ENRG 312: Measurement and Data Acquisition System Notes

Mechatronics and Measurement Systems

What is a Mechatronics System?

Mechatronics is a rapidly developing interdisciplinary field of engineering dealing with product design that integrates mechanical and electronic components coordinated by a control architecture. The primary disciplines involved are mechanics, electronics, controls, and computer engineering. Mechatronic systems are referred to as smart devices and require engineers to design and select analog/digital circuits, microprocessor-based components, mechanical devices/structures, sensors, actuators, and controls to achieve the desired goal.

Mechatronics System Components

Typical components include:

• Actuators: Produce motion or cause action.

• Sensors: Detect system parameters, inputs, and outputs.

• Digital Devices: Control the system.

• Conditioning and Interfacing Circuits: Provide connections between control circuits and input/output devices.

• Graphical Displays: Provide visual feedback to users.

Measurements System

A fundamental part of many mechatronic systems is a measurement system, which typically includes:

1. Transducer: A sensing device that converts a physical input into an output, usually a voltage. Often referred to as a sensor.

2. Signal Processor: Performs filtering, amplification, and signal conditioning on the transducer output.

3. Recorder: An instrument, computer, hard-copy device, or display that maintains the sensor data for monitoring or processing.

Measurement systems are traditionally used to measure physical and electrical quantities like mass, temperature, pressure, capacitance, and voltage. They can also locate things or events, such as motion from an earthquake. A measurement system is often integrated into a control system, emphasizing the importance of measurement in control. Measurement systems can also be used as stand-alone devices for data acquisition in laboratory or field environments.

Digital Thermometer Example

• Thermocouple: Transducer that converts temperature to a small voltage.

• Amplifier: Increases the magnitude of the voltage.

• A/D Converter: Changes the analog signal to a coded digital signal.

• LEDs: Display the value of the temperature.

Basic Electrical Elements

Importance

Electric circuits and components are important for understanding and designing all elements in a mechatronic system, especially discrete circuits for signal conditioning and interfacing. Practically all mechatronic and measurement systems contain electrical circuits and components.

Passive Electrical Elements

There are three basic passive electrical elements:

• Resistor (R)

• Capacitor (C)

• Inductor (L)

These elements require no additional power supply, unlike active devices such as integrated circuits.

Ideal Energy Sources

There are two types of ideal energy sources:

• Voltage Source (V)

• Current Source (I)

The passive elements are defined by their voltage-current relationships.

Resistor

A dissipative element that converts electrical energy into heat. Ohm’s law describes the voltage-current relation for an ideal resistor.

$V = IR$

A real resistor has a limited power dissipation capability designated in watts and may fail if this limit is exceeded. If a resistor’s material is homogeneous and has a constant cross-sectional area, such as a cylindrical wire, then the resistance is given by

$R = \frac{\rho L}{A}$

where:

• $\rho$ is the resistivity

• $L$ is the wire length

• $A$ is the cross-sectional area

Example

Determine the resistance of a copper wire 1.0 mm in diameter and 10 m long. $R = \frac{\rho L}{A}$ For Copper: $R = \frac{(1.72 \times 10^{-8} \Omega \cdot m)(10 m)}{\pi (0.5 \times 10^{-3} m)^2} = 0.219 \Omega$

Capacitor

A passive element that stores energy in the form of an electric field. The simplest capacitor consists of a pair of parallel conducting plates separated by a dielectric material. The capacitor’s voltage-current relationship is defined as:

$i = C \frac{dv}{dt}$

Inductor

A passive energy storage element that stores energy in the form of a magnetic field. The simplest form of an inductor is a wire coil. The inductor’s voltage-current relationship can be expressed as:

$v = L \frac{di}{dt}$

Voltage and Current Sources and Meters

Understanding and utilizing various instruments, including an oscilloscope, multimeter, power supply, and function generator, are essential.

• Oscilloscope: A type of electronic test instrument that graphically displays varying signal voltages.

• Multimeter: An electronic measuring instrument that combines several measurement functions in one unit. A typical multimeter can measure voltage, current, and resistance.

