The post How Does Static Electricity Work? appeared first on Electrical A2Z.

]]>The term static means standing still or at rest, and **static electricity** refers to an electric charge at rest. The result of this buildup of static electricity is that objects may be attracted to each other or may even cause a spark to jump from one to the other.

One of the simplest ways to produce static electricity is by **friction.** For example, a static charge can be produced by rubbing a balloon with a piece of wool, as illustrated in **Figure 1**.

The process causes electrons to be pulled from the wool to the balloon. As a result the balloon ends up with an excess of electrons and a negative charge. At the same time, the wool loses electrons, creating a shortage of electrons and a positive charge. Note that the charged atoms remain on the surface of the material.

Static electricity is different from **current electricity** that flows through metal wires. Most often the materials involved in static electricity are nonconductors of electricity.

**Figure 1** Producing static electricity by friction.

Static electricity is formed when we accumulate extra negatively charged electrons and they are discharged to an object or person. Take, for example, the rubber soles of your shoes and that wool carpet in the living room. When you walk across the carpet, your body builds up a negative charge of extra electrons it can’t get rid of through the insulating soles of your shoes. Then when you reach for the doorknob, you may experience a **static electric shock** caused by electrons moving from your hand to the metal doorknob (**Figure 2**).

Static electricity occurs more often during the colder seasons because the air is drier, and it’s easier to build up electrons on the skin’s surface. In warmer weather, the moisture in the air helps electrons move off of you more quickly so the static charge is not as great.

**Figure 2** Static electric shock.

The first law of electric charges is illustrated in **Figure 3** and states that **like charges repel and unlike charges attract.** An invisible **electrostatic field** exists in the space between and around charged balls.

When two like-charged bodies are brought together, their electric fields repel one body from the other. When two unlike-charged bodies are brought together, their electric fields attract one body to the other.

**Figure 3** Law of electric charges.

The electrostatic field around two charged objects can be represented graphically by lines referred to as **electrostatic lines of force** (**Figure 4**).

Lines are directed away from positively charged objects and toward negatively charged objects. **The attractive or repulsive force that is exerted between two charged particles is directly proportional to the strength of their charges and inversely proportional to the square of distance between the two charges.** This means that the bigger the charges, the more will be the force; the more distance they are apart, the less the force between them.

**Figure 4** Electrostatic field around two charged objects.

If two strongly charged bodies (one positive and one negative) are moved near to each other, before contact is made you actually see the equalization of the charges take place in the form of an arc. **Lightning** is a perfect example. Cloud-to-ground lightning occurs when the electric charge travels between a negatively charged cloud base and the positively charged ground. A single bolt of lightning delivers about **1 trillion watts** of electricity.

**Electrostatic discharge (ESD)** is the rapid discharge of static electricity from one object to another of a different potential. This is a very rapid event that happens when two objects of different potentials come into direct contact with each other.

Electrostatic discharge is one of the main causes of device failures in the **semiconductor** industry. Static electricity, so low that you can’t feel it, can wreck havoc with today’s large-scale microelectronic devices.

Methods of protection against ESD include **prevention** of static charge buildup and **safe dissipation** of any charge buildup (**Figure 5**). Packaging materials such as static shielding bags, conductive bags, and electrostatic discharge containers and boxes provide direct protection of devices from electrostatic discharge.

Antistatic footwear and wrist straps when properly worn and grounded keep the human body near ground potential, thus preventing hazardous discharge between bodies and objects.

**Figure 5** Methods of protection against ESD.

The three common ways for a neutral object to become charged are by friction, conduction, or induction. Charging by **conduction** or contact occurs when a neutral object is placed in contact with an already-charged object.

If the object is negatively charged, electric repulsion will push some of the excess electrons from the charged to the neutral object. If the object is positively charged, electric attraction will pull some electrons from the neutral object to the charged one.

In the example shown in **Figure 6**, when a rod (that has an excess of electrons) touches a neutral ball, the charge distributes itself over both objects. When they are separated, the ball will now be electrically charged.

