$30.00

## Description

**Objectives**

The purpose of the lab is to study some of the opamp configurations commonly found in practical applications and also investigate the non-idealities of the opamp such as finite Gain-Bandwidth product and slew rate limitations. The circuits studied will include an integrator, a differentiator, a non-inverting amplifier and a unity-gain buffer.

**2 **** ****Introduction**

**2.1 **** ****Integrator**

The circuit in Fig. 1 is the lossless inverting integrator, which generates an output signal that corresponds to the integral of the input signal over time.

_{C}

Figure 1: Inverting integrator circuit

In the frequency domain, the output voltage is described as:

_{1}

1

^{V}^{o} ^{=} ^{−} _{sR} _{C} ^{V}^{i } ^{(1)}

Notice that the gain V_{o} /V_{i} is theoretically infinite at DC (s = 0), hence any small DC signal at the input will saturate the opamp output over time. In a practical integrator circuit, a large resistor in parallel with the capacitor is required to prevent the capacitor from storing charge due to offset currents and voltages at the input. This configuration is known as “lossy” integrator or a first-order lowpass circuit, which is shown in Fig. 2. The output voltage is now given by the following expression:

V = − V (2)

__ ____ __^{R}_{2} ^{/}^{R}_{1}__ ____ __

^{o } ^{1} ^{+} ^{sR}_{2} ^{C } ^{i}

The DC gain is now finite and determined by the ratio of the two resistors.

R_{2}

Figure2:Lossyintegratorcircuit

**2.2 **** ****Di****f****ferentiator**

As the counterpart of the integrator, the differentiator takes the derivative of the input signal. This configuration is shown in Fig. 3.

R_{1}

Figure 3: Inverting differentiator circuit

The output voltage in Fig. 3 can be obtained as

V_{o} = −sCR_{1} V_{i } (3)

Ideally, the output voltage becomes very large at very high frequencies. However, due to opamp’s gain limitation at high frequencies, differentiator ’s gain also starts to decrease, deviating from the ideal behavior. To limit the high-frequency gain to a known value, a pseudo differentiator circuit as shown in Fig. 4 can be used.

R_{2}

Low frequency input signals dominate integrator ’s output due to higher gain, whereas high frequency signals are more dominant for differentiators for the same reason. Due to the limited high frequency capabilities of the opamps, differentiators are more difficult to build in practice. In addition, integrator functionality is tolerant to noise at the input, whereas addition of noise to the input signal makes the differentiation operation unreliable. Therefore, in most of the practical systems such as filters, integrators are used instead of differentiators.

**2.3 **** ****Non-Idealities**** ****of**** ****the**** ****Opamp**

**2.3.1 **** ****Finite**** ****Gain-Bandwidth**** ****(GBW)**** ****Product:**

The open-loop gain of the opamp decreases when frequency increases following the roll-off of a single pole system. However, the product of open-loop DC gain and the 3-dB frequency (bandwidth) is a constant, which is defined as the Gain-Bandwidth product (GBW). For the UA741, GBW is about 1.2MHz. Finite GBW of the opamp limits the bandwidth of closed-loop amplifier configurations.

Figure 5: Non-inverting amplifier configuration

2.3.2 Slew Rate

An ideal opamp is capable of following the input signal no matter how fast the input changes. In a real 741, the output rise/fall transient cannot exceed a maximum slope; the maximum rate of change of the output voltage as a function of time is defined as the slew rate. Applying signals with transients that exceed this limit results in distorted output signals. To avoid distortion due to slew rate limitations, maximum rate of change of output must be kept less than the slew rate specifications of the opamp. The slew rate can be measured by applying a large square waveform at the input. The slope of the rise time at the output is the slew rate.

Figure 6: Unity-gain buffer configuration

**3 **** ****Calculations**

**3.1 **** ****Lossy**** ****Integrator**

**1.**** **For the lossy integrator in Fig. 2, derive the time-domain equation for the output in terms of the input.

**2.**** **Find R_{1} to have a low-frequency gain of -22 if R_{2} = 22k Ω and C = 220nF , and calculate the 3-dB frequency.

**3.**** **Sketch the magnitude and phase Bode plots for the transfer function V_{o} /V_{i} .

**4.**** **Calculate V_{o} (t ) for V_{i} (t ) = 0.5 sin(2π1000t ).

**5.**** **Sketch the output waveform if the input is a 500mV 1kHz square wave signal.

**3.2 **** ****Pseudo**** ****Di****f****ferentiator**

**1.**** **For the first order high-pass filter in Fig. 4, derive the time-domain equation for the output in terms of the input.

**2.**** **Find R_{2} to have a high-frequency gain of -22 if R_{1} = 1k Ω and C = 33nF , and calculate the 3-dB frequency.

