Tarun RepalaEE 3101

Experiment #6:

Operational AmplifierI. ABSTRACT

This lab investigates the characteristics of non-ideal operating behaviors of differential operational amplifier circuits using the LM741 op amp. The experiment looks into finding the low frequency response, cutoff frequency, bandwidth, and quality factor of a variety of amplification circuits using operational amplifiers. The close loop gain was determined theoretically and experimentally. The frequency response of four different gain circuits were observed by finding the bandwidth of the circuit. We observed three different active filters, which were low pass, slanted band reject, and band pass.

II. INTRODUCTION

An operational amplifiers have three terminals: a non-inverting input marked with a plus sign, an inverting input marked with a minus sign, and an output port for voltage or current amplification. The voltage at the inverting input is equal to the voltage at the non-inverting input. The current at each of the inputs is always zero.

The operating characteristics are investigated in the first part of the lab such as frequency response, dynamic range, and slew rate. A low frequency sine wave is used to determine an open loop gain. Then the voltage gain vs. frequency for amplifier having gains of 1, 5, 20, and 40 are measured. The slew rate of the op amp was determined with a square wave input.

This lab investigates some of the non-ideal operating behavior of the op amp. The emphasis is on the frequency response characteristics. Op amps used in the design of active filters is studied. The op amps are to be operated from +/-15V supplies. The transfer characteristics and its cutoff frequency of various circuit configurations are investigated to understand the active filters.

III. EXPERIMENT

Operating Characteristics

1. Using the circuit below to determine the low frequency (i.e. dc) open loop gain Av of the 741 op amp in your lab kit. Use a low frequency (less than 5 Hz) sine wave or triangle wave signal for the input Vin.

The Fig. 1 circuit below is used to measure the low frequency open loop gain of op amp.

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Fig 1. Test circuit to measure open loop gain

The equation for the open loop voltage of the test circuit is derived. An assumption of the op amp is that it has an infinite input resistance, zero output resistance and a finite voltage gain. The node connecting the three 100k resistors is defined as node A and node connecting the inverting input of amplifier to the 10 and 100k resistors as node B. The following derivation can be calculated:

Vin−Va100 k

=Va−Vb100 k

+ Va−Vout100 k

Vin=3Va−Vb−Vout

Va−Vb100 K

=Vb10

Va=Vb∗104

Vin=(3∗104−1 ) Vb−Vout

Vout=Av¿

Vb=−VoutAv

Vin=[−3∗104

Av−1]Vout

Av=−3∗104 VoutVout +Vin

The negative sign comes from the 180 degree phase shift between the input and output signals. To experimentally test the case, Fig. 1 was setup with a Vdd of +/- 15V. The input and output waveforms are graphed and the waveforms are 180 degrees out of phase. Using the derived equation above, the experimental open loop gain was −2.95∗104 V /V which is only slightly lower than expected.

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2. Construct an inverting amplifier with a voltage gain of 5 and investigate the linearity and dynamic range of its output as a function of op amp supply voltages for an input sinusoidal signal at 1 kHz.

The voltage supply for op amp is Vdd. To obtain a gain of 5, we use the circuit shown below and Ri is 2k and Rf is 10k.

Fig 2. Inverting amplifier with gain of Rf/Ri

Using an input sine signal of 4.2Vpp at a frequency of 1kHz. The output waveform for 5 different values of Vdd are shown below.

Vdd (V) Vout (Vpp) Vout (Vmean)+/-1.0 0.78 0.51+/-2.0 1.45 0.60+/-3.0 3.33 0.59+/-4.0 5.35 0.50+/-5.0 7.25 0.53

Table 1. Output voltages Vpp and Vmean at different Vdd supply values

The output peak-to-peak values increase when power supply values increase. This is expected since Vdd values will cause a clipping in the output signal if they are too small. Vmean seems to be around the same value.

3. Measure the voltage gain vs. frequency for an amplifier having gains of 1, 5, 20 and 40.

To obtain circuits that have these voltage gains, use circuit in figure 2 and change Ri and Rf. For voltage gain of 1, use Ri=Rf=2k ohms. For gain of 5, use Rf=10k ohm and Ri = 2k ohms. For the 20 gain, use Ri = 1k ohms and Rf = 20k ohms. Finally for 40 gain, use Ri=1k and Rf=40k ohms. All circuits use +/-15V from the power supply for the Vdd terminals.

