❖ Abstract:

In this experiment study how to construct differentiators and integrators using Op-
Amps and study delves into the realm of operational amplifier (op-amp)
differentiators, highlighting their remarkable functionality and practical applications.
To design and simulate a Differentiator circuit and observe output with different input
waveforms. Through rigorous experimentation and analysis, we investigate the
dynamic behavior and response characteristics of various differentiator circuits,
shedding light on their ability to accurately differentiate time-varying signals.

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 Table of Contents

Contents

❖Introduction: 
❖Objectives: 
❖Equipments:  
❖Theory:  
❖Methodology:  
❖Procedures:  
❖Observations:  
❖Calculations:  
❖Results: 
❖Discussion: 
❖Conclusion: 

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t

❖ Introduction:
The operational amplifier (op-amp) is a fundamental building block in the realm of analog
electronics, enabling precise signal processing and amplification. One intriguing application
of op-amps lies in their ability to perform differentiation, a mathematical operation essential
for analyzing rapidly changing signals. The differentiator op-amp circuit presents a versatile
tool for engineers and researchers, offering a means to extract valuable information from
time-varying signals in various domains such as communications, control systems, and
biomedical instrumentation. This experimental study aims to investigate the behavior and
performance characteristics of differentiator op-amp circuits. By designing and implementing
differentiator configurations, we seek to understand their response to input waveforms of
varying frequencies and amplitudes. Through rigorous experimentation and analysis, we aim
to unravel the intricacies of op-amp differentiators, examining parameters such as gain,
bandwidth, linearity, and noise. Ultimately, this research endeavors to provide valuable
insights into the practical utilization of differentiator op-amp circuits, empowering engineers
to harness their potential in diverse applications.

❖ Objectives:
1. Analyze the dynamic response of differentiator op-amp circuits to various
input waveforms.
2. Determine the frequency response and bandwidth limitations of differentiator
op-amp circuits.
3. Investigate the influence of component values on the accuracy of
differentiation.
4. Evaluate the impact of noise and signal distortion on the differentiation
performance of op-amp circuits.

❖ Equipments:
1. 741 op-amp.
2. DC power supply &DMM.
3. Oscilloscope.
4. Functional generator.
5. Resistors, capacitors and set of wires.

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 ❖ Theory:

The basic Differentiator Amplifier circuit is the exact opposite to that of the Integrator
operational amplifier circuit that we saw in the previous experiment. Here, the
position of the capacitor and resistor have been reversed and now the Capacitor, C is
connected to the input terminal of the inverting amplifier while the Resistor, R1 forms
the negative feedback element across the operational amplifier. This circuit performs
the mathematical operation of Differentiation that is it produces a voltage output
which is proportional to the input voltage's rate-of-change and the current flowing
through the capacitor. Or in other words the output voltage is a scaled version of the
derivative of the input voltage. The capacitor blocks any DC content only allowing AC
type signals to pass through and whose frequency is dependent on the rate of change
of the input signal. At low frequencies the reactance of the capacitor is "High"
resulting in a low gain (R1/Xc) and low
output voltage from the op-amp.

Figure 1 The Differentiaator op-amp Circuit

❖ Methodology:

• The Differentiator Amplifier:

The basic operational amplifier differentiator

circuit produces an output signal which is the
first derivative of the input signal In the
differentiator amplifier circuit, the position of
the capacitor and resistor have been reversed
and now the reactance, XC is connected to the
input terminal of the inverting amplifier while the
resistor, Rƒ forms the negative feedback element across the
operational amplifier as normal.
Figure 2 Differentiator op-amp circuit

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This operational amplifier circuit performs the mathematical operation
of Differentiation, that is it “produces a voltage output which is directly proportional
to the input voltage’s rate-of-change with respect to time. In other word the faster or
larger the change to the input voltage signal, the greater the input current, the greater
will be the output voltage change in response, becoming more of a “spike” in shape.
As with the integrator circuit, we have a resistor and capacitor forming an RC
Network across the operational amplifier and the reactance ( Xc ) of the capacitor
plays a major role in the performance of a Op-amp Differentiator

