sign indicates that there is 180° phase shift in the output with respect to the input, Thus, it can be shown that in an ideal situation the voltage across the resistor will be proportional to the derivative of the voltage across the capacitor with a gain of RC. Operation Input signals are applied to the capacitor C. Capacitive reactance is the important factor in the analysis of the operation of a differentiator. Capacitive reactance is Xc = . Capacitive reactance is inversely proportional to the rate of change of input voltage applied to the capacitor. At low frequency, the reactance of a capacitor is high, and at high frequency reactance is low. Therefore, at low frequencies and for slow changes in input voltage, the gain, , is low, while at higher frequencies and for fast changes the gain is high, producing larger output voltages. If a constant DC voltage is applied as input, then the output voltage is zero. If the input voltage changes from zero to negative, the output voltage is positive. If the applied input voltage changes from zero to positive, the output voltage is negative. If a square-wave input is applied to a differentiator, then a spike waveform is obtained at the output. The active differentiator isolates the load of the succeeding stages, so it has the same response independent of the load. Frequency response The transfer function of an ideal differentiator is , and the Bode plot of its magnitude is: Advantages A small time constant is sufficient to cause differentiation of the input signal Limitations At high frequencies: this simple differentiator circuit becomes unstable and starts to oscillate; the circuit becomes sensitive to noise, that is, when amplified, noise dominates the input/message signal. Practical differentiator In order to overcome the limitations of the ideal differentiator, an additional small-value capacitor C1 is connected in parallel with the feedback resistor R, which prevents the differentiator circuit from oscillating, and a resistor R1 is connected in series with the capacitor C, which limits the increase in gain to a ratio of . Since negative feedback is present through the resistor R, we can apply the virtual ground concept, that is, the voltage at the inverting terminal = voltage at the non-inverting terminal = 0. Applying nodal analysis, we get Therefore, Hence, there occurs one zero at and two poles at and . Frequency response From the above plot,