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Operational amplifier –
integrator and differentiator
1. Objectives
The aim of the exercise is to get to know the circuits with operational amplifiers suitable for linear signal transformation. The scope of the exercise includes the design and measurement of the basic parameters of the integrator and differentiator. NOTICE: estimated preparation time can be as long as 3 to 6 hours.
2. Components and instrumentation
The exercise examines the properties of an integrator and differentiator. These systems, built using operational amplifiers, are discussed in the following sections. 2.1. Integrator
The integrator performs the function of:
𝑈𝑂𝑈𝑇(𝑡)^ = ∫ 𝑈𝐼𝑁(𝑡)𝑑𝑡 (1)
Schematic diagram of a perfect integrator is shown in Fig. 2 In time domain the capacitor current can be expressed as:
But the input current:
Because of KCL:
so:
𝑈𝑂𝑈𝑇(𝑡)^ = −
Transfer function in frequency domain can be written as::
In the system of Fig. 2, there is no DC feedback, which in practice means saturation of the operational amplifier. Therefore, an additional resistor R1 was introduced (Fig.2). Such a system is called a lossy integrator.
C
UOUT
R
UIN
IC IIN
RG
EG
RLoad
TV[dB]
a) b)
f (log scale)
-20dB/dec
Transfer function of a perfect integrator
Fig. 1. Basic (perfect) integrator circuit a) schematic diagram, b) transfer function (absolute value) in dB vs. frequency in logarithmic scale.
a) b)
RC
1 1 RC
1
R
R 1
AVOPAMP
-
C
UOUT
R
UIN
IC IIN
RG
EG
RLoad
TV
2 πfT
Rd
R 1
Transfer function of a real opamp
Transfer function of a real integrator
Fig. 2. Real integrator: a) schematic diagram, b) transfer function (abs) - fT gain bandwidth of a opamp.
The Rd resistor in the system in Fig. 3 is used to minimize the offset error,
𝑅𝑑 =
where RG the output resistivity of generator (usually Rg= 50 Ω in the lab).
Transfer function of the real integrator shown in Fig.2 can be expressed in frequency domain as:
𝑇𝑉(𝑗𝜔)^ = −
As it results from the course of the transfer function of this system (Fig.3), the correct integration takes place for ω (slope - 20 dB / dec):
Where fT is the operational amp gain bandwith.
This, for sinusoidal waves, corresponds to period of sinewaves of:
So R 1 can be taken R 1 = 10*4.8k≈51k
2.2. Differentiator
An perfect differentiator system performs the function:
𝑈𝑂𝑈𝑇(𝑡)^ =
Basic schematic diagram of a differentiator is shown in Fig.4..
Analyzing this circuit in time domain we can put:
𝑈𝑂𝑈𝑇(𝑡) 𝑅 and^ 𝐼𝐼𝑁^ =^ 𝐶^
𝑑𝑈𝐼𝑁(𝑡) 𝑑𝑡.^ According to KCL:
𝐼𝐼𝑁 + 𝐼𝐶 = 𝐶
= 0 (15)
hence:
𝑈𝑂𝑈𝑇(𝑡)^ = −𝑅𝐶
A perfect transfer function can be written as:
𝑇𝑉(𝑓)^ = −𝑗𝜔𝑅𝐶
-
R
C
b) TV [dB]
UIN UOUT RLoad
IIN
IR
f (log scale)
20dB/dec Perfect transfer function
Fig. 4. Basic differentiator; a) schematic diagram, b) transfer function
The system performs the function of differentiation at pulsations at which the slope of the transfer function TV(ω) is equal + 20 dB/dec.
The basic differentiator system has many disadvantages: tendency to instability, decrease in gain for higher frequencies associated with the frequency response of the opamp, very low input impedance at high frequencies, large input voltage noise. These defects can be reduced by introducing an additional R 1 resistor in the system. The diagram of the modified differentiator system is shown in Fig. 7
R
C
a) b)
RC
1 R 1 C
1
Rd
RG
EG
R 1
AVOL
2 fT
UIN UOUT RLoad
R/R 1
TV [dB] Tranfer function of the OpAmp Perfect transfer function of an differentiator
Real transfer function of the differentiator
Fig. 5. Modified differentiator; a) schematic diagram, b) transfer function (abs).
Resistor R d let minimize the offset of the opamp:,
𝑅𝑑 = 𝑅 (18)
Transfer function can be written as (perfect opamp):
𝑇𝑉(𝑓)^ = −
As it results from the course of the Transfer function of this system (Fig. 7), the differentiation of sinusoidal signals takes place at frequencies f
𝑓 ≪
≪ 𝑓𝑇, (20)
where fT gain bandwidth of opamp.
In practice R 1 , should be chose to fulfill the relation:
So:
(21a)
where fT is the gain bandwidth of opamp (for TL061 – 1MHz).