• Op-amp Differentiator Circuit:

The input signal to the differentiator is applied to the capacitor. The capacitor blocks
any DC content so there is no current flow to the amplifier summing point, X resulting
in zero output voltage. The capacitor only allows AC type input voltage changes to
pass through and whose frequency is dependent on the rate of
change of the input signal. At low frequencies the reactance
of the capacitor is “High” resulting in a low gain (Rƒ/Xc ) and
low output voltage from the op-amp. At
higher frequencies the reactance of
the capacitor is much lower resulting
in a higher gain and higher output
voltage from the differentiator
amplifier.
However, at high frequencies an op-
amp differentiator circuit becomes
unstable and will start to oscillate. This
is due mainly to the first-order effect, which determines the
frequency response of the op-amp circuit causing a second-
order response which, at high frequencies gives an output voltage far
higher than what would be expected. To avoid this the high frequency gain of the
circuit needs to be reduced by adding an additional small value capacitor across the
feedback resistor Rƒ.
Ok, some math’s to explain what’s going on!. Since the node voltage of the operational
amplifier at its inverting input terminal is zero, the current, i flowing through the
capacitor will be given as:

The charge on the capacitor equals Capacitance times Voltage across the capacitor

Thus the rate of change of this charge is:

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but dQ/dt is the capacitor current, i

from which we have an ideal voltage output for the op-amp differentiator is given as:

Therefore, the output voltage Vout is a constant –Rƒ*C times the derivative of the
input voltage Vin with respect to time. The minus sign (–) indicates a 180o phase
shift because the input signal is connected to the inverting input terminal of the
operational amplifier.
One final point to mention, the Op-amp Differentiator circuit in its basic form has
two main disadvantages compared to the previous operational amplifier integrator
circuit. One is that it suffers from instability at high frequencies as mentioned above,
and the other is that the capacitive input makes it very susceptible to random noise
signals and any noise or harmonics present in the source circuit will be amplified
more than the input signal itself. This is because the output is proportional to the
slope of the input voltage so some means of limiting the bandwidth in order to
achieve closed-loop stability is required.

• Op- amp Differentiator Waveforms:

If we apply a constantly changing signal such as a

Square-wave, Triangular or Sine-wave type signal to
the input of a differentiator amplifier circuit the
resultant output signal will be changed and whose
final shape is dependant upon the RC time constant
of the Resistor/Capacitor combination.

Figure 3 Waveforms of differentiator op-amp

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❖ Procedures:
1. Connect the circuit as shown in fig below.
2. Apply a symmetrical triangular wave of 2Vp-p amplitude and 1KHz frequency.
3. Connect the input and output of the circuit to channel 1and channel 2 of the DSO
respectively and observe the waveforms.
4. Take screenshots of the waveforms from DSO.
5. Repeat the same for the sine-wave.
6. Compare the practical values with theoretical values in case of the sine-wave.

Figure 4 triangle input & square output waveforms of differentiator op-amp

Figure 5 sine input & cosine output waveforms of differentiator op-amp

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 ❖ Observations:
1. Observe the output waveform from CRO. A square wave will generate a triangular
wave and sine wave will generate a cosine wave.
2. Measure the frequency and the voltage of the output waveform in the CRO.
3. Calculate

4. Compare the calculated output voltage with

the experimentally observed voltage from the output waveform.
5. Observe outputs of the differentiator circuit using different input waveforms.

❖ Calculations:

Differentiator circuit has been taken and the required parameters values is being noted down
below:
1. Input Voltage: 1V
2. Frequency: 1 kHz
3. Capacitor: 0.1 uF
4. Resistors of: 150Ω & 1.5kΩ

• For High frequency: at 1 kHz

𝒅 𝑽𝒊𝒏
𝑽𝒊𝒏 = 𝟏 𝒔𝒊𝒏(𝟐𝒇𝝅𝒕) 𝒔𝒐 𝑽𝒐 = −𝑹𝒇 𝑪
𝒅𝒕
𝒅(𝟏𝒔𝒊𝒏(𝟐𝝅𝒇𝒕))
= −(𝟏. 𝟓 ∗ 𝟏𝟎𝟑 ∗ 𝟎. 𝟏 ∗ 𝟏𝟎−𝟔 ) ∗
𝒅𝒕

