**David E. Thompson, Professor**

**University**** of ****Idaho**

In signal conditioning, perhaps the most useful and flexible active device available is the Operational amplifier, or op-amp. The op-amp was named because of its ability to perform several mathematical operations (add, subtract, multiply by constant, differentiate, integrate, and more). For many applications, it has been superseded by the digital computer, but op-amps are an essential element of many instrumentation solutions.

Figure 1 portrays a K2-W tube-based general purpose computing Op-Amp from George A. Philbrick Research first introduced in 1952. The op-amp is shown with and without its bakelite shell. The first solid-state monolithic op-amp (not shown) was manufactured by Fairchild Semiconductors and sold for $300 in 1963 as the µA702. It had a tendency to burn out when loaded, contained just nine transistors, and sales were primarily Military and Aerospace consumers. This was followed up by the µA709 with higher gain, a larger bandwidth, lower input current, more user-friendly supply voltage requirements, and a $70 price. Today op-amps sell for pennies in large quantities, and are infinitely more powerful and useable than their predecessors.

The op-amp is constructed from several transistor stages and commonly employs a differential-input stage, an intermediate-gain stage and a push-pull output stage. The differential amplifier consists of a matched pair of bipolar transistors or FETs. The push-pull amplifier has the potential to provide large currents to the op-amp’s load and hence we state that the op-amp has a small output impedance. It typically also has an extremely high input impedance and thus limits the current drain from sensors and other instrumentation elements to very small levels (nano- to pico-amps). This ability to serve as a buffer to shield a sensor from the load of a measurement system is important, but the ability to manipulate the signal to improve its readability and reduce noise is much more important.

The op-amp is a highly linear device with its output voltage
proportional to the input voltage, or *V _{out}*

Like most highly successful and widely used integrated
circuits, costs can be quite small. The
cheap models operate (with external components limiting the overall gain to 1)
from DC to about 10MHz, while the high-performance models may operate up to
1 GHz. A popular device ($0.18 each)
is the LM411 dual op-amp whose open loop gain is shown in **Error!
Reference source not found.**.
Note that this is a log-log plot of gain (output/input) versus
frequency, and shows that the response curve is flat down to very low
frequencies and even to a zero frequency input voltage (DC). The gain drops 1 decibel/decade from around 2
Hz. Electrically, this model amplifier
has an input bias current of about 50pA and total harmonic distortion of
0.02%. Op-amps are integrated circuits (IC’s)
and usually available in an 8-pin dual, in-line package (DIP) as well as other
forms. Some ICs have more than one
op-amp on the same chip to improve the density on printed circuit boards.

Before proceeding we define a few terms:

**linear****
amplifier** - the output is directly proportional to the amplitude of
input signal.

**open-loop****
gain, A** - the voltage gain without feedback (about 10^{6}).

**closed-loop****
gain, G** - the voltage gain with negative feedback.

**negative****
feedback** - the output is connected to the inverting input forming a
feedback loop (usually through a *feedback resistor*, R_{f}).

Figure 3.a shows a complete diagram of an operational
amplifier. A more common shorthand version of the diagram is shown in **Error! Reference source not
found.**b, where missing parts are assumed to exist that
result in different gains for different inputs. The inverting input means that
the output signal will be of opposite polarity with the signal applied to this
input. For harmonic signals, that means
a 180 degrees phase shift. On the diagram, typical power supply voltages are V+
= +15Vdc, and, V- = -15Vdc.

For dual supply amplifiers, V_{cc}
is typically, but not necessarily, ±15V. The positive and negative voltages are
necessary to allow the amplification of both positive and negative signals
without introducing a bias to the voltages.
Since many systems must interface to computers, many op-amps have been
designed to operate from a single supply voltage at +5Vdc and the reference
voltage for alternating signals raised to
approximately 2.5vdc.

For a linear amplifier (and a differential amplifier) the open-loop gain is A and the output is given as

_{} Equation
1

Figure
3. Operational amplifier a) Schematic diagram, b) Shorthand.

As implied in Figure 3 and
Equation 1, two conditions must be satisfied for linear operation:

- The actual input voltage at the amplifier must operate within a very narrow voltage range since A is so large, or

_{}

- For no clipping or saturation, the output voltage swing must be restricted to

_{}

Figure 4. Input-output relationship for an open-loop op-amp.

