# Operational Amplifiers (Op-Amps)

## Contents |

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.

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 *V*_{out} = *A*(*V*^{ + } − *V*^{ − }), where *V*^{ + } and *V*^{ − } are the voltages at the noninverting and inverting inputs, respectively. *A* 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*.

2. The input current draw is zero—no current can flow in or out of the input terminals. In practice, the input impedance is on the order of 10^{6} to 10^{12} 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 National Semiconductor's Knowledge Base

__ Input Offset Voltage__: In a real op-amp, there will be a slight voltage difference between the inputs. This voltage difference can change with temperature.

__ Input Bias Current__: This is the average current that flows through the two inputs.

__ Saturation__: 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 ±15V power source, the output will will generally be able to swing between ±13.5V—but you just might get a chip that can only output ±12V. Op-amps that are designed to be able to output voltages very close thier rails are refered to as being

**rail to rail**.

__ Slew Rate__: The output voltage cannot change instantaneously; the maxiumum rate of change possible for the output voltage is known as the

**slew rate**.

__ Rise Time__: 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.

__ Common Mode Gain__: 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.

[Click here for the data sheet of the LM471]

[Click here for the data sheet of the LM148 quad 471]

[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 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.

### 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.

We can put this into our push-pull follower to reduce the amount of current needed to drive the circuit and eliminate crossover distortion.

### Inverting Amplifier

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:

*V*_{A} = *V*_{B} = 0

Since the op-amp inputs cannot draw current, all the current will go through *R*_{1} and *R*_{2} to get to *V*_{out}. This enables us to write:

*V*_{in} = *i**R*_{1} and *V*_{out} = − *i**R*_{2}

Combining these equations to eliminate *i* gives us

We now choose *R*_{1} and *R*_{2}; their ratios will specify our gain. We typically choose values between 1k and 100k—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

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

*i* = *V*_{in} / *R*_{2}

*V*_{out} = *V*_{in} + *i**R*_{2}

Substituting the first equation into the second yields:

### Summer

This summer circuit will output the inverted sum of the input voltages. A good choice for *R* would be about 10kΩ.

*V*_{out} = 0 − (*i*_{1} + *i*_{2} + *i*_{3})*R*

*i*_{1} = *V*_{1} / *R* *i*_{2} = *V*_{2} / *R* *i*_{3} = *V*_{3} / *R*

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

### Differential Amplifier

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

### Integrator

We can build a circuit that will integrate voltage:

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

There is a problem with this circuit though—the integrator is only good if the *V*_{o}*u**t* 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** *R*_{s} across the capacitor to bleed off any long-term charges that store up in the capacitor. As a rule of thumb, *R*_{s} should be greater than 10*R*_{1}.

Input bias current flowing through *R*_{1} and *R*_{s} can generate a small DC offset, and we can try to cancel it out by adding another resistor *R*_{2} between the non-inverting input and the ground such that

### Differentiator

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

### Controllers

## References

Hayt, William H. Jr., Jack E. Kemmerly, and Steven M. Durbin. __Engineering Circuit Analysis__. 6th ed. New York:McGraw-Hill, 2002.