To obtain the symmetric APF, let R1 = R; R5 = 2R4; and R6 = R5. The desired transfer function is then:

The purpose of all-pass filters is to generate a phase shift with frequency with no attenuation; the phase of the APF described earlier can be shown to be:

7.2.4 Generalized Impedance Converter AFs

Often one wishes to design AFs that work at very low frequencies, e.g., below 20 Hz. Such filters can find application in conditioning low-frequency physiological signals such as the ECG, EEG, and heart sounds. If a conventional biquad AF is used to make a high-pass filter whose fn = 1 Hz, for example, it is necessary to obtain very large capacitors that are expensive and take up excessive volume on a PC board. For example, the capacitor required for fn = 1 Hz and R3 = 100 kΩ is:

The GIC circuit, shown in general format in Figure 7.8, allows the size of a small capacitor, e.g., 0.001 μF, to be magnified electronically, transformed

into a large, equivalent, very high-Q inductor, or made into a D-element that can be shown to have the impedance, ZD(ω) = 1/(ω2 D), where D is deter-

mined by certain Rs and Cs as shown next.

Find an expression for the driving point impedance of the general GIC circuit, Z11 = V1/I1. Clearly, from Ohm’s law:

© 2004 by CRC Press LLC

I1

Z1

V1

V2 IOA

Z2 I2

V1

Z3 I3

IOA V3

Z4 I4

V1

Z5 I5

FIGURE 7.8

General architecture for a generalized impedance converter (GIC) circuit. The Zk can be resistances or capacitances (1/jωCk), depending on the filter requirement.

Also, from the ideal op amp assumption and Ohm’s law, it is possible to write the currents:

From the Equation 7.40, Equation 7.41, and Equation 7.42, it is easy to show:

and the GIC driving point impedance is given compactly as:

© 2004 by CRC Press LLC

FIGURE 7.9

Magnitude and phase of Z11(f) looking into a GIC emulation of a 0.1-HY inductor. In this MicroCap simulation, TL072 op amps were used. Smooth line is magnitude.

Now if Z2 or Z4 = 1/jωC (a capacitor) and the other elements are resistors, Z11 has the form:

that is, the GIC emulates a low-loss inductor over a wide range of frequency. The inductance of the emulated inductor is:

Figure 7.9 illustrates the magnitude and phase response of an inductive Z11 vs. frequency. Using MicroCap‘, 20 log Z11(f ) and – Z11(f ) vs. f are

plotted. The circuit used in this simulation is the same as in Figure 7.8, with Z1 = Z3 = Z4 = Z5 = 1 kΩ and Z2 = 1/jωC2; C2 = 0.1 μF.

Equation 7.47 shows that the simulated inductance is 0.1 Hy. (Op amp output current saturation effects are not included in this simple simulation.) Note that Z11(f ) increases linearly with frequency until about 180 kHz, at which its phase abruptly goes from the ideal +90 to –270∞, and 20 log Z11(f ) exhibits a tall peak at approximately 1 MHz. Clearly, the circuit model emulates a 0.1-Hy inductor below 180 kHz. A simple nonsaturating, two-time- constant model of the TL082 op amp was used in the MicroCap simulation.

Because the GIC inductive Z11 is referenced to ground, it must be used in active filters that require inductors with one end tied to ground. Figure 7.10(A) illustrates an inductive GIC used in a simple high-Q, quadratic BPF.

© 2004 by CRC Press LLC

−

A

B

FIGURE 7.10

(A) Circuit of an RLC band-pass filter using the GIC inductor. The GIC circuit allows emulation of a very large, high-Q inductor over a wide range of frequencies and is particularly well suited for making filters in the subaudio range of frequencies. (B) The actual BPF.

Figure 7.10(B) illustrates a simple R–L–C BPF suitable for circuit analysis. A node equation gives the BPF’s transfer function:

© 2004 by CRC Press LLC

FIGURE 7.11

Circuit showing how a GIC “D” element can make a low-frequency, low-pass filter.

The GIC “D element” Z11 is a frequency-dependent negative resistance (FDNR) that can be made by putting capacitors in the Z1 and Z3 positions in the GIC circuit. Thus, the driving point impedance is real and negative:

Figure 7.11 shows that the D element can make a quadratic LPF. Write the node equation for Vo in terms of the Laplace complex variable, s:

By using relation Equation 7.44 for V2/VS, it is possible to realize notch and all-pass filters from the basic GIC circuit. Figure 7.12 and Figure 7.13 illustrate examples of these filters. Readers can develop their transfer functions as exercises.

© 2004 by CRC Press LLC