Experiment 11
Operational Amplifiers I
Ideal op-amps
An op-amp is a differential amplifier with an inverting V− input and non-inverting V+ input. The output
voltage Vo is given by the difference of these two input voltages times the open loop gain Av :
Vo = Av ∗ (V+ − V− )
A standard way to derive approximate theoretical equations for the circuits involving op-amps is to assume
that the op-amp is an ideal device having the following electrical characteristics:
1. the inputs draw no current, hence i+ = i− = 0 and the input impedance Z+ = Z− = ∞,
2. the output can supply an infinite amount of current, hence Zo = 0,
3. the open loop gain, or voltage amplification Av = ∞.
4. The opamp adjusts the output voltage so that V− = V+ . This follows from Equation 11.1 since
Vo cannot exceed the finite power supply voltage. This equivalence is used to determine the gain
equation for an (ideal) op-amp circuit.
The LM358 op-amp
The LM358 consists of a pair of general purpose
operational amplifiers capable of amplifying signals
ranging from 0 Hz (DC) to 1 MHz. The chip can
operate using a dual power supply of up to ±15 V
down to a single 3 V battery. It can be used in
mixed analog/digital circuits that typically operate
from a single 3-5V power supply.
? From the LM358 data sheet, determine the
values of Av , the input bias current and output source current. Does the LM358 approximate the characteristics of an ideal op-amp?
Figure 11.1: LM358 pinout
? The slew rate dVo /dt defines the maximum rate of change in Vo . What is the LM358 slew rate? Does
the frequency response of the amplifier depend on the amplitude of the signal?
Figure 11.2: Operational amplifier power supply options
Open loop operation
Since Av of the op-amp is very large, a tiny voltage difference between the inputs causes the output to swing
between the power supply limits, or saturate. This effect can be used to implement a voltage comparator
or level detector. One input is set to a reference voltage determined by R1 and R2 . The output will change
state based on the voltage at the other input.
Use EWB to simulate the circuit in Figure 11.3. The voltage divider resistors set V+ . Their choice
must satisfy the input bias current requirement of the op-amp and can be in the range of 10 to 100
KΩ. Monitor the output voltage as you vary V− . What is the smallest voltage change that will
saturate the op-amp output to the limits of the power supply? Estimate the open loop gain. Swap
the inputs to the op-amp. What do you observe? In what ways is this op-amp switch an improvement
over the transistor switch discussed in the previous lab?
The circuit of Figure 11.3 modulates the output between two voltages by comparing the value of Vin to a
reference voltage at the other input. In essence, this is a
1-bit analog-to-digital converter. To observe signal modulation, first remove the resistors from the op-amp.
To one input of an op-amp connect a 0-5V triangle
wave of frequency f0 . This is typically a waveform
of set frequency and amplitude known as the carrier
signal or carrier. To the other input connect a 05V sine wave of a frequency around f0 /50. This is
the modulating signal.
Figure 11.3: Voltage comparator
? Describe the dynamics of the output waveform. What is being modulated? How do changes to the
frequency/amplitude/offset of the modulating signal affect the output signal?
Closed loop operation
Application of feedback from Vo to V− causes the op-amp
to conform to Rule 4 mentioned in the introduction. This
arrangement, shown in Figure 11.4, is known as a voltage
follower or unity gain amplifier.
How might this op-amp arrangement be useful?
What is being amplified? Derive the gain equation.
Figure 11.4: Voltage follower
An analog memory cell
It is sometimes necessary to temporarily store a voltage. This is required when converting a voltage to
a digital value, or to implement an analog delay. Figure 11.5 shows the schematic of a typical track and
hold circuit. When the switch is closed, Vo tracks Vin . With the switch open, the capacitor is effectively
isolated from Vin and Vo reflects the voltage stored in the capacitor.
? What opamp characteristics are desirable in this type of circuit? What are the benefits/limitations
imposed upon the circuit by the resistor?
? The value of C chosen is critical to the proper operation of a track and hold circuit. What determines
the value for C and hence the range of operation of this circuit?
Figure 11.5: Sample and hold circuit
Set Vin to a 1 Hz sine wave. Describe the output as the switch is opened and closed. In track mode,
what type of circuit is represented? Estimate a practical Vin bandwidth. As you increase the Vin
frequency, what do you observe?
Connect a multimeter to the output of your track and hold circuit. With the circuit in hold mode,
determine the discharge rate of the capacitor. How long before Vo drops by 1%?
Note: A typical circuit would incorporate an analog switch or FET to perform the sample/hold switching
and more importantly, the switch would be controlled by some electronically generated pulse. The contacts
of a mechanical switch oscillate for about 5 ms as they open or close. This contact ’bounce’ genarates a
series of pulses that would wreak havoc in a typical analog or digital circuit.
Analog computation
Figure 11.6: Two op-amp solution of Y = mX + b
The op-amp was originally designed to perform mathematical operations form basic addition to multiplication, exponentialtion and the solution of differential equations. The electrical behaviour of resistors,
capacitors and diodes are used to this end. While not as precise as digital devices, analog computers are
very fast and simple to implement and do not require data conversion to and from the digital domain.
Figure 11.6 shows a two op-amp circuit that can be used to evaluate the equation
Y = mX + b ≡ m ∗ (X + b/m).
The first adds an offset b/m to Vin . The second op-amp sets the gain, or slope m.
Derive the transfer function for the two op-amp circuit of Figure 11.6.
Figure 11.7: Single op-amp solution of Y = mX + b
It may not seem readily apparent, but the circuit of Figure 11.7 also evaluates Y = mX + b using a
single op-amp. Due to the feedback path, the op-amp adjusts the output Vo so that V− = V+ . Because of
the very large input impedance of the op-amp, no appreciable current flows into the op-amp inputs and it
can be removed, leaving for analysis two separate circuits.
Show that the equation below is valid and that it does represent the equation Y = mX + b:
Vo =
Rf + R
∗ (V1 + V2 )
A practical example
It is often useful to convert a transducer output voltage to a voltage range that quantitatively represents
the actual quantity that the sensor measures.
Suppose that you wish to build an analog thermometer calibrated to display temperature on a voltmeter
in units of 100mV/◦ C so that 0◦ C displays 0V, 10◦ C displays 1V, and so on. Suppose that the temperature
sensor used is an LM61 temperature-to-voltage converter. The output voltage of this device corresponds
to 600mV at 0◦ C and varies linearly at a rate of 10mV/◦ C.
Determine the transfer function parameters required to properly display the LM61 output as tem!✠
perature on the voltmeter.
For each of the circuits of Figure 11.6 and Figure 11.7, use the coordinate pairs evaluated above
to determine the fitting parameters for the transfer function. Evaluate appropriate resistance and
reference voltages, then build the circuit.
Sweep the input voltage representing the output of the LM61 and verify that the two circuits give
identical results.
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