**Figure 6** Charging by conduction.

Charging by **induction** involves an already-charged object that is **brought close** to but does not touch the neutral object.

In the example shown in **Figure 7**, the negatively charged rod is brought close to an electrically neutral ball. The electrons on the ball are repelled and move to the opposite side of the ball. Touching the negative right side of the ball drains electrons from the ball to ground, giving the ball a net positive charge. The rod is then removed, leaving a positively charged ball.

**Figure 7** Charging by induction.

There are several useful applications for the forces of attraction between charged particles. For these applications, static electric charges are normally produced by a **high-voltage DC** (direct current) source.

**Electrostatic paint spraying** (**Figure 8**) uses static electricity to attract the paint to the target, reducing paint wastage and improving coverage of the target. The paint and target part are charged with **opposite charges** so the two attract and the paint sticks to the target. This process produces a uniform cover of paint with excellent adhesion.

**Figure 8** Electrostatic paint spraying.

Electrostatic **air cleaners** or **precipitators** use positively and negatively charged plates to remove dirt particles from the air. **Figure 9** illustrates the operation an electronic air cleaner used in a home-heating system to clean the air as it circulates through the furnace.

- The dirty air passes through a paper filter that removes large dust and dirt particles from the air.
- The air then moves through an electrostatic precipitator consisting of two oppositely charged, high-voltage grids.
- The precipitator works by giving a
**positive charge**to particles in the air and then attracting them with a**negatively charged**grid. - Finally, the air passes through a carbon filter, which absorbs odors from the air.

**Figure 9** Electrostatic air cleaner.

**Review Questions**

- Define
*static electricity.* - Explain why rubbing a balloon with a piece of wool results in a negative charge on the surface of the balloon.
- State the law of electrostatic charges.
- How is the force between two charged particles affected by the strength of their charges and the distance between them?
- Define
*electrostatic discharge*. - Electrostatic discharge is one of the main causes of device failure in the ________________ industry.
- Name three ways for a neutral object to become charged.
- Charging by ________________ involves an already-charged object that is brought close to but does not touch the neutral object.
- Summarize the operation of an electrostatic paint spraying process.
- Summarize the operation of an electrostatic air cleaner.
- Why does static electricity occur more often during the colder seasons?

**Review Questions – Answers**

- Static electricity is the accumulation of electrical charges on the surface of a material, usually an insulator.
- Rubbing a balloon with a piece of wool results in a negative charge on the surface of the balloon because electrons are pulled from the wool to the balloon.
- Like charges repel and unlike charges attract.
- The attractive or repulsive force that is exerted between two charged particles is directly proportional to the strength of their charges and inversely proportional to the square of the distance between the two charges.
- Electrostatic discharge is the rapid discharge of static electricity from one object to another of a different potential.
- semiconductor
- A neutral object can become charged by friction, conduction, or induction.
- induction
- An electrostatic paint spraying process operates by charging the paint and the part to be painted with opposite charges. The paint is attracted to the part. When the paint hits the part, the charge is neutralized and no more paint is applied to that area.
- The electrostatic air cleaner operates by charging the particles in the air with positive charges and then attracting them to a negative grid.
- Because the air is drier, and it’s easier to build up electrons on surfaces.

The post How Does Static Electricity Work? appeared first on Electrical A2Z.

]]>The post Average and RMS Value of Alternating Current and Voltage appeared first on Electrical A2Z.

]]>**Figure 1** shows the magnitude and polarity of an AC voltage. Starting at zero, the voltage rises to maximum in the positive direction. It then falls back to zero. Then it rises to maximum with the opposite polarity and returns to zero.

**Figure 1.** Current and voltage of alternating current.

The current wave is also plotted on the graph. It shows the flow of current and the direction of the flow. Above the zero line, current is flowing in one direction. Below the zero line, the current is flowing in the opposite direction.

The graph in **Figure 1** represents instantaneous current and voltage at any point in the cycle. But what is a cycle? A **cycle** is a sequence or chain of events occurring in a period of time. An **AC cycle** can be described as a complete set of positive and negative values for AC.