• Function Generator: Electronic test equipment used to generate different types of electrical waveforms over a wide range of frequencies.

When analyzing electrical networks on paper, we usually assume that sources and meters are ideal.

• An ideal voltage source has zero output resistance and can supply infinite current.

• An ideal current source has infinite output resistance and can supply infinite voltage.

• An ideal voltmeter has infinite input resistance and draws no current.

• An ideal ammeter has zero input resistance and no voltage drop across it.

Real-world sources and meters have output/input impedances that deviate from these ideal characteristics.

Semiconductor Physics

Semiconductor Materials

• Conductors (e.g., metals) allow large currents to flow easily due to their conduction band.

• Insulators have tightly bound valence electrons, preventing easy electron movement even when an electric field is applied.

• Semiconductors have properties between conductors and insulators.

Semiconductor Properties

• Conductive materials have a large number of free electrons and low resistivity, facilitating current flow.

• Insulator materials have minimal to no free electrons and large resistivity, preventing current flow.

• Semiconductor materials have some free electrons and intermediate resistivity.

When a voltage is applied across a semiconductor, some valence electrons jump to the conductance band and move in the electric field to produce a current, although smaller than in a conductor.

Doping Semiconductors

• N-type Semiconductor: Charge carriers are electrons.

• P-type Semiconductor: Charge carriers are holes.

• PN-Junction: Formed by creating a p-type region adjacent to an n-type region.

The interaction between n-type and p-type semiconductor materials (PN-junction) is the basis for most semiconductor electronic devices, such as diodes, transistors, thyristors, and integrated circuits.

PV Solar Cell Application

When a p-type and an n-type semiconductor are placed in physical contact:

• A depletion zone (or space charge region) is created at the junction.

• Atoms in one side of this region are depleted of electrons (positively charged ions), and in the other side, they are depleted of holes (negatively charged ions), resulting in an induced electric field.

• Photons from the sun can be considered individual particles carrying a certain amount of linear momentum.

Semiconductor Diodes

Diode Structure

• Anode: The p-type side of the diode.

• Cathode: The n-type side of the diode.

• Depletion Region: Created at the PN junction when electrons from the n-type silicon diffuse to occupy holes in the p-type silicon.

• Contact Potential: The voltage difference across the depletion region, typically 0.6–0.7 V for silicon.
  Forward Bias: Occurs when a voltage source is connected to the PN junction with the positive side connected to the anode and the negative side to the cathode, forming a complete circuit.
  Reverse Bias: Occurs when the anode is connected to the n-type silicon and the cathode to the p-type silicon, enlarging the depletion region and inhibiting diffusion of electrons, thus limiting current. A small reverse saturation current does flow (on the order of $10^{-9}$ to $10^{-15}$ A).

Diode Functionality

• A PN junction passes current in only one direction, acting as a silicon diode or rectifier.

• Analogous to a fluid check valve, which allows fluid to flow only in one direction.

• PN junctions also occur in more advanced devices like transistors and integrated circuits.

Diode as Rectifier

A diode is useful as a rectifier, where it passes only the positive half or the negative half of an AC signal.

Half-Wave Rectifier Circuit

When $V<em>{in}$ is positive, the diode is reverse biased and equivalent to an open circuit. No current flows through the resistor, and the output $V</em>{out}$ equals $V_{in}$.

When $V<em>{in}$ is negative, the diode is forward biased and equivalent to a short circuit. No voltage drop occurs across the diode, and $V</em>{out}$ is 0 V.

Full-Wave Rectifier Circuit

Adding a capacitor can enhance the output signal, reducing ripple. A larger capacitor value results in less ripple in the output signal.

Real vs. Ideal Diodes

The current-voltage characteristic curve for a semiconductor diode illustrates:

• A nonlinear increase in current as the forward bias voltage approaches 0.7 V.

• A real diode requires about 0.7 V of forward bias to enable significant current flow.

