View source for Operational Amplifiers (Op-Amps)

__TOC__
Op-amps and transistors are the staples of analog circuit design.  Internally, op-amps consist of many transistors, capacitors, and resistors; all crammed onto a small integrated circuit.

The symbol for an op-amp:[[Image:op_amp_symbol.gif]]

Each op-amp has an inverting input, a non-inverting input, and an output.  In practice, op-amps also must be powered, but these leads are often omitted from schematics.

==Ideal Op-Amps==
Op-amps are generally very well described by their ideal model.  There are several basic rules for ideal op-amps:

1.  The output voltage satisfies <math>V_{out}=A ( V^+ - V^- )</math>, where <math>V^+</math> and <math>V^-</math> are the voltages at the noninverting and inverting inputs, respectively. <math>A</math> is typically very large, such that we can consider it to be infinite.  In practice, the output voltage cannot go beyond the power supply '''rails'''.

A consequence of this is that if there is any connection from the output to the ''inverting'' input, the op-amp will do its best to keep the voltages at the two inputs equal.  This is called '''negative feedback'''.  ''(Note that the op-amp won't track voltages if you connect the output back to the positive input).''

2. The input current draw is zero&mdash;no current can flow in or out of the input terminals.  In practice, the input impedance is on the order of <math>10^6</math> to <math>10^{12}</math> ohms.

3. The output impedance is zero.  This means that there is no limit on the current the op-amp can source or sink.  In practice, read the specs to find the limit.

==Real Op-Amps==
Here are some of the more important differences between ideal and real op-amps.  These characteristics for an op-amp can usually be found in the data sheets from the manufacturer.  Explantions for the terms in the spec sheets can be found at [http://search.national.com/home/html/entry.html National Semiconductor's Knowledge Base]

<u>'''Input Offset Voltage'''</u>: In a real op-amp, there will be a slight voltage difference between the inputs.  This voltage difference can change with temperature.

<u>'''Input Bias Current'''</u>: This is the average current that flows through the two inputs.

<u>'''Saturation'''</u>: The output voltage is bounded by the positive and negative power supplies, known as '''rails'''.  In fact, many op-amps will only go up to a few volts short of rails.  If we look at the '''Output Voltage Swing''' in the data sheet for the LM411, we see that under a &plusmn;15V power source, the output will will generally be able to swing between &plusmn;13.5V&mdash;but you just might get a chip that can only output &plusmn;12V.  Op-amps that are designed to be able to output voltages very close thier rails are refered to as being '''rail to rail'''.

<u>'''Slew Rate'''</u>: The output voltage cannot change instantaneously; the maxiumum rate of change possible for the output voltage is known as the '''slew rate'''.

<math>SR=\frac{\Delta V}{\Delta t}</math>

<u>'''Rise Time'''</u>: The '''rise time''' of an op-amp is the time it takes for the output voltage to go from 10% to 90% of its final value, under a certain set of specified conditions set by the manufacturer.

<u>'''Common Mode Gain'''</u>: The '''common-mode voltage''' is the DC voltage shared by both pins(since they try to be the same).  Ideally, the output voltage of the op-amp should only depend on the voltage difference between the inputs, but real op-amps don't have such pefectly linear gains.  The effect that the common-mode voltage has on the gain is known as the '''common-mode gain'''.

==The 741 and 411 Type Op-Amps==
The 741 (bipolar) and 411 (FET) families of op-amps are popular chips made by many manufacturers.  The model number is often preceded by a modifier/identifier, such as "LM741" or "LF411".  These op-amps also come in varieties where you get multiple op-amps on a single package.

[[media:LM741.pdf|Click here for the data sheet of the LM741]]

[[http://www.national.com/ds/LM/LM148.pdf Click here for the data sheet of the LM148 quad 741]]

[[http://www.national.com/ds/LF/LF411.pdf Click here for the data sheet of the LF411]]


For now, we can ignore the "offset" pins.  These are used to make very fine adjustments in the reference voltages.

For more information on how to read the op-amp data sheets, try going to [http://search.national.com/home/html/entry.html National Semiconductor's Knowledge Base]

==Op-Amp Applications==
===Comparator===
Because of its huge gain, the op-amp is very sensitive to voltage differences between its inputs.  A few millivolts are enough to saturate it either way.  We take advantage of this property to make a voltage comparator, which will output either a high or low depending on the input.

[[Image:opamp_voltage_comparator.gif]]

===Voltage Follower===
The voltage follower doesn't amplify the voltage because the output is connected back to the inverting input.  However, it can be used as a buffer to isolate circuits or be used as a current amplifier.

[[Image:opamp_voltage_follower.gif]]

Combining the voltage follower with the
[[Diodes_and_Transistors#Push-Pull_Follower|push-pull follower]], we get the benefits of both:
zero crossover distortion (due to the op amp) and large output currents (due to the push-pull
transistors).

