terms as 1+jAv/v 1 Band1+2zAjv/v 2 B+Ajv/v 2 B^2 become unity as vapproaches zero,
at very low frequencies the sinusoidal transfer function G(jv)can be written
In many practical systems,lequals 0, 1, or 2.
1.For l=0, or type 0 systems,
or
The low-frequency asymptote is a horizontal line at 20 logKdB. The value of K
can thus be found from this horizontal asymptote.
2.For l=1, or type 1 systems,
or
which indicates that the low-frequency asymptote has the slope –20dBdecade.
The frequency at which the low-frequency asymptote (or its extension) intersects
the 0-dB line is numerically equal to K.
3.For l=2, or type 2 systems,
or
The slope of the low-frequency asymptote is –40dBdecade. The frequency at
which this asymptote (or its extension) intersects the 0-dB line is numerically equal
to
Examples of log-magnitude curves for type 0, type 1, and type 2 systems are shown
in Figure 7–87, together with the frequency to which the gain Kis related.
The experimentally obtained phase-angle curve provides a means of checking the
transfer function obtained from the log-magnitude curve. For a minimum-phase system,
the experimental phase-angle curve should agree reasonably well with the theoretical
phase-angle curve obtained from the transfer function just determined. These two phase-
angle curves should agree exactly in both the very low and very high frequency ranges.
If the experimentally obtained phase angle at very high frequencies (compared with the
corner frequencies) is not equal to –90°(q-p), where pandqare the degrees of the nu-
merator and denominator polynomials of the transfer function, respectively, then the
transfer function must be a nonminimum-phase transfer function.
1 K.
20 log @G(jv)@ = 20 logK- 40 logv, for v 1
G(jv)=
K
(jv)^2
, for v 1
20 log @G(jv)@ = 20 logK- 20 logv, for v 1
G(jv)=
K
jv
, for v 1
20 log @G(jv)@ = 20 logK, for v 1
G(jv)=K, for v 1
vlimS 0 G(jv)=
K
(jv)l
Section 7–9 / Experimental Determination of Transfer Functions 487