The alternating current in your home changes direction 120 times per second. It has a frequency of 60 cycles per second (60 cps). **Frequency**, measured in cycles per second, or hertz (Hz), is the number of complete cycles occurring per second. If 60 cycles occur in one second, then the time period for one cycle is 1/60 of a second, or 0.0166 seconds. This is the **period of the cycle**. Refer again to **Figure 1**. The maximum rise of the waveform shows the amplitude of the wave, including the peak (highest point) voltage and current.

We learned that induced current in a rotating wire in a magnetic field flowed first in one direction and then in the other direction. This was defined as an alternating current. **Two points to remember are**:

• The frequency of this cycle of events increases as rotation speed increases.

• The amplitude of the induced voltage depends on the strength of the magnetic field.

**Vectors**

When solving problems involving alternating currents, vectors are used to depict the magnitude and direction of a force. A vector is a straight line drawn to a scale that represents units of force. An arrowhead on the line shows the direction of the force. The length of the vector shows the magnitude.

The development of an AC wave is shown in **Figure 2**. This wave is from a single coil armature, represented by the rotating vector, making one revolution through a **magnetic field**.

Assume that the peak induced voltage is 10 volts. Using a scale in which one inch equals five volts, the vector is two inches, or 10 volts, long. Vectors of this nature are assumed to rotate in a counter- clockwise direction.

**Figure 2.** The development of a sine wave. On the left is the rotating phasor. On the right is one cycle of the sine wave.

The time base in **Figure 2** is a line using any convenient scale. It shows the period of one cycle or revolution of the vector. The time base is grouped into segments that represent the time for certain degrees of rotation during the cycle.

**For example**, at 90 degrees rotation, one quarter of the time period is used. At 270 degrees rotation, three quarters of the time period is used. The wave is developed by plotting voltage amplitude at any instant of revolution against the time segment. The developed wave is called a **sine wave**.

The instantaneous induced voltages are proportioned to the sine of the angle θ (theta) that the vector makes with the horizontal. The instantaneous voltage may then be found at any point of the cycle by making use of the following equation:

$e={{E}_{\max }}\times \sin \theta $

(Notice that the lowercase e was used to represent the instantaneous voltage instead of the usual uppercase. By convention, instantaneous values are represented with lowercase variables.)

To apply this equation, assume that an **AC generator** is producing a peak voltage of 100 volts. What is the instantaneous voltage at 45 degrees of rotation?

$e=100V\times \sin {{45}^{o}}=100V\times 0.707=70.7V$

A study of the differences between an AC wave and a direct current raises a key question. What is the actual value of the AC wave? The voltage and current vary constantly and reach peak value only twice during a cycle.

Often, the average value of the wave is needed. The average value is the mathematical average of all the instantaneous values during one half-cycle of the alternating current. The formulas for computing the average value from the peak value (max) of any AC waves are:

$\begin{matrix}{{E}_{avg}}=0.637\times {{E}_{\max }} \\or \\{{I}_{avg}}=0.637\times {{\operatorname{I}}_{max}} \\\end{matrix}$

If E_{avg} or I_{avg} is known, the conversion to find E_{max} or I_{max} can be made using the following equations.

$\begin{matrix}{{E}_{\max }}=1.57\times {{E}_{avg}} \\or \\{{I}_{\max }}=1.57\times {{\operatorname{I}}_{avg}} \\\end{matrix}$

A more useful alternating current value is the **effective value**. The term effective value comes from scientists finding the AC heating effect equivalent of a direct current.

A specified volume of water was heated using a specified voltage level of dc. Then, the same quantity of water was heated using AC. The AC voltage that produced equivalent heating to the DC voltage was the effective value. The formulas to find the effective value of any AC voltage or current are:

$\begin{matrix}{{E}_{eff}}=0.707\times {{E}_{\max }} \\or \\{{I}_{eff}}=0.707\times {{\operatorname{I}}_{\max }} \\\end{matrix}$

Where E_{max} and I_{max} are the peak values of the ac signal. If E_{eff} or I_{eff} are known, the conversion to find the peak values can be made by using the following equations.

