Bandwidth and Stability Equipment • • • •
Oscilloscope Function Generator (FG) Breadboard 9V Batteries
• Resistors: 1 MΩ, 100 kΩ, 9.1 kΩ, 1 kΩ • Capacitors: 0.01 µF, 0.1 µF, 100 pF • LF411 or LF351 OpAmp
Goals of the Experiment Theory When performing linear operations with an OA one must take into consideration several limitations of the amplifier. One of these is the frequency response of the amplifier. The manufacturer must limit op-amp internal gain to curve 1 in Figure 9.1 to avoid oscillation due to unavoidable internal delays. As the amount of negative feedback is decreased, producing higher amplification, the gain of the system becomes more dependent on the amplifier and less of a function of the feedback network. Therefore, the bandwidth of the OA which has a negative feedback loop depends on the (open loop) internal gain of the amplifier. The drop in gain at high frequencies in the open-loop bandwidth characteristic (curve 1) and closed-loop (curve 2) is shown in Figure 9.1. The dashed line [1] shows the open-loop gain Aopen which may begin at near a million, but begins to fall at frequencies above a few tens of Hertz and reaches one at a frequency of f unity The double dashed line [2] shows the closed loop gain, as set by the feedback network. It is constant until it intersects the open loop gain line, then both open loop and closed loop gains follow the same characteristic reduction with frequency. The product of gain and bandwidth is roughly a constant for any OpAmp and is given in the manufacturer’s specification sheet for the device. Note that by convention, bandwidth is defined by the point where the gain, A = (0.707) Aclosed A practical consequence is that we must not be greedy for too much gain in a single op-amp stage. Three op-amps, each with gain 10 fold will provide gain of 1000 total, with bandwidth of 300000 Hz, while a single op-amp with gain 1000 provides bandwidth of only 3000 Hz.
Figure 9.1: A plot of opamp gain, in both open ans closed loops, as a function of frequency. [1] open loop 6
log10 ( Gain)
5 4 [2] closed loop
3 2
−6 dB/Octave slope
1
1
2
3
4
5
log10 ( f )
6 f unity
Experimental Procedure 1. Measure −3 dB (i.e. Half-power) point • Connect the circuit as shown in Figure 9.2 • For each resistor value in the Table 9.1, using 500 Hz input, adjust the function generator amplitude until the opamp output is undistorted. • Use the oscilloscope to measure the ratio of the output to input amplitudes at 500 Hz (mid frequency). • Determine the frequency at which the gain is 0.707 (−3dB) of the mid frequency gain. • Record the results in Table 9.1. Table 9.1: Measure the gain and -3dB frequency so that the gain bandwidth product can be calculated. R1 (kΩ) 9.1 100 1000
A0 Gain at 500 Hz
fc −3dB Frequency (
)
A0 f c Gain Bandwidth Product (
Average A0 f c :
2
)
Figure 9.2: Circuit diagram of a simple inverting amplifier. 1 MΩ
1 kΩ
2
−
3
6
+
Vout
Vin
2. Calculate slope and bandwidth • Superimpose the results of gain versus frequency for various gain conditions on the log plot reproduced below. • Calculate the slope of the amplifier’s open-loop "roll-off" in dB per octave form Figure 9.3. • Calculate A0 f c , the gain-bandwidth product, by averaging the three measurements. This is the unity gain bandwidth. • Determine the unity-gain bandwidth of the OA from the OA spec sheet. • Compare the spec shet A0 f c to the calculated value, and find the percent error. • Record the results in Table 9.1.