• When a real diode is reverse biased, it can withstand a reverse voltage up to a limit known as the breakdown voltage, where the diode will fail as the reverse current increases precipitously.
  Important specifications that differentiate diodes are the maximum forward current and the maximum reverse bias voltage where breakdown occurs.

Zener Diode

• Breakdown voltage is less (usually 5v, 10v, 12v, etc.)

• Condition to operate safe is that current passing through Zener diode is less than $I_{zmax}$

  • Higher than $I_{zmax}$ will damage the diode

• When connected in reverse biased:

  • It passes current when $V<em>s$ supply is greater than $-V</em>z$

  • it acts as voltage regulator (usually 5v) if current passing through it is less than $I_{zmax}$

Voltage Regulator

Drawbacks of Zener diode voltage regulator:

• The output voltage cannot be set to a precise value, and

• regulation against source ripple and changes in load is limited.

Special semiconductor devices are designed to serve as voltage regulators, such as the three-terminal regulator designated as the 78XX, where the last two digits (XX) specify a voltage with standard values: 5 (05), 12, or 15 V. The 78XX can deliver up to 1 amp of current and is internally protected from overload. They are accurate, reject ripple on the input, reject voltage spikes, have roughly a 0.1% regulation, and are quite stable, making them useful in mechatronic system design.

For regulated voltage sources with a value not provided in a manufacturer’s standard sequence, a three-terminal regulator designed to be adjustable by the addition of external resistors can be used. Example: The LM317L can provide an adjustable output with two external resistors. The output voltage is given by:

$V<em>{out} = 1.25 \left(1 + \frac{R</em>2}{R_1}\right)$

Light-Emitting Diode (LED)

LEDs are diodes that emit photons when forward biased. The intensity of light is related to the amount of current flowing through the device. An LED has a voltage drop of 1.5 to 2.5 V when forward biased, somewhat more than small signal silicon diodes. A series current-limiting resistor in the circuit is crucial to prevent excess forward current, which can quickly destroy the diode. Usually, a 330 $\Omega$ resistor is included in series with an LED when used in digital (5 V) circuit designs.

Bipolar Junction Transistor (BJT)

BJT Basics

• BJTs consist of three adjacent regions of doped silicon, each of which is connected to an external lead.

• There are two types of BJTs: NPN (most common) and PNP transistors.

• Note the direction of the arrow in the transistor symbol is towards the negative region

NPN BJT

• NPN: consists of a thin region or layer of p-type silicon sandwiched between two regions or layers of n-type silicon.

• Three leads are connected to the three regions, and they are called the collector, base, and emitter.

• Base acts as valve to allow/control the current flow from Collector to Emitter using $I_B$, with three cases:

  • No current at the G ($I_B$ = 0), no current pass from C to E.

  • The more $I_B$ current, the greater the “gate” opens to allow for current to flow

  • Until it saturates and the gate/valve is fully open

• $I_B$: Control Current

• $I_C$: Load Current

As the base current is gradually increased, the base-to-emitter diode of the transistor begins to conduct when $V<em>{BE}$ is about 0.7 V. At this point, $I</em>C$ begins to flow and is roughly proportional to $I_B$. The transistor operation has three regions:

• Cutoff region (where no collector current flows)

• Active region (where collector current is proportional to base current)

• Saturation region (where collector current is strictly controlled by the collector circuit)

When designing a transistor switch, we need to guarantee that the transistor is fully saturated when it is on.

Why not using mechanical switch instead?

• Mechanical switch needs mechanical energy/force to operate, while transistor needs low electrical signal to operate

• Speed of switching in transistor are fast.

What I can do with a transistor as switch?

• Turn on/off devices/loads such as: Light, motor, heater, solenoid valve, etc.

• By using small control current ($I<em>B$, around 1 mA), you can control large current ($I</em>{CE}$, around 10 A) through the load/device.

Common emitter configuration

notice how the base-to-emitter forward bias voltage $V<em>{BE}$ and the collector-to-emitter voltage drop $V</em>{CE}$ do not change much after the transistor is saturated, even when input voltage $V_{in}$ is increased well above the minimum required for saturation.