The following are properties of an ideal amplifier, which to a good approximation are obeyed by an operational amplifier:

- Large forward transfer function (high open loop gains, low output impedance).
- Virtually nonexistent reverse transfer function.
- Large input impedance, (any signal can be supplied to the op-amp without loading problems).
- Small output impedance, (the power supplied by the op-amp is not limited).
- Wide bandwidth.
- Seemingly infinite gain, A>>>1.

If these approximations are followed, two rules can be used to analyze op-amp circuits:

**Rule 1:** The input currents are negligible.

**Rule 2:** The input voltages are essentially equal.

To apply these rules requires negative feedback.

Feedback is used to control and stabilize the amplifier gain. The open-loop gain is too large to be useful since even the slightest input noise will causes the circuit to clip and/or saturate. Stabilization is obtained by feeding the output back into the input (closed negative feedback loop). In this way the closed-loop gain does not depend on the amplifier characteristics.

Consider the circuit shown in Figure 5. One can sum voltages and currents using Thevanin’s Theorem results in the following equations.

Figure 5. Inverting amplifier with currents and voltages shown.

See if you can identify the loops used to generate these equations.

_{} Equation
2

_{} Equation
3

As shown in Figure 4, we can also write following equation for the
linear region of the Op-Amp:

_{} Equation
4

Summing the currents at the location of *e _{b}* yields (since

_{} Equation
5

Solving these equations simultaneously yields the gain equation for the amplifier as shown in Equation 4.

_{} Equation
6

For large open loop gain, *A>>1*, this reduces to *G
= -R _{f}/R_{i}*. This is the classic gain for an inverting
operational amplifier. One can use this
general derivation approach for any configuration of components that modify the
gain of an operational amplifier.

Figure 6
shows a non-inverting amplifier, sometimes referred to as a *voltage
follower*.

Figure 6. Circuit and shorthand diagram for a non-inverting, unity-gain amplifier.

Applying our rules to this circuit we have E_{in}=E_{out}
and i_{in}=0.
The amplifier has a unit closed-loop gain, G=1, and does not change the
sign of the input signal (no phase change for a harmonic signal). This configuration is often used to buffer
the input to an amplifier since the input resistance is high,
there is unit gain and no inversion. The *buffer amplifier* is also used
to isolate a signal source from a load.

The use of inverting amplifiers is often of the greatest
utility and one can achieve a non-inverting amplifier by cascading two
inverting amplifiers. The general
schematic for an inverting amplifier is shown in Figure 7. Here, the
gain, G = - R_{f}/R_{i} allowing us
to multiply a signal by a constant.

Figure 7. A simple inverting op-amp circuit.

Similarly, one may use a modification of this circuit to add a number of voltages at different gains.

Figure 8. Summation of voltages.

What other operations can we perform with
op-amps? How about integration and
differentiation as shown in Figure 9a and Figure 9b,
respectively.

Figure 9. Integrator and differentiator circuits.

In attempting to manipulate signals with frequency content, it is important to note that Gain is not a simple number, but a complex number with both amplitude and phase that are frequency dependent. We will cover this in detail later. If we consider an op-amp configured as a simple amplifier, what is the frequency range of operation for a gain of 1000? The answer is labeled as the DC-100X curve in Figure 10.

Figure 10. DC and AC coupled amplifier harmonic response curves.

Similarly, if a small capacitor is put in front of such an
amplifier and the gain adjusted to G=1, the response curve will be the curve
AC-1X as shown. As an engineer, when you
calibrate an instrumentation system, it is important to vary the input
frequency and determine the shape of the gain curve from Gain, *G = E _{out}/E_{in}*
as well as any phase changes. We will
discuss determining the complete harmonic response of an instrument later.

In any practical circuit, there may be deviations from the ideal op-amp. In general, these are often negligible, but for precision circuits we may need to add circuitry to improve the result.

Even though modern op-amps are great, they may still have a bothersome offset or bias voltage inherent to the integrated circuit. In a voltage follower circuit, often a feedback resistor is used as shown in

Figure 11 along with a resistor to ground to introduce a variable gain. Note that we still have negative feedback for stability.