$\begin{matrix}{{E}_{\max }}=1.414\times {{E}_{eff}} \\or \\{{I}_{\max }}=1.414\times {{\operatorname{I}}_{eff}} \\\end{matrix}$

The effective value is also called the **rms value (root mean square)**. It gets this name because the value represents the square root of the average of all currents squared between zero and maximum of the wave. The currents are squared, so the power produced can be compared to direct current. **Watt’s law** states: P = I^{2}R.

Using the 0.707 factor, the value of a direct current can be found that will equal the alternating current. **For example**, a peak AC current of 5 amperes produces the same heating effect in a resistance as a DC current of 3.53 amperes. Plugging the values into the equation:

I_{eff} = 0.707 × 5 amperes of AC = 3.53 amperes DC

Note that average and effective values can be applied to either voltage or current waves.

A few waveforms can be drawn on the same time base to show the phase relationship between them. In **Figure 3**, waveforms E and I show the voltage and current in a given circuit. The current and voltage rise and fall at the same time. They cross the zero line at the same point. The current and voltage are in phase. The in phase condition only exists in the **purely resistive circuit**.

**Figure 3.** These current and voltage waves are in phase.

Many times the current will lead or lag the voltage**, Figure 4**. When the current wave leads or lags the voltage wave, the two waves are said to be out of phase. This creates a phase displacement between the two waves. Displacement is measured in degrees. The phase displacement is equal to the angle θ between the two polar vectors.

**Figure 4.** These current and voltage waves are out of phase.

The ac generator is like the **DC generator** in many respects with one key exception. The commutator is omitted. The ends of the armature coils are extended out to slip rings. Brushes sliding on the slip rings provide connection to the coils at all times. The current in the externally connected circuit is an alternating current.

In large commercial generators, the magnetic field is rotated and the armature windings are placed in slots in the stationary frame, or stator, of the generator. This method allows for the generation of large currents in the armature while avoiding sending these currents through moving or sliding rings and brushes.

The rotating field is excited through slip rings and brushes by a small DC generator mounted on the same shaft as the rotating magnetic field. This small DC generator is called the **exciter.** The DC voltage is needed for the magnetic field. Commercial power generators convert many different items (such as moving water, coal, oil, wind, nuclear energy) into electricity. The force mechanism that is used to turn the generator is called the prime mover, **Figure 5.**

**Figure 5.** The prime mover, exciter, and three-phase alternator share a common rotating shaft. The exciter provides electrical energy for the alternator.

**The Alternator or AC Generator**

The AC generator (also called an alternator) is used in the charging system of all U.S. automobiles. **Figure 6** shows the inside of the unit, including a built-in voltage regulator to control output. The output is rectified from alternating current to direct current for charging the battery and other electrical devices in the car.

Manufacturers say the alternator has some advantages over the DC generator. These advantages include higher output at lower speeds, as well as trouble-free service.

**Figure 6.** A typical AC generator (alternator) is shown in external and cutaway views.

The post Average and RMS Value of Alternating Current and Voltage appeared first on Electrical A2Z.

]]>The post Electrodynamometer Type Wattmeter Working Principle appeared first on Electrical A2Z.

]]>A circuit diagram of a simple electrodynamometer movement is shown in **Figure 1**. This arrangement creates a **wattmeter**, a meter that measures the **instantaneous power**. A moving coil, with the proper **multiplier resistance**, is connected across the voltage in the circuit.

The coils of the electromagnet are connected in series with the circuit under measurement. The action between the two magnetic fields is proportional to the product of the voltage and the current. Deflection of the indicating needle is read on a calibrated scale in watts.