2.2.1. Time domain equations
Given the input of the differentiator from Fig.5, a triangular signal with a peak-to-peak value of UINpp = 2UINm and frequency f, at the output of the system we get a rectangular signal with a peak-to-peak value of UOUTpp = 2UOUTm - Fig.6.
For 0≤t≤T/2 Input signal can be described as:
𝑈𝐼𝑁(𝑡) =
𝑡 − 𝑈𝐼𝑁𝑚 (22)
3.2. Problems
- Derive the formula for the output voltage of the integral circuit in the time domain.
- Sketch the transfer function (amplitude and phase) of the perfect and real integrator (amplification in [dB], logarithmic frequency axis).
- Sketch the transfer function (amplitude and phase) of the perfect and real differentiator (amplification in [dB], logarithmic frequency axis).
- Derive formulas for the amplitude of the output of the integrator when excited by a rectangular signal with a given amplitude
- Derive formulas for the amplitude of the output of the differentiator when excited by a triangle signal with a given amplitude
3.3. Detailed preparation
Before classes, students receive a project task from the tutor like that described in examples above. Solution of the task should include:
- Design task, diagram and calculation of system elements. Values of passive elements should be selected from normalized series of values - resistors choose from a series of E24(5%), capacitors from the values available in the laboratory (360p, 1n, 1n5, 3n3, 4n7, 6n8, 10n, 15n, 22n, 100nF).
- Computer simulations (e.g. in LTspice – for AC analysis, the frequency on logarithmic axis, gain in dB).
- Sketch of the layout of the elements on the PCB.
4. Contest of the report
4.1. Assemble of the circuit
- Bearing in mind that every passive element is made with a certain accuracy, before proceeding with assembly of the system, measure the actual values of used elements using the meter available on the stand.
- Apply the actual values of the elements to the prepared circuit diagram.
- Assemble the circuit board on the PCB.
4.2. Integrator
4.2.1. Time domain measurements (square – triangle waveforms)
- Assemble the measuring system according to the diagram from Fig. 7, the tested system should be supplied with ± 12V voltage
- Provide a rectangular signal from the generator with parameters compliant with the requirements of the design task. On the output voltage oscillogram, measure its amplitude UOUTm and compare with that calculated. If necessary, correct the values of the elements so as to obtain an output signal of parameters given in the task.
- By changing the signal frequency from the generator, measure the UOUTm = UOUTm (1/f=T) (T is the period) relationship (the output signal should be triangle, use “amplitude” measurement rather than “peak-peak), 7) Measur and plot the relation UOUTm = UOUTm (T) (lin-lin axes) and compare with the theoretical one (eq.13). NOTE: The UOUTm = UOUTm (T) graph cannot be called “the gain” of the system or its” transfer function”
INTEGRATOR
OR
DIFFERENTIATOR
GENERATOR
SCOPE
SUPPLY
UIN UOUT
+ - + -
Fig. 7. Block diagram of measurement setup.
4.2.2. Frequency domain (sine waveform)
- Using sine signals measure the transfer function (amplitude and phase). Use frequency range 10Hz to 5MHz (log axis). Amplitude of the signal should be chosen small enough to achieve undisturbed output signal and high enough to obtain real results. Measurements should be done by means of oscilloscope using RMS values (Cyc-RMS).
- Put the results on the relevant graphs obtained in computer simulation
4.3. Differentiator
4.3.1. Time domain measurements (triangle – square waveforms)
- Assemble the measuring system according to the diagram from Fig. 7, the tested system should be supplied with ± 12V voltage
- Provide a triangle waveform from the generator with parameters compliant with the requirements of the design task. On the output voltage oscillogram, measure its peak-to-peak value UOUTpp = 2UOUTm and compare with that calculated. If necessary, correct the values of the elements so as to obtain an output signal of parameters given in the task.
- By changing the signal frequency from the generator, measure the UOUTm = UOUTm (f) relationship (the output signal should be squerwave, use “amplitude” measurement rather than “peak-peak”), 7) Measure and plot the relation UOUTm = UOUTm (f) (lin-lin axes) and compare with the theoretical one (eq.23).
NOTE: The UOUTm = UOUTm (f) graph cannot be called “the gain” of the system or its” transfer function”
4.3.2. Frequency domain (sine waveform)
- Using sine signals measure the transfer function (amplitude and phase). Use frequency range 10Hz to 1 MHz (log axis). Amplitude of the signal should be chosen small enough to achieve undisturbed (observe on scope !) output signal and high enough to obtain real results. Measurements should be done by means of oscilloscope using RMS values (Cyc-RMS).
- Put the results on the relevant graphs obtained in computer simulation
4.4. Report should include
- Title page.
- Calculation for task (p. 3.3),