Transistor Regions

• In active region, BJT transistor acts as Amplifier: A small base current controls a larger collector current, and therefore the BJT functions as a current amplifier. This characteristic can be approximated with the following equation:

  • $\beta = \frac{I<em>C}{I</em>B}$

  • $\beta$: amplification factor known as the beta for the transistor

• In the saturated region, the BJT transistor acts as an On-Off Switch

• In the cut-off region, the BJT transistor does not conduct ($I<em>B = I</em>C = 0$)

Example: Transistor as a switch

When $V<em>{in}$ is less than 0.7 V, the BE junction of the transistor is not forward biased (V{BE} < 0.7 V), and the transistor does not conduct ($I<em>C = I</em>E = 0$). You can therefore assume that the collector-to-emitter circuit can be replaced by a very high impedance or, for all practical purposes, an open circuit. This state is referred to as the cutoff or OFF state of the transistor.

When the BE junction is forward biased ($V<em>{BE} = 0.7$V), the transistor conducts. Then, current passes through the CE circuit, and $V</em>{out}$ is close to ground potential. This state, modeled by the forward-biased diode (illustrated in Figure b) is referred to as the saturated or ON state of the transistor. The resistor $R<em>B$ is required in this circuit to limit the base current because the BE junction essentially behaves like a diode. The relationship between the base current and $R</em>B$ is given by

$I<em>B = \frac{V</em>{in} - V<em>{BE}}{R</em>B}$

Note that

• The transistors used in power applications, called power transistors, are designed to conduct large currents and dissipate more heat.

• Power transistors are the basis for interfacing low-output current devices such as integrated circuits and computer ports to other devices requiring large currents.

• Relays, which mechanically make and break connections, are an alternative to transistors, BUT they cannot switch as fast as transistors and don’t last as long, but they are very easy to use and can switch DC as well as AC power.

Example: LED Switch

Objective is to turn a dashboard LED on or off with a digital device having an output voltage of either 0 V or 5 V and a maximum output current of 5 mA. The LED requires 20–40 mA and has a 2 V voltage drop when forward biased.

When the digital output is 0 V, the transistor is in cutoff, and the LED is OFF.

When the digital output is 5 V, the transistor is in saturation, and the base current is

$I<em>B = \frac{V</em>{in} - V<em>{BE}}{R</em>B}$

Example: Motor Control

• For NPN, when the digital output (GPIO) is 0 V, the transistor is in cutoff, and the motor is OFF.

• For NPN, when the digital output is 5 V, the transistor is in saturation, and the motor is ON.

Note:

• Choose the correct transistor model that handles the load current.

• A diode is used parallel to the motor to save the transistor from the dissipated energy of the motor coils

Optoisolator

An opto-isolator consists of an LED and a phototransistor separated by a small gap. The light emitted by the LED causes current to flow in the phototransistor circuit. With no common ground, the opto-isolator creates a state of electrical isolation between the input and output circuits by transmitting the signal optically rather than through an electrical connection. A benefit of this isolation is that:

• The output is protected from any excessive input voltages that could damage components in the output circuit.

• because the supplies and grounds are separate, any fluctuations or disturbances that might occur in the output circuit have no effect on the control signals on the input side.

Transistors - FET

field-effect transistor

• (FET) operates on a different principle than the BJT but serves a similar role in mechatronic system design.

• is also an important component in the design of digital integrated circuits.

• Both the BJT and FET are three-terminal devices.

• Both BJTs and FETs operate by controlling current between two terminals using a voltage applied to a third terminal.

with a FET, the electric field produced by a voltage on one electrode controls the availability of charge carriers in a narrow region, called a channel, through which a current can be made to flow.

Therefore, a FET can be described as a transconductance amplifier, which means the output current is controlled by an input voltage.

The control electrode in the FET, called the gate, is analogous to the base of the BJT. In contrast to the BJT base, the FET gate draws no direct current (DC).

