Differentiator

11.5.1. Basic Circuit

If the resistor and capacitor of the integrator in Fig. 11.7 are interchanged, we obtain the differentiator shown in Fig. 11.14. The application of KCL to the summing point yields the relationship

11.7.1. Logarithm

A logarithmic amplifier must produce an output voltage that is proportional to the logarithm of the input voltage. It is therefore possible to make use of the diode characteristic

In this equation, Is is the saturation leakage current, is the thermal voltage, and n is a correction factor between 1 and 2. For the forward-biased diode, when I_A>> I_S, (11.8) can be approximated with good accuracy by

Hence, the forward voltage is

which is the required logarithmic function. The simplest way of using this relationship for computation of the logarithm is shown in Fig. 11.20, where a diode is incorporated into the feedback loop of an operational amplifier. This amplifier converts the input voltage V_i to a proportional current I_A = V_i / R₁. At the same time, the voltage V_o = —V_AC appears at its low-impedance output. Therefore

or

at room temperature.

The usable range is limited by two effects. The diode possesses a parasitic series resistance, across which a considerable voltage is present at high currents, leading to errors in the computation of the logarithm. In addition, the correction factor n is current-dependent. A satisfactory accuracy can therefore only be achieved over an input voltage range of one or two decades.

The unfavorable effect of the varying correction factor n can be eliminated by replacing the diode D by a transistor T, as shown in Fig. 11.21. If I_C >> I_CS, for the collector current we can write

and hence

The output voltage of the transistor logarithmic amplifier shown in Fig. 11.21 is therefore:

11.7.2. Exponential Function

Figure 11.24 shows an exponential function amplifier whose design is analogous to that of the logarithmic amplifier in Fig. 11.21. When a negative voltage is applied to the input, the current flowing through the transistor may be written as

and the output voltage is therefore

Дата добавления: 2014-01-11