Figure 11. Non-inverting amplifier with variable gain.

For this circuit, the gain, **G = 1+(R _{f}/R_{i})**. This is especially useful for circuits which can
only have a single supply voltage (typically +5v in digital applications). The input impedance is simply R

An improvement in the circuit shown in Figure 6 allows us to compensate for the small voltage
offsets, E_{off}, resulting from the bias
currents, i_{b}, at
the input of a real amplifier as shown in Figure
12.

Figure 12. Inverting amplifier with bias compensation.

One is frequently faced with the necessity to perform a differential operation. The schematic for this operation is shown in Figure 13. If you study this configuration, you will note that both the output signal and the ground reference are feedback in this application.

Figure 13. A balanced differential amplifier.

There are a growing number of modern sensors that produce an
output current that is proportional to the input variable being measured (measurand). A simple current-to-voltage converter is shown
in Figure 14. Applying our
ideal amplifier rules gives and output voltage E_{out}
= i_{in}R_{f}.

Figure 14. Current-to-voltage converter.

The use of current-producing sensors is helpful in lengthy cable runs where the voltage losses due to the resistance of the cable will affect the measurements.

The
output of an integration circuit as shown in **Figure
15** is stable because it effectively integrates out any
input noise. On the other hand, a small
DC offset error on the input will cause the output to grow continually with time
until the op-amp saturates. When the
output reaches either the positive or negative power supply rail, the op-amp
cannot drive the output higher, and the output is said to “saturate”. It is therefore important to avoid even the
smallest offset error on the input to an integrator circuit.

Figure 15. An integrator circuit with
a gain of -1/R_{i}C_{f}.

The output of a differentiation circuit is often unusable because of input noise. Because the derivative of a signal includes high frequency noise and the output of a differentiator is proportional to its frequency, we have to limit the bandwidth of the circuit.

Figure 16. A practical differentiator circuit.

The frequency response of this system is shown in as
depicted in Figure 16. This looks
like a combination of a differentiator and an integrator circuit, which it is.
You will note that the maximum gain is limited by G_{max}
= -R_{f}/R_{i}. The slope of an ideal differentiator on the
log-log plot is +1 since d/dt(e^{j}^{wt}) = jwe^{j}^{wt}.

Figure 17. Frequency response of ideal integrator (red),
differentiator (blue), and amplifier(gold) and their
combination (green).

We can combine the above inverting, summing, offset, differentiation and integration circuits to build an analog computer that can solve differential equations. However, in modern applications, differentiator and integrator circuits are mainly used to condition signals. Another important application is supporting digital data acquisition.

There are times when we wish to grab a voltage and hold it
for some period of time. This is
supported by the Sample-Hold configuration shown in Figure 18. The
capacitor, *C _{s}*, is the
sampling capacitor, and it’s leakage is a key factor
in how long the sample is retained.

Figure 18. Sample-Hold amplifier configuration.

The circuit in Figure 19 is an instrumentation amplifier with an overall gain of 90 and uses matched 0.1% resistors. This design was provided by Analog Devices who sells a single IC Instrumentation Amp to replace such a circuit. Their product (AD623) uses only a single external resistor to set the gain and can be studied in more detail at the URL below:

http://www.analog.com/UploadedFiles/Application_Notes/16110895AN-539.pdf

This example has excellent common-mode voltage and noise rejection, and an excellent application is to amplify the output from a Wheatstone bridge.

The following are some internet website references to common op-amps.

**National
Semiconductor**

Company website

Low Power Dual PDIP (8) OpAmps ($0.18)

http://www.national.com/pf/LM/LM358.html

Low Drift & Offset Dual JFET MDIP (8) OpAmps ($0.45)

http://www.national.com/pf/LF/LF411.html

Low Drift & Offset Dual JFET MDIP (8) OpAmps ($0.55)

http://www.national.com/pf/LF/LF412.html

**Analog Devices**

Company website

AD623, single or dual supply,
rail-to-rail instrumentation amplifier ($1.25)

http://www.analog.com/en/prod/0,,759_782_AD623%2C00.html

OP284, dual amp, single or dual
supply, rail-to-rail operational amplifier ($2.70)

http://www.analog.com/en/prod/0,,759_786_OP184%2C00.html