A conducting channel, whose conductivity is controlled by the gate, lies between the drain, which is analogous to the BJT collector, and the source, which is analogous to the BJT emitter.

field-effect Transistor (FET)

There are two families of FETs:

• metal-oxide-semiconductor FETs (MOSFETs),

  • Depletion-mode MOSFETs.

  • Enhancement-mode MOSFET,

• Junction field-effect transistors (JFETs).

  • Each of these families is available in:

    • p-channel and n-channel varieties.

    • focus primarily on the widely used n-channel enhancement mode MOSFET.

    • Close analogy to the NPN BJT transistor.

Metal-Oxide-Semiconductor FETs (MOSFETs)

N-channel Enhancement-mode metal-oxide-semiconductor FETs (MOSFETs) has:

• a p-type substrate, and

• n-type source and drain that form PN junctions with the substrate.

• There is a thin silicon dioxide layer insulating the gate from the substrate.

Operation principal:

when a positive DC voltage is applied to the gate, an electric field formed in the substrate below the gate repels holes in the p-type substrate, leaving a narrow layer or channel in the substrate in which electrons predominate.

Case 1:

If the gate is grounded (Vg=0), no drain-to-source current Id flows for a positive drain voltage Vdd because the drain PN junction is reverse biased and no conducting channel has formed. In this state, the MOSFET mimics a very large resistor (~ $10^8$ - $10^{12}$ $\Omega$) and no current flows between the drain and source. The MOSFET is said to be in cutoff region.

Case 2:

As $V<em>{gs}$ is gradually increased beyond a gate-to-source threshold voltage Vt, the n-channel begins to form. Vt depends on the particular MOSFET considered but a typical value is about 2 V. Then as $V</em>{ds}$ is increased from 0, conduction occurs in the n-channel due to a flow of electrons from source to drain. With a positive $V<em>{gs}$ larger than Vt , as $V</em>{ds}$ is increased from 0, we enter the active region, also called the ohmic region, of the MOSFET. In this region, as $V<em>{gs}$ is further increased, the conduction channel grows correspondingly, and The MOSFET appears to function like a variable resistor whose resistance is controlled by $V</em>{gs}$.

Case 3:

when ($V<em>{gs} - V</em>t$) reaches $V<em>{dd}$, there is no longer an electric field at the drain end of the MOSFET. Therefore, the width of the n-channel shrinks to a minimum value close to the drain, resulting in what is called pinch-off. This pinch-off limits a further increase in drain current, and the MOSFET is said to be in the saturation region. In saturation, the current is almost constant with further increases in $V</em>{ds}$. The drain-to-source resistance, called Ron, is minimal (usually less than 5 $\Omega$) as it enters the saturation region

Characteristic Family of Curves for The N-Channel Enhancement-Mode MOSFET

voltage on the gate $V<em>g$ (which is also the gate-to-source voltage $V</em>{gs}$, because the source is grounded) was gradually increased from 0 to 10 V, more gradually in the ranges of interest. Notice that for this MOSFET, the threshold voltage where conduction begins (I_{ds} > 0), is about 3.5 V.

Example: Testing and simulating a power MOSFET circuit

Simulate it in Multisim software and explore the three regions of operation for the n-channel MOSFET transistor. Note (for the MOSFET used, model IRF6201PBF):

The active region is between:

• $V_g$ = 1.2v to 1.3v

• (at $V<em>g$ = 1.235v ➔ $V</em>{ds}$=2.813)

Why MOSFET Transistors

MOSFETs are used to make excellent high-current voltage-controlled switches. signals can be gated (blocked or passed) in circuits, driven by DC motors, current sources used in the internal design of integrated circuits (ICs) like microprocessors.

Applications of MOSFET Transistors

Switching Application: Switching Power to A Load Circuit

Note that:

Case 1: When (V_g <= 0) for the MOSFET to be cutoff so that no current is delivered to the load.

Case 2: When ($V<em>g - V</em>{tt} \approx V_{dd}$), the MOSFET enters saturation resulting in nearly full voltage Vs across the load (because Ron is small).

