
Steady-state error. The error when the time period is large and the transient response has decayed, leaving the continuous response.
Test input signal.An input signal used as a standard test of a system’s ability to respond adequately.
Transient response.The response of a system as a function of time.
Transfer function in the frequency domain.The ratio of output to the input signal where the input is sinusoid. It is expressed as .