Page 52 - Understanding Automotive Electronics

2735 | CH 2 Page 39 Tuesday, March 10, 1998 10:55 AM

THE SYSTEMS APPROACH TO CONTROL AND INSTRUMENTATION 2

The input fluctuates through a complete cycle beginning at zero, increasing
positively to a maximum, then moving symmetrically negatively back to zero in T
seconds. It then repeats periodically in intervals of T seconds. The frequency f of
the input is the number of cycles in one second, or
f = 1/T
In mathematical terms, a sinusoidal input x (t) is given by the following equation:
x (t ) = X sin(2πft )

where
X = amplitude of the input
f = frequency
t = time
The output of an electronic system from a sinusoidal input is also
sinusoidal at the same frequency, as depicted by the graphs in Figure 2.7, in
which the solid curve is the input and the dashed curve the output.
The frequency response of an electronic system is defined as the ratio of the
peak amplitude of the output (denoted Y) to the peak amplitude of the input
(denoted X ). Notice that there is a time delay in the response of the system
relative to the input, corresponding to the difference in zero crossing of the
input and output.
In formal mathematical terms, the output y(t) is given by
y (t) = Ysin[2πf (t – d )]

where
Y = amplitude of the output
d = time delay
The system frequency response, denoted H, is formally defined as follows:
H = Y /X

The delay is expressed as a fraction of the period, called the phase, and is usually
denoted by the Greek letter phi (φ):
φ = 2π d/T

= 2πfd
For an electronic system that is ideal with respect to its dynamic
performance, the frequency response is a constant for all frequencies. In
practice, the frequency response varies with frequency. A graph of the
frequency response of a typical electronic system is shown in Figure 2.8.
In this example, the frequency response is close to ideal (i.e., constant) over
a range of frequencies from frequency f to f . For frequencies below f or above f ,
1
2
2
1
the response drops off. In the example of a hi-fi amplifier, f might be about 20
1
Hz and f might be about 20 kHz. The range of frequencies over which the
2
frequency response is nearly constant is called the system bandwidth. The
UNDERSTANDING AUTOMOTIVE ELECTRONICS 39