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Figure 9.3: The open loop frequency response of a LF351 op-amp. This information was taken from the LF351 data sheet. Open Loop Frequency Response
Open Loop Voltage Gain (dB)
120 100 80 60 40 20 0 100
101
102
103
104
105
106
107
Frequency (Hz)
Table 9.2: Compare the calculated gain bandwidth product to the one provided by the manufacture. Average A0 f c : ( ) Slope in dB per Octave:
Manufacturer’s spec A0 f c : ( )
4
Percent Error: (%)
OpAmp Current to Voltage Converter There are two primary causes of un-expected opamp behaviour: • Inadequate power supply bypassing (easily fixed with lots of capacitors, close to pins). • Circuit details which add delay to the feedback loop. 1. Excess delay in the feedback loop is usually the result of stray capacitance at the inverting input pin. This might be an unavoidable aspect of the detector, or it may be excess wire length. Using the circuit already wired previously, select gain 100, and apply a 50 mV square wave at 1 kHz. Figure 9.4: Circuit diagram of an inverting amplifier with a capacitor added in parallel. 1 MΩ
0.01 µF 1 kΩ
2
−
3
Vout
6
+
Vin Place a 0.01 µF capacitor from the minus input (pin 2) to ground, i.e. across the 1 kΩ resistor. Sketch the resulting output in Figure 9.5. CH1 Sensitivity (V/div)
CH2 Sensitivity (V/div)
Time Base
Figure 9.5: Draw the sine wave before and after passing through the integrator. Notice that the problem can be fixed by adding a 100 pF capacitor across the 100 kΩ resistor. This is the only solution if the input capacitance cannot be eliminated - but note that the bandwidth is now 5
reduced because of the lowpass cutoff frequency of the 100 kΩ resistor and 100 pF capacitor after adding the 100 pF capacitor, sketch the resulting waveform in Figure 9.5. Figure 9.6: Circuit diagram of an inverting amplifier with a capacitors added in parallel to the resistors. 100 pF 1 MΩ
0.01 µF 1 kΩ
2
−
3
Vout
6
+
Vin 2. The other common cause of extra delay is capacitance directly on the output of the op-amp. This often is not actually in the form of a capacitor but in the form of a long piece of transmission line which is supposed to carry the output signal to its destination. Remember that a coax wire is at least 100 pF per meter of length. To see this effect remove the 0.01 and 100 pF capacitors from your circuit. Keep the squarewave at 50 mV and 1 kHz. Simply place a 0.1 µF capacitor from output to ground and sketch the resulting waveform in Figure 9.5. Figure 9.7: Circuit diagram of a inverting amplifier with capacitance at the out put of the opamp. 1 MΩ
1 kΩ
2
−
3
Vout
6
+
0.1 µF
Vin Simulates a length of coaxial cable without termination resistors The cure for this problem is to terminate the transmission line at both ends, as it ought to have been to begin with. 6
Figure 9.8: Circuit diagram of a inverting amplifier with capacitance at the out put of the opamp. 1 MΩ Coaxial Cable 50 Ω 1 kΩ
2
CH2
−
3
+
CH1
6
50 Ω
Vin
While these capacitances are obviously large, when using high frequency op-amps with 100 megahertz bandwidths just tens of picofarads will cause these effects. For this reason a special class of opamps with current feedback have become popular. Their bandwidth does not have to be reduced at high gain, but the inverting input is not as versatile. All high frequency op-amps must be backterminated with a resistor equal to the characteristic of the transmission line carrying the output. Special construction techniques can help reduce stray capacitance. CH1 Sensitivity (V/div)
CH2 Sensitivity (V/div)
Time Base
Figure 9.9: Sketch the waveform before and after it passes through the coaxial cable.
Your Guide to Success • A power supply bypass capacitor (0.01 u f ) on each plus and minus supply at each op-amp. • Minimize (or isolate) capacitance at inverting input pin. If unavoidable, compensate with equivalent (1/gain) feedback capacitance. 7
• Minimize (or isolate) capacitance at the output. Couple to transmission line with back terminating resistor. • If high gains are required, distribute over several op-amps stages. • Some high frequency op-amps are under-compensated and must be used at or above some specified gain or they will oscillate. (e.g. LF357, LM6365 ). • Use a single point as local ground in high gain or high bandwidth stages. • See capacitance everywhere at high frequencies and shield stages from each other. • Current feedback topology is an option but be aware of differences in setting gain (e.g. LM6181). • High performance op-amps are relative; to achieve one high rating, other ratings are compromised: – High input impedance - lower speed and precision LF411 – High precision - lower speed and lower input impedance LM11 – High voltage output - lower speed, lower precision and input impedance, heating LM343 – High current output - lower speed, lower precision and input impedance, heating LM759 – High bandwidth - lower precision and input impedance, tendency to oscillate LM6361
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