4.  The differentiator or differentiating amplifier is as shown in figure.  This circuit will perform the mathematical operation of differentiation on the input voltage waveform and the output is a derivative of the input signal.  The differentiator can be constructed from the basic inverting amplifier by interchanging resistance R and C as shown in figure.

5. Apply the KCL at node V2 ; Ic = IB + IF........ (1) IB = 0.: Ic = IF....... (2) We know that, Ic = C (dVc / dt)...... (3) The voltage across C is given by, Vc = Vin – V2......(4)

6. Substituting this into equation (3) we get, Ic = C (Vin – V2).......(5) IF = (V2 – Vo) / RF........(6) We know that Ic = IF, C (Vin – V2) = (V2 – Vo) / RF.(7) Using the concept of virtual ground we can write V1 = V2 = 0, So now equation (7) is written as,.: C (Vin) = (– Vo) / RF.:Vo = - RF C (Vin)......(8)

8.  The gain of basic differentiator is (Rf/Xc). As Xc decreases with increase in frequency, the gain will increase with increase in frequency.  The increase in gain at high frequencies will make the circuit unstable. Also the input impedance Xc reduces with increase in frequency.  This makes the circuit very much susceptible to high frequency nose. When amplified, this noise can completely override the output signal.

10.  Both the instability and high frequency noise problems can be corrected by using the practical differentiator circuit shown in fig.  R and Cf have been added to the basic differentiator circuit.  Due to reduction in Xc at high frequencies, the gain of the basic differentiator increases drastically.  To stabilize it, we have connected a resistor R in series with C. This will control the amount of increase in gain at high frequency.

11.  Sharp output  Gain can be controlled  Sharp frequency Response

12.  This circuit is badly affected by noise.  It is less stable. There is a possibility of oscillations.  Gain increase with increase in frequency.  Output is affected by the parameter of op- amp.

13.  In the P-I-D.  As a high pass filter.  In the wave shaping circuit to generate narrow pulses corresponding to any sharp change in the input signal.

16.  The op-amp is used as inverting amplifier. It introduce the phase shift of 180 0 between its input & output.  The output of inverter amplifier is applied at the input R-C phase shift network, this network is introduce a phase shift at 180 0.  This feedback network introduce the signal at its input and feed it to the amplifier input. The level of attenuation is decided by the feedback function β.  The gain of the inverting amplifier is decided by the value of R f & R i.  It can be prove that the value of the feedback function β at the frequency of oscillators is β=1/29.

17.  Gain of inverting amplifier, A= R f /R i R f /R i ≥ 29  Frequency 0f oscillation of R-C circuit, f = 1/2π√6 RC  The equation show that the frequency of oscillators of the RC phase shift depends on the feedback network.  The RC phase shift oscillators generally used over the frequency range 100 Hz to 100 kHz.

18.  Simplicity of network.  Useful for frequency in the audio range.  A sinewave output can be obtained

19.  Poor frequency stability.  Difficult to get a variable frequency output.

20. It uses a feedback network which consists of resistors and capacitors. Description The wien bridge has four arms. The arm AD which consists of the series combination of R1 and C1 and the arm CD which consists of the parallel combination of R2& C2 are called as the frequency sensitive arms.

22. This is because the components connected in these arms decide the oscillator frequency. The resistors R3 & R4 are used to generate a reference voltage which remains constant independent of frequency. The ac input voltage is applied between points A and C of the bridge. The ac output of the bridge is obtain between points B and D of the bridge.

23.  The feedback factor or gain of the feedback network (ß) is defined as : ß = V f /V in At the oscillator frequency : Phase shift introduced by Wien bridge = 0 Feedback factor ß = 1/3

25.  The circuit in which the output voltage waveform is the “integration” of the input voltage waveform is called as an integrator or integrating amplifier.  Two integrator circuits :- 1. Ideal 2. practical

27. Apply the KCL at node V2 ; I1 = IB + IF........ (1) Due to high I/P impedance Ri of the OP-Amp ; IB will be negligible as compare to IF. So IB = 0,.: I1 = IF....... (2).:I1 = (Vin – V2)/ R1

28. Here IF = IC = CF [d(V2-Vo) / dt] Put the value of I1 & IC in equation (2); (Vin – V2) / R1 = CF [d(V2-Vo) / dt ]....... (3) According to “virtual ground”... V1 = V2 = 0. So, Now equation (3) is written as ;.: Vin / R1 = -CF [d(Vo) / dt].:Vin / -R1CF = d(Vo) / dt...... (4)

29. Now integrating the equation (4) we get,.......(5). Where C is the integration constant and it is proportional to the output voltage Vo at t = 0 sec. Conclusion:- Equation (5) indicates that the O/P voltage is negative integration of the I/P voltage.

31.  In the ideal integrator the capacitor is connected in the feedback path, Hence its reactance will decide the gain of inverting amplifier as, Avf = -Cf / R1 = -Xf / R1 Where Xf is the reactance of the capacitor Cf.  In the absence of any ac input, the capacitive reactance will be infinite. Hence the gain of ihe integrator will be infinite.

32.  Another problem with the ideal integrator is that the gain of the amplifier which is infinite at dc(f=0), goes on decreasing with increase in frequency (due to reduction in Xf).Hence the integrator cannot integrate high frequency input signal.  Because of all these problems, the ideal integrator is generally not used in practice.

34.  Due to the inclusion of resistance Rf across Cf, the low frequency gain does not become infinite.  The value of Rf is very high.  The expression for the O/P voltage remains same as the obtained for the ideal integrator.  At high frequency Xcf is very small hence it will shunt the feedback resistance Rf and the high frequency gain of the integrator will be very low.

35.  Low distortion  Better Linearity  Gain can be controlled  Sharp frequency Response  Less effect of noise  This circuit is highly stable, so less possiblity of oscillation.

36.  It can operate as an integrator over a short frequency.  Op-amp parameters affect the output.  Gain reduces with increase in frequency.

37.  In a triangular wave or ramp generators.  In the ADC.  In the integral type controllers in a closed loop control system.  In analog computers to solve differential equation.  As a low pass filter.  In the communication circuits for recovering the modulating signal.