 Hello everyone. In today's session, we are going to learn about op-ang as a differentiator or a differentiation amplifier. At the end of this session, students will be able to analyze and describe a differentiation amplifier implemented using operational amplifier. So these are the contents of my today's presentation. The differentiator or a differentiation amplifier performs mathematical operation of differentiation. Its output voltage waveform is a derivative of input voltage waveform. The differentiator is most commonly used in signal wire shaping circuits to detect high frequency component in an input signal. It is also used as a rate of change detector in frequency modulation, FM modulators. This figure shows figure one that is basic differentiation amplifier. So basic differentiator or differentiation amplifier can be constructed from basic inverting amplifier with negative feedback, in which input resistor R1 is replaced by the capacitor C1. The input voltage to be differentiated is applied to inverting input through capacitor C1. RF is a feedback resistor. So IB1 and IB2 are input bias current, are approximately zero, flowing through inverting and non-inverting input of operational amplifier. ROM is an offset minimizing resistor connected to non-inverting input. Its value is approximately equal to the value of the feedback resistor RF. The output is taken across load resistor RL. That is the output is taken across the feedback resistor RF. Because the non-inverting input is connected to ground through offset minimizing resistor. So voltage V1 is equal to zero volt. So by the concept of virtual ground, the voltage at inverting input that is V2 is also zero volt. So inverting input is acting as a virtual ground. So output that is differentiated voltage waveform is taken across RF or RL. We can verify the working of this basic differentiator by getting the equation for output voltage. Let us write Kirchhoff's current equation at node V2. So that is current flowing through capacitor IC is equal to IB plus IF, where IF is current flowing through feedback resistor RF. Since the input bias current IB is approximately zero, so the capacitor current IC is equal to the feedback current IF. So current flowing through capacitor C1 flows through feedback resistor RF. So let us differentiate this expression current in terms of voltage. So using a relation, basic relation of C is equal to Q by V for capacitor. We can write the voltage across capacitor C1 is the derivative of V in minus V2 with respect to time dt. So that is equal to V2 minus V0 upon RF. Since V1 is equal to zero volts and V1 is equal to V2 is equal to zero volts by the concept of virtual ground, we can write C1 into derivative of V in with respect to time. So that is equal to minus V0 upon RF. So finally the output voltage equation V0 equal to minus RF C1 derivative of V in with respect to time dt. Thus the output voltage V0 is equal to RF C1 times the negative instantaneous rate of change of input voltage V in with time. The student can pause video here and think over this question and try to write the answer. Why the basic differentiator shown in figure one will not do the proper differentiation of input signal waveform? Because the differentiator performs a reverse operation of integration. The sine wave input will produce cosine wave output or a triangular wave input will produce a square wave output. If input is square wave signal output should be a spy wave signal. But this basic differentiator has some practical problems. So the gain of this circuit that is feedback resistance upon the input impedance that is RF upon XC1 increases with increase in frequency at a rate of 20 dB per decade. So with respect to time the output voltage is continuously increases at the rate of 20 dB per decade. So this makes the circuit unstable. So output of differentiator does not reach to a finite value within finite time. So circuit becomes unstable. Also the input impedance XC1 decreases with increase in frequency. So this makes circuit very sensitive to high frequency noise. So after amplification the noise signal completely overrides the desired differentiated output signal. So both stability, circuit stability and the high frequency noise problem can be corrected by the addition of two components. Register R1 and capacitor CF in the basic differentiator circuit as shown in Figure 2. So Figure 2 shows a practical differentiator circuit and this capacitor CF is connected in parallel to feedback resistor RF and resistor R1 is connected in series with capacitor C1 at the input side. And the op-amp is biased with dual DC power supply plus 15 volt or minus 15 volt and output is taken across load resistor RL. So that is coming in parallel with a parallel combination of RF CF. So output voltage is now a derivative of input voltage waveform. So as op-amp is working in inverting mode, so output is negative. Negative RF C1 times the derivative of input voltage with respect to time. Now this figure shows input output waveforms for differentiator circuit. So if input is a sine wave signal, so output is a cosine wave signal. So when input is a sine wave signal at zero level, so output cosine wave signal at peak level, so output is sine wave signal at peak level. So output is cosine wave signal at zero level. If input voltage waveform is a square wave signal in that case the output signal waveform is a spike wave signal. So, when input voltage is constant, so the derivative of constant is 0, so output voltage is 0. When input voltage changes from one voltage level to another voltage level, so we are getting the output. So, in this way the output is spike wave voltage signal for square wave input voltage signal. Now, this figure shows a frequency response for basic operational amplifier in open loop and basic differentiator and closed loop response of practical differentiator circuit. So, it is a graph of a voltage gain versus a frequency. So, gain is expressed in dB plotted on y axis and frequencies plotted on x axis. So, frequency is increased in multiples of 10. So, on x axis some relative frequency is plotted. So, frequency response of basic op-amp in open loop that is shown by a solid line and frequency response of a basic differentiator circuit. So, that is shown by this dotted line plus this solid line. So, from frequency f to fb the gain of basic differentiator increases at the rate of 20 dB per decade. So, after frequency fb gain is continued to increase and the plot shown by a dotted line that is for practical differentiator. So, the gain decreases at the rate of 20 dB per decade after frequency fb. So, in frequency response f a is a frequency at which the gain is 0 dB and it is given by f a equal to 1 upon 2 pi rfc1. The fc is the unity gain bandwidth of fm, f is some relative operating frequency. From frequency response from f to fb the gain increases at a rate of 20 dB per decade after fb the gain decreases at a rate of 20 dB per decade. So, this 40 dB per decade change in gain caused by rfc1 and rfcf combination. So, the gain limiting frequency fb is given by fb equal to 1 upon 2 pi rfc1 where rfc1 equal to rfcf. The rfc1 and rfcf help to reduce significantly the effect of high frequency noise and offsets. Also rfc1 and rfcf makes the circuit more stable by preventing increase in gain with increase in frequency. So, rfc1 and rfcf, rfcf well it should be selected such that fb is greater than f a and less than fc where f a equal to 1 upon 2 pi rfc1 and fb equal to 1 upon 2 pi rfcf or 1 upon 2 pi rfc1. The input signal will be differentiated properly if the time period t of the input signal is larger than or equal to rfc1. That is the time period of input signal to be differentiated should be greater than or equal to rfcf. The design of guidelines for practical differentiator are select f a equal to highest frequency of input signal to be differentiated. Assume value of c1 less than 1 micro farad and calculate value of rf. This is the reference. Thank you.