**3.**** **Sketch the magnitude and phase Bode plots for the transfer function V_{o} /V_{i} .

**4.**** **Calculate V_{o} (t ) for V_{i} (t ) = 0.1 sin(2π1000t ).

**5.**** **Sketch the output waveform if the input is a 100mV 1kHz triangular wave signal.

**3.3 **** ****Finite**** ****GBW**** ****Limitations**

**1.**** **For the non-inverting amplifier in Fig. 5, assume R_{1 } = 1k Ω. Find the values of R_{2 } to set the low-frequency gain to 23, 57, and 83.

**2.**** **Find the transfer function V_{o} /V_{i} (s ) for each gain value.

**3.**** **Sketch the magnitude Bode plots of the three transfer functions on the same plot.

**3.4 **** ****Slew **** ****Rate**** ****Limitations**

**1.**** **For the unity-gain buffer in Fig. 6, find the transfer function V_{o} /V_{i} (s ), and sketch the magnitude Bode plot.

**2.**** **If the slew rate of the opamp is 0.5V /µs , find the maximum frequency of the 1V sine wave input signal before the output is distorted.

**3.**** **If the slew rate of the opamp is 0.5V /µs , find the maximum amplitude of the 75kHz sine wave input signal before the output is distorted.

**4 **** ****Simulations**

**For**** ****all**** ****simulations,**** ****provide **** ****screenshots**** ****showing**** ****the**** ****schematics **** ****and**** ****the**** ****plots**** ****with**** ****the**** ****simulated**** ****values **** ****prop- ****erly**** ****labeled.**

**4.1 **** ****Lossy**** ****Integrator**

**1.**** **Draw the schematics for the lossy integrator in Fig. 2 with the calculated component values.

**2.**** **Obtain the magnitude and phase **Bode**** ****plots**** **of the transfer function using **AC**** ****simulation**. Measure the low- frequency gain, 3-dB frequency, and the magnitude and phase of the transfer function at 1kHz.

**3.**** **Apply a 1kHz 500mV sine wave signal to the input V_{i } and obtain the **time-domain**** ****waveforms**** **for the in- put and output voltages using **transient **** ****simulation**. Measure the magnitudes of the input and the output voltages, and the phase difference between them.

**4.**** **Apply a 1kHz 500mV square wave signal to the input V_{i} and obtain the **time-domain**** ****waveforms**** **for the input and output voltages using **transient**** ****simulation**. Measure the peak-to-peak voltage of the output.

**4.2 **** ****Pseudo**** ****Di****f****ferentiator**

**1.**** **Draw the schematics for the pseudo differentiator in Fig. 4 with the calculated component values.

**2.**** **Obtain the magnitude and phase **Bode**** ****plots**** **of the transfer function using **AC**** ****simulation**. Measure the low- frequency gain, 3-dB frequency, and the magnitude and phase of the transfer function at 1kHz.

**3.**** **Apply a 1kHz 100mV sine wave signal to the input V_{i } and obtain the **time-domain**** ****waveforms**** **for the in- put and output voltages using **transient **** ****simulation**. Measure the magnitudes of the input and the output voltages, and the phase difference between them.

**4.**** **Apply a 1kHz 100mV triangular wave signal to the input V_{i} and obtain the **time-domain**** ****waveforms**** **for the input and output voltages using **transient**** ****simulation**. Measure the peak-to-peak voltage of the output.

**4.3 **** ****Finite**** ****GBW**** ****Limitations**

**1.**** **Draw the schematics for the non-inverting amplifier in Fig. 5.

**2.**** **Obtain the magnitude and phase **Bode**** ****plots **** **of the transfer functions using **AC**** ****simulation**** **with **parameter ****sweep**, where the resistor R_{2} is swept to realize the three gain values 23, 57, and 83. Group all three traces in one plot as in your sketch.

**3.**** **Measure the low-frequency gain and 3-dB frequency for each value of R_{2} .

**4.4 ****Slew **** ****Rate**** ****Limitations**

**1.**** **Draw the schematics for the unity-gain buffer in Fig. 6 with a load of 2.2k Ω in parallel with 100pF at the output.

**2.**** **Obtain the magnitude and phase **Bode**** ****plots**** **of the transfer function using **AC**** ****simulation**. Measure the low- frequency gain, 3-dB frequency, and the magnitude of the transfer function at 75kHz and 150kHz.

**3.**** **Apply a 75kHz 1V sine wave signal to the input V_{i } and obtain the **time-domain**** ****waveforms**** **for the input and output voltages using **transient**** ****simulation**. Perform **Fourier**** ****simulation **to measure the **total**** ****harmonic ****distortion **** ****(THD)**** **on the output waveform.