The controlling parameter for the MOSFET is gate voltage $V<em>g$. Recall that with the BJT, the controlling parameter is base current $I</em>B$.

With the BJT, one must ensure adequate base current to saturate the BJT. Using the MOSFET, the current drawn by the gate is essentially 0, so current sourcing is not a concern. However, one needs to calculate the drain current $I_d$ and power dissipation to select a MOSFET capable of switching the desired current for the load. Also, as with a BJT, if the load is inductive, a flyback diode is necessary to prevent damage to the MOSFET when it is switched off

Comparison between BJT and MOSFET

Analog Signal Processing Using Operational Amplifiers

Operational Amplifier

Operational amplifier circuits is an IC that are made of multiply transistors and resistors. A component-level diagram of the common 741 op amp.

Operational amplifier circuits are important for interfacing analog components in a measurement/mechatronic system.

When designing measurement systems, it is essential that engineers develop a basic understanding of the acquisition and processing of electrical signals. Usually signals come from transducers/sensors, which convert physical quantities (e.g., temperature, strain, displacement, flow rate) into currents or voltages, but are not in the form we would like them to be.

Analog Signal

The transducer output is usually described as an analog signal, which is continuous and time varying but might be:

• Too small, usually in the millivolt range

• Too “noisy” , usually due to electromagnetic interference

• Containing the wrong information, sometimes due to poor transducer design or installation

• Having a DC offset, usually due to the transducer and instrumentation design

Operational Amplifier Circuits

Many of these problems can be remedied, and the desired signal information can be extracted through appropriate analog signal processing. The simplest and most common form of signal processing is amplification, where the magnitude of the voltage signal is increased.

Other important forms include signal inversion, differentiation, integration, addition, subtraction, and comparison.

Amplifier

Ideally, an amplifier increases the amplitude of a signal without affecting the phase relationships of different components of the signal.

When choosing or designing an amplifier, one must consider:

• size

• cost

• power consumption

• input impedance

• output impedance

• gain

• bandwidth

Physical size depends on the components used to construct the amplifier. Most amplifiers are designed to have a large input impedance so very little current is drawn from the input. Most amplifiers are designed to have a very small output impedance so the output voltage will not change much as the output current changes.

Model

An amplifier can be model as a two-port device, with an input and output voltage referenced to ground.

$A<em>v = \frac{V</em>{out}}{V_{in}}$

• Av is the voltage gain of an amplifier defines the factor by which the voltage is changed

• The operational amplifier, or op amp, is a low-cost and versatile integrated circuit consisting of many internal transistors, resistors, and capacitors manufactured into a single chip of silicon. It can be combined with external discrete components to create a wide variety of signal processing circuits. The op amp is the basic building block for many analog circuits.

Ideal Model for the Operational Amplifier

Op amp is a differential input, single output amplifier It is assumed to have infinite gain. The two inputs are called the inverting input, labeled with a minus sign, and the noninverting input, labeled with a plus sign. The

symbol is sometimes used in the schematic to denote the infinite gain and the assumption that it is an ideal op amp. The voltages are all referenced to a common ground.

the op amp is an active device requiring connection to an external power supply Usually plus and minus 15 V. Since the op amp is an active device, output voltages and currents can be larger than the signals applied to the inverting and noninverting terminals

Feedback

An op amp circuit usually includes feedback from the output to the negative (inverting) input. This so-called closed loop configuration results in stabilization of the amplifier and control of the gain. When feedback is absent in an op amp circuit, the op amp is said to have an open loop configuration. This configuration results in considerable instability due to the very high gain, and it is seldom used.