𝑽𝒐 = (𝟏. 𝟓 ∗ 𝟏𝟎𝟑 ∗ 𝟎. 𝟏 ∗ 𝟏𝟎−𝟔 ) ∗ 𝟐𝟎𝟎𝟎π 𝒄𝒐𝒔(𝟐𝟎𝟎𝟎π𝒕)=

𝑽𝒐 𝟎.𝟗𝟓
𝑽𝒐 = 0.95(volt)cos(2000πt) , 𝑨𝒗 = = = 𝟎. 𝟗𝟓
𝑽𝒊𝒏 𝟏

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 • For Low frequency: at 1 Hz

𝒅 𝑽𝒊𝒏
𝑽𝒊𝒏 = 𝟏 𝒔𝒊𝒏(𝟐𝒇𝝅𝒕) 𝒔𝒐 𝑽𝒐 = −𝑹𝒇 𝑪
𝒅𝒕
𝒅(𝟏𝒔𝒊𝒏(𝟐𝝅𝒇𝒕))
= −(𝟏. 𝟓 ∗ 𝟏𝟎𝟑 ∗ 𝟎. 𝟏 ∗ 𝟏𝟎−𝟔 ) ∗
𝒅𝒕

𝑽𝒐 = (𝟏. 𝟓 ∗ 𝟏𝟎𝟑 ∗ 𝟎. 𝟏 ∗ 𝟏𝟎−𝟔 ) ∗ 𝟐 ∗ 𝟏π 𝒄𝒐𝒔(𝟐π𝒕)=

𝑽𝒐 𝟎.𝟗𝟓∗𝟏𝟎−𝟑
𝑽𝒐 = 0.95*𝟏𝟎−𝟑 (volt)cos(2πt) , 𝑨𝒗 = = = 𝟎. 𝟗𝟓 ∗ 𝟏𝟎−𝟑
𝑽𝒊𝒏 𝟏

❖ Results:

• For High frequency:

High freq(kHz) Vin(V) Vout(v) Av

1 1 0.95 0.95

• For Low frequency:

High freq(Hz) Vin(V) Vout(v) Av

1 1 0.95*𝟏𝟎−𝟑 0.95*𝟏𝟎−𝟑

The integrator circuit design output waveforms have been studied.

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 ❖ Discussion:
The experimental results demonstrate the versatile nature of differentiator op-amp circuits in
accurately differentiating time-varying signals. The dynamic response analysis revealed that
the circuits exhibited excellent differentiation performance across a wide range of input
waveforms, including sinusoidal, square, and triangular signals. The frequency response
characterization indicated that the bandwidth limitations of the differentiator op-amp circuits
were primarily determined by the internal op-amp characteristics and the values of external
components. Furthermore, the influence of component values on differentiation accuracy was
evident, with careful selection of resistors and capacitors resulting in improved precision.
However, it was observed that extreme values of components could introduce signal distortion
and compromise the linearity of the output. The evaluation of noise and signal distortion
effects highlighted the importance of noise reduction techniques and careful circuit design to
minimize their impact on differentiation accuracy.

❖ Conclusion:

The mathematical operation differentiation can be realized by an electronic circuit called

a differentiator which is based on an operational amplifier working in inverting configuration
with a reactive component in its input branch. In the first section, we present the ideal
differentiator which is a simple circuit to theoretically establish how a differentiator works.
In particular, we pinpoint that due to the capacitor in the input branch, the circuit is
equivalent to an inverting op-amp with a gain of 0 at low-frequency and a gain tending to
infinite values in high-frequency regime. The consequence of this electrical behavior is that an
ideal differentiator can be characterized by a simple transfer function T=+jx, which means
that its related Bode diagram is a linear plot with a slope of +20 dB/decade. However, the ideal
differentiator cannot be designed in practice due to the infinite gain the circuit is supposed to
have when increasing the frequency. As a consequence, the differentiation operation is limited
up to a certain frequency when the output will start to saturate.

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