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The corner frequencies of each of the gains are obtained by the output voltage where the midband gain is satisfied is multiplied by 0.707. The resulting voltage is found by altering the frequency to the cutoff or 3db down frequency.

Frequency (hz) Vin (v) Vout (V) Gain1000 1.08 1.08 1

100000 1.08 1.12 1.037037037150000 1.08 1.04 0.962962963170000 1.08 1 0.925925926200000 1.08 0.88 0.814814815230000 1.08 0.8 0.740740741250000 1.08 0.76 0.703703704280000 1.08 0.68 0.62962963400000 1.08 0.52 0.481481481500000 1.08 0.44 0.407407407

Frequency (hz) Vin (v) Vout (V) Gain1000 1.08 5.2 4.814814815

10000 1.08 5.4 515000 1.08 5.6 5.18518518520000 1.08 5.6 5.18518518530000 1.08 5.4 535000 1.08 5 4.6296296337000 1.08 4.8 4.44444444445000 1.08 4.2 3.88888888952000 1.08 3.44 3.18518518560000 1.08 3.12 2.888888889

100000 1.08 1.84 1.703703704200000 1.08 1 0.925925926

Table 2. Freq response of 1 gain Table 3. Freq. response of gain 5

Frequency (hz) Vin (v) Vout (V) Gain1000 0.26 4.2 16.15385

10000 0.26 4.4 16.9230820000 0.26 4.2 16.1538530000 0.26 3.6 13.8461535000 0.26 3.4 13.0769240000 0.26 3.2 12.3076950000 0.26 2.8 10.7692380000 0.26 2 7.692308

100000 0.26 1.8 6.923077

Frequency (hz) Vin (v) Vout (V) Gain1000 0.26 8 30.76923

10000 0.26 7.8 3015000 0.26 7 26.9230820000 0.26 6.2 23.8461525000 0.26 5.4 20.7692330000 0.26 4.6 17.6923135000 0.26 4.2 16.1538540000 0.26 3.8 14.6153850000 0.26 3 11.5384670000 0.26 2.4 9.230769

100000 0.26 1.8 6.923077

Table 4. Freq. response of gain 20 Table 5. Freq. response of gain 40

The cutoff frequency for 1 gain is 230kHz, for gain 5 is 48kHz, for gain 20 is 42kHz and for gain 40 is 23kHz. The bandwidth decreases as the gain increases.

4. Determine the slew rate of the op amp using a unity-gain voltage follower with a square wave input signal vs(t) at 50 kHz.

The effect of different sinusoidal input waveforms with various amplitudes and frequencies is observed. The value of the maximum slew rate is determined by differentiating the input signal.

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SR=2 πf V bp

The first signal observed is square wave input signal with 1 V bp with a frequency of 50kHz. The unity gain voltage follower in figure 2 with Ri=Rf=10k is set. Using the above equation, the theoretical slew rate is 3.58¿104 dv/dt. The graph below shows the output waveform of the signal. The experimental slew rate 9.8¿104 dv/dt, which is more than double the theoretical value.

Figure 3. Slew rate measurements for 1Vbp/50kHz

The next signal with sine wave input of 2.0Vbp with frequency of 100kHz. The theoretical slew rate is 1.17∗106 dv/dt and the experimental slew rate is found to be 7.43∗105dv/dt. The experimental slew rate is smaller than theoretical value.

The other two sine waves investigated have an amplitude of 1.5Vbp and a frequency of 75kHz and the other has amplitude of 0.5Vbp and frequency of 25kHz. The theoretical slew rates are

7.59∗105 dv /dt and 9.12∗104 dv /dt for 75kHz and 25kHz respectively. The experimental slew

rates were 6.57∗105 dv /dt and 1.0∗105 dv /dt respectively. The 75kHz signal, the experimental was a bit smaller than the theoretical. For the 25kHz signal, the experimental was larger. It is noticed that as the frequency and amplitude decrease, the slew rate decrease as well.