[[Image:opamp_voltage_follower_with_pushpull.gif]]

===Inverting Amplifier===
[[Image:opamp_inv_amp.gif]]

In the amplifier circuit above, we use feedback to regulate our gain.  The result is an amplifier that will invert the input signal and apply a gain to it.

Since the non-inverting input is grounded and there is negative feedback, the voltage at the inverting input is also at 0V, so:

<math>V_A=V_B=0</math>

Since the op-amp inputs cannot draw current, all the current will go through <math>R_1</math> and <math>R_2</math> to get to <math>V_{out}</math>.  This enables us to write:

<math>V_{in}=iR_1</math> and <math>V_{out}=-iR_2</math>

Combining these equations to eliminate ''i'' gives us

<math>V_{out}=-\frac{R_2}{R_1}V_{in}</math>

We now choose <math>R_1</math> and <math>R_2</math>; their ratios will specify our gain.  We typically choose values between 1k and 100k&mdash;if our resistances are too small, the circuit will waste power; if our resistances are too large, the tiny bit of current leaking through the op-amp may start to become noticeable.

===Non-Inverting Amplifier===

[[Image:opamp_non_inv_amp.gif]]

This amplifier is similar to the inverting amplifier, except it will not invert the signal.  We calculate the gain as follows:

<math>i=V_{in}/R_2</math>

<math>V_{out}=V_{in}+iR_2</math>

Substituting the first equation into the second yields:

<math>V_{out}=V_{in}\left(1+\frac{R_2}{R_1}\right)</math>

===Summer===
====Summer With Gain====
[[Image:opamp_summer_gain.gif]]

This summer circuit will output the inverted sum of the input voltages, after applying gains to the inputs.  A good choice for the resistors would be about 10k&Omega;.

Our equations:

<math>V_{out}=0-(i_1+i_2+i_3)R</math>

<math>i_1=V_1/R_1</math>&nbsp;&nbsp;&nbsp; <math>i_2=V_2/R_2</math>&nbsp;&nbsp;&nbsp; <math>i_3=V_3/R_3</math>

<math>V_{out}=-R_F\left(\frac{V_1}{R_1}+\frac{V_2}{R_2}+\frac{V_3}{R_3}\right)</math>

====Summer Without Gain====
If we do not want the summer to apply any gain to the inputs, simply choose all the resistor values to be the same.

[[Image:opamp_summer.gif]]

This summer circuit will output the inverted sum of the input voltages.  A good choice for ''R'' would be about 10k&Omega;.

<math>V_{out}=0-(i_1+i_2+i_3)R</math>

<math>i_1=V_1/R</math>&nbsp;&nbsp;&nbsp; <math>i_2=V_2/R</math>&nbsp;&nbsp;&nbsp; <math>i_3=V_3/R</math>

<math>V_{out}=-(V_1+V_2+V_3)\,</math>

'''Question:''' How could you build a simple digital-to-analog converter by using different input resistances?

===Differential Amplifier===
[[Image:opamp_differential_amp.gif]]

This circuit will amplify the voltage difference between V2 and V1.

<math>V_{out}=\frac{R_2}{R_1}(V_2-V_1)</math>

===Integrator===
We can build a circuit that will integrate voltage:

[[Image:opamp_integrator1.gif]]

The output signal is a scaled and inverted integral of the input signal:

<math>V_{out}=-\frac{1}{RC}\int_0^t V_{in}(\tau)d\tau</math>

There is a problem with this circuit though&mdash;the integrator is only good if the <math>V_{out}</math> is less than the maximum output voltage of the op-amp.  Our integrator is thus not very useful for low frequency signals, becuase the charge will store up on the capacitor and eventually saturate the op-amp.  Even if we have a high frequency signal, any DC offset will add up in the capacitor over time.  We can remedy this problem by adding a '''shunt resistor''' <math>R_s</math> across the capacitor to bleed off any long-term charges that store up in the capacitor.  As a rule of thumb, <math>R_s</math> should be greater than 10<math>R_1</math>.

[[Operational_Amplifiers_%28Op-Amps%29#Real_Op-Amps| Input bias current]] flowing through <math>R_1</math> and <math>R_s</math> can generate a small DC offset, and we can try to cancel it out by adding another resistor <math>R_2</math> between the non-inverting input and the ground such that

<math>R_2=\frac{R_1R_s}{R_1+R_s}</math>

[[Image:opamp_integrator2.gif]]

===Differentiator===
[[Image:opamp_differentiator.gif]]

The differentiator will scale and invert the derivative of the input signal:

<math>V_{out}=-RC\frac{dV_{in}}{dt}</math>

===Controllers===
==References==
Hayt, William H. Jr., Jack E. Kemmerly, and Steven M. Durbin. <u>Engineering Circuit Analysis</u>. 6th ed. New York:McGraw-Hill, 2002.