Page 251 - Analog and Digital Filter Design

248 Analog and Digital Filter Design

R
Figure 9.3 - =;=” R,
First-Order All-Pass I
Design

The values of the capacitor and inductor are given by the following equations:

2R
C=-- 2 L=-
o.R o
Where L is a center-tapped inductor, each half-winding = Ll4.

The equations for the equalizer assume that the pole location has been denor-
malized by scaling it for the required frequency. The frequency is the same
as the passband cutoff of the filter being equalized. In the case of quadrature
networks, which will be described in this chapter, it is the passband center
frequency.
The action of the first-order equalizer can be explained by considering its
behavior at very high and very low frequencies, with reference to Figure 9.3.
Let us consider the input to be at the left-hand side and the output to be at
the right-hand side. At low frequencies, the inductor’s reactance is high and the
capacitor’s reactance is low. The inductor is effectively a short circuit and the
capacitor an open circuit, so the output signal will be in phase with the input
signal.

At high frequencies, the inductor’s reactance is high and the capacitor’s reac-
tance is low. Now the capacitor is effectively a short circuit and the inductor can
be considered a transformer with the center tap grounded. Because the start of
the “primary” winding goes to the input and the “finish” of the secondary goes
to the output, the output is anti-phase with the input.

The symbol for first-order equalizers that is often given in textbooks is
shown in the left-hand side of Figure 9.4. This diagram does not convey (to me,
at least!) the true nature of the circuit. It actually represents a balanced circuit:
the broken lines depicting a mirror image of the components shown. The full
circuit diagram shown on the right-hand side reveals that it is actually a bridge
circuit.