Ideal Op Amp Assumptions

1. It has infinite impedance at both inputs; hence, no current is drawn from the input circuits. Therefore:

2. It has zero output impedance. Therefore, the output voltage does not depend on the output current.

Operational Amplifier Configurations

Common types of op amp circuits:

• Inverting amplifier circuit

• Noninverting amplifier circuit

• Summer/Difference op amp circuits

• Integrator amplifier circuit

• Differentiator amplifier circuit

• Sample and Hold amplifier circuit

• Comparator amplifier circuit

Combination of Operational Amplifier Configurations

Inverting Amplifier

An inverting amplifier is constructed by connecting two external resistors to an op amp. This circuit inverts and amplifies the input voltage. the resistor $R_F$ forms the feedback loop. This feedback loop always goes from the output to the inverting input of the op amp, implying negative feedback.

To analyze this circuit, we use Kirchhoff’s laws and Ohm’s law.

replace the op amp with its ideal model Applying Kirchhoff’s current law at node C and utilizing assumption 1 (that no current can flow into the inputs of the op amp) Since the two inputs are assumed to be shorted in the ideal model, C is effectively at ground potential.

The voltage gain of the amplifier is determined simply by the external resistors both RF and R, and it is always negative. The reason this circuit is called an inverting amplifier is that it reverses the polarity of the input signal. This results in a phase shift of 180 degree for periodic signals.

The voltage at node C is $V_{in}$ because the inverting and noninverting inputs are at the same voltage. Therefore, applying Ohm’s law to resistor R:

$i<em>{in} = \frac{V</em>{in}}{R}$

Substituting $i_{in}$ into Vout equation:

$V<em>{out} = - \frac{R</em>F}{R} V_{in}$

Dividing $V<em>{out}$ by $V</em>{in}$ yields the input/output relationship:

$\frac{V<em>{out}}{V</em>{in}} = - \frac{R_F}{R}$

Non-Inverting Amplifier

This circuit amplifies the input voltage without inverting the signal. The voltage at node C is $V_{in}$ because the inverting and noninverting inputs are at the same voltage. Therefore, applying Ohm’s law to resistor R:

$i<em>{in} = \frac{V</em>{in}}{R}$

Applying it to resistor $R_f$:

$V<em>{out} = V</em>{in} \left(1 + \frac{R_F}{R}\right)$

The noninverting amplifier has a positive gain greater than or equal to 1. useful in isolating one portion of a circuit from another by transmitting a scaled voltage without drawing appreciable current. Voltage gain for noninverting op amp.:

$\frac{V<em>{out}}{V</em>{in}} = 1 + \frac{R_F}{R}$

Buffer/Follower Amplifier (Special Case)

If $R<em>F = 0$ and $R = \infty$ in the noninverting op amp circuit, the resulting circuit can be represented as shown. This circuit is known as a buffer or follower because it has a high input impedance and low output impedance. This circuit is useful in applications where you need to couple to a voltage signals without loading the source of the voltage. The high input impedance of the op amp effectively isolates the source from the rest of the circuit. $V</em>{out} = V_{in}$

Summer/Difference Amplifier

The summer op amp circuit is used to add analog signals. The difference op amp circuit is used to subtract analog signals

Integrator Amplifier

If the feedback resistor of the inverting op amp circuit is replaced by a capacitor, the result is an integrator circuit.

Referring to the analysis for the inverting amplifier

$V<em>{out}(t) = -\frac{1}{RC} \int V</em>{in}(t) dt$

Differentiator Amplifier

If the input resistor of the inverting op amp circuit is replaced by a capacitor, the result is a differentiator circuit.

$V<em>{out}(t) = -RC \frac{dV</em>{in}(t)}{dt}$

Sample and Hold Circuit

A circuit which samples an input signal and holds onto its last sampled value until the input is sampled again. is used extensively in analog-to-digital conversion where a signal value must be stabilized while it is converted to a digital representation. The sample and hold circuit consists of a voltage-holding capacitor and a voltage follower. With switch S closed:

During sampling, the capacitor voltage C is connected to the voltage follower during sampling. When the switch is opened, the capacitor C holds the input voltage corresponding to the last sampled value, because negligible current is drawn by the follower. Therefore:

$V<em>{out} = V</em>{in}$

Comparator Amplifier

The comparator circuit is used to determine whether one signal is greater than another

• The comparator is an example of an op amp circuit where there is no negative feedback and