**4.**** **Repeat the previous step for 75kHz 2V sine wave, and 150kHz 1V sine wave input signals.

**5 **** ****Measurements**

**For**** ****all**** ****measurements,**** ****provide **** ****screenshots**** ****showing**** ****the**** ****plots**** ****with**** ****the**** ****measured**** ****values **** ****properly**** ****labeled.**

**5.1 **** ****Lossy**** ****Integrator**

**1.**** **Build the lossy integrator in Fig. 2 with the simulated component values.

**2.**** **Obtain the magnitude and phase **Bode**** ****plots **** **of the transfer function using the **network **** ****analyzer**. Measure the low-frequency gain, 3-dB frequency, and the magnitude and phase of the transfer function at 1kHz.

**3.**** **Apply a 1kHz 500mV sine wave signal to the input V_{i} and obtain the **time-domain**** ****waveforms**** **for the input and output voltages using the **scope**. Measure the magnitudes of the input and the output voltages, and the phase difference between them.

**4.**** **Apply a 1kHz 500mV square wave signal to the input V_{i} and obtain the **time-domain**** ****waveforms**** **for the input and output voltages using the **scope**. Measure the peak-to-peak voltage of the output.

**5.**** **Remove the resistor R_{2} and observe what happens to the output signal. Comment on the result.

**5.2 **** ****Pseudo**** ****Di****f****ferentiator**

**1.**** **Build the pseudo differentiator in Fig. 4 with the simulated component values.

**2.**** **Obtain the magnitude and phase **Bode**** ****plots **** **of the transfer function using the **network **** ****analyzer**. Measure the low-frequency gain, 3-dB frequency, and the magnitude and phase of the transfer function at 1kHz.

**3.**** **Apply a 1kHz 100mV sine wave signal to the input V_{i} and obtain the **time-domain**** ****waveforms**** **for the input and output voltages using the **scope**. Measure the magnitudes of the input and the output voltages, and the phase difference between them.

**4.**** **Apply a 1kHz 100mV triangular wave signal to the input V_{i} and obtain the **time-domain**** ****waveforms**** **for the input and output voltages using **scope**. Measure the peak-to-peak voltage of the output.

**5.3 **** ****Finite**** ****GBW**** ****Limitations**

**1.**** **Build the non-inverting amplifier in Fig. 5 with R_{1} = 1k Ω and R_{2} corresponding to the gain of 23.

**2.**** **Obtain the magnitude and phase **Bode**** ****plots**** **of the transfer function using the **network**** ****analyzer**.

**3.**** **Measure the low-frequency gain and 3-dB frequency.

**4.**** **Repeat the steps 1, 2, and 3 using R_{2} corresponding to the gain values of 57 and 83.

**1.**** **Build the unity-gain buffer in Fig. 6 with a load of 2.2k Ω in parallel with 100pF at the output.

**2.**** **Obtain the magnitude and phase **Bode**** ****plots **** **of the transfer function using the **network **** ****analyzer**. Measure the low-frequency gain, 3-dB frequency, and the magnitude of the transfer function at 75kHz and 150kHz.

**3.**** **Apply a 75kHz 1V sine wave signal to the input V_{i } and obtain the **time-domain**** ****waveforms**** **for the input and output voltages using the **scope**. Measure the **total**** ****harmonic**** ****distortion **** ****(THD)**** **on the output waveform using the **spectrum**** ****analyzer**.

**4.**** **Repeat the step 3 for 75kHz 2V sine wave, and 150kHz 1V sine wave input signals.

**6 **** ****Report**

**1.**** **Include calculations, schematics, simulation plots, and measurement plots.

**2.**** **Prepare a table showing calculated, simulated and measured results.

**3.**** **Compare the results and comment on the differences.

**7 **** ****Demonstration**

**1.**** **Build the circuits in Figs. 2, 4, 5 and 6 on your breadboard, and bring it to your lab session

**2.**** **Your name and UIN must be written on the side of your breadboard.

**3.**** **Submit your report to your TA at the beginning of your lab session.

**4.**** **For the lossy integrator in Fig. 2:

• Show the frequency response using the network analyzer.

• Show the time-domain output for a square-wave input using the scope.

**5.**** **For the pseudo differentiator in Fig. 4:

• Show the frequency response using the network analyzer.

• Show the time-domain output for a triangle-wave input using the scope.

**6.**** **For the non-inverting amplifier in Fig. 5:

• Use R_{1} = 1k and R_{2} determined by your TA (within 30k-50k), and estimate the gain and bandwidth.

• Show frequency response using the network analyzer and measure the gain and bandwidth.

**7.**** **For the unity-gain buffer in Fig. 6:

• Show the time-domain output for a +2V to -2V square-wave input using the scope.

• Measure the slew rate from the slope of the output waveform.