Active Filters

5. Construct the circuit shown below and determine its transfer characteristic and its cutoff frequency f3dB.

The circuit below is constructed. The cutoff frequency was found in the same manner as previously. The output voltage where the max gain was present was multiplied by 0.707 and it is used to find cutoff frequency. The equations for gain and cutoff frequency are as follows:

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Ao=−R2

R1

ω3 dB=1

C R2

Fig 4. Low pass filter

The theoretical gain and cutoff frequency are -1.96 v/v and 1.59kHz respectively. While configuring the circuit, 2.94Vpp gave a cutoff of 1.8kHz. This is a bit higher than expected with an error of 13.2%. The PSPICE simulation shows that the circuit is a low pass filter. The 3db cutoff frequency for the simulation is 1.5kHz, which is about 5% difference from theoretical calculation.

Fig 5. PSPICE simulation of low pass filter

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6. Construct the circuit shown in figure below and determine its transfer characteristic and its cutoff frequency 3dB.

There are two capacitors in this circuit, one is connected to the output and the other is connected to the non-inverting input. The theoretical cutoff frequency is found using the following

equation: ω3db=1

1.414 RC.

The cutoff frequency is calculated to be 1.59kHz. Setting up the circuit, the output of 1.58Vpp is at the cutoff frequency of 1.5kHz. The error is 5.66%, which is smaller than the previous circuit. The circuit seems to be a band reject filter when it was tested through a range of frequencies from 100Hz to 100kHz.

Fig 6. Band reject filter

This filter is different from figure 5 since inverting input of the 741 is directly connected to the output.

7. Construct the circuit shown in figure below. Determine its transfer characteristic and the center frequency at which the gain is a maximum.

This is the last active filter circuit. Theoretically, the mid-band gain H should be 1 with the center frequency and bandwidth equations given below:

ωo=√1+ R

Rr

√2 RC∆ ω= 1

RC

The center frequency and bandwidth are calculated to be 1.5kHz and 723.4Hz respectively. When the circuit is tested, the experimental mid-band gain is .95, a center frequency of 1.5kHz,

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and upper and lower frequencies 1.9kHz and 1.18kHz. This is a bandwidth of 720Hz. The error of the gain is 4.6%

Fig 7. Band pass filter

The next thing to determine is how the quality factor is effected by changing the value of R1 and R2. The quality factor can be determined using:

Q=ωo

∆ ω

The theoretical and experimental quality factor of 2.05 is achieved using the equation above. The quality factor is determined using R1 values of 50K and 10K ohms where R2=2 R1. With R1

=50K ohm and R2=100K ohm configuration, the theoretical center frequency is 946Hz and bandwidth is 318Hz. When the circuit is tested, a mid-band gain of .93, a center frequency of 950Hz, and lower and upper frequencies of 810Hz and 1.14kHz respectively. This gives a bandwidth of 330Hz with an error for the bandwidth of 3.68%. The theoretical quality factor is 2.97 and experimental is 2.88, which gives a 3.03% error.

For R1=10 K ohm and R2= 20K ohm configuration, center frequency is expected to be 2.34kHz with a gain of 1. The bandwidth is 1.6kHz and quality factor is 1.5. Testing the circuit, the gain is .93, corner frequencies of 1.7kHz and 3.3kHz, a bandwidth of 1.6kHz, and a quality factor of 1.5. The error is .6% for the bandwidth and 2.04% error for quality factor.

Analyzing the configurations, there is a direct relation between the quality factor and resistance. As R1 increases, the quality factor increases. The theory supports this since quality factor is equal to center frequency over the bandwidth. The quality factor is related to approximately the square root of the resistance.

IV. CONCLUSION

In this experiment, circuits were constructed and tested with op amps to investigate the non-ideal behavior of 741 op amp. The low frequency closed loop gain was determined both theoretical and experimental. The experimental gain was within 2% of the expectation. The dynamic range

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of the op amp was determined by varying the inputted power supply voltage at Vdd. There was a linear increase in output peak to peak voltage as the supply voltage increased but not with the output mean. The frequency response of the four different gain circuits were determined. The bandwidth of circuit increased as the gain decreased. The slew rate of op amp was investigated for four different input signals and it is concluded that with an increasing amplitude and frequency for the input waveform, the slew rate increased. Three different active filters circuits: low pass, band reject, and band pass were constructed. The cutoff frequencies of these transfer functions were found experimentally. By taking a closer look at band pass filter, the effect of changing the resistance value on the quality factor was observed. It was observed that increasing the resistance led to increasing quality factor.