**Figure 1.** Diagram of a wattmeter.

While wattmeter measures the instantaneous power used in a circuit, a watt-hour meter measures the amount of power used in a given time. It is installed by a power company on the outside of a home or business. Since a watt-hour is a very small unit, standard utility meters read in kilowatt-hours (kWh), or 1000 watt-hours.

\[\text{number of kW=}\frac{E\times I}{1000}\]

Electric power consumed is purchased at current rates per kWh.

The watt-hour meter is a complicated type of **induction motor**. It uses field coils in series with the line current and also field coils connected across the line voltage. An aluminum disk rotates within these fields at a rate proportional to the power consumed. The disk is geared to an indicating dial. The dial shows the amount of power used.

To read the watt-hour meter, see **Figure 2**. Dial A, on the right, reads from 0–10 units of kilowatt hours. For each revolution of dial A, dial B advances one. For each revolution of dial B, dial C advances one. For each revolution of dial C, dial D advances one. Mathematically, dial A represents units of one; dial B represents units of ten; dial C represents units of one hundred; dial D represents units of one thousand.

**Figure 2.** The kilowatt hours indicated equal 4255. When the dial indicator rests between numbers, the lower number is selected.

Be careful when reading the meter. Note that two of the dials read clockwise (A and C), and two of the dials read counterclockwise (B and D). This arrangement is because the gears for the dials are connected directly to each other.

Read the meter illustrated in **Figure 2**:

Dial A points to 5.

Dial B points between 5 and 6.

Dial C points between 2 and 3.

Dial D points between 4 and 5.

Therefore, the correct reading would be 4255 kWh. Notice that when the indicating arrow is between numbers, the lower number should be used.

At times it is difficult to determine if the arrow has reached the number to which it is pointing or if the arrow has just approached the number. To make your determination, look to the dial immediately to the right of the dial in question. If the arrow on the dial to the right is between 9 and 0, then the number is still the lower value. If the arrow is past the 0, then the number is the larger value.

Use **Figure 3** as an example. The arrow in dial D appears to be pointing to the number 4. To determine if it is a 4 value or still a 3, look at the dial immediately to its right, dial C. Dial C has not quite reached 0. It is still between the 9 and the 0. Thus, dial D is actually still a 3.

To practice reading meters, why not read the meter at your home each day for several days and compute the power used?

The power consumed is found by subtracting the previous reading from the present reading. The difference between the two readings is the number of kilowatt- hours of power used over the length of time between your two readings.

**Figure 3.** This watt-hour meter reads 3970 kWh. To determine if dial D is really a four, you must look at dial C. Since dial C is between the nine and the zero, dial D is still three. It will become a four when the dial C indicator passes zero and rests between the zero and one. Dial B indicates seven. Dial A verifies that dial B is a seven and not still a six because dial A indicator is at rest between the zero and one.

The post Electrodynamometer Type Wattmeter Working Principle appeared first on Electrical A2Z.

]]>The post Electric Current, Voltage, and Resistance Overview | Three Basic Electrical Quantities appeared first on Electrical A2Z.

]]>• **current**: is the directed flow of charge through a conductor.

• **Voltage**: is the force that generates the current.

• **Resistance**: is an opposition to current that is provided by the material, component, or circuit.

Electric Current, Voltage, and resistance are the three primary properties of an electrical circuit. The relationships among them are defined by the fundamental law of circuit operation, called **Ohm’s law**.

As you know, an outside force can break an electron free from its parent atom. In copper (and other metals), very little external force is required to generate free electrons. In fact the thermal energy (heat) present at room temperature (22^{0}C) can generate free electrons. The number of electrons generated varies directly with temperature. In other words, higher temperatures generate more free electrons.

The motion of the free electrons in copper is random when no directing force is applied. That is, the free electrons in copper are random when no directing force is applied. That is, the free electrons move in every direction, as shown in **Figure 1**. Since, the free electrons are moving in every direction, this net flow of electrons in any direction is zero.

**Figure 1** Random electron motion in copper

**Figure 2** Illustrates what happens when an external force causes all of the electrons to move in the same direction. In this case, a negative potential is applied to one end of the copper and a positive potential is applied to the other. As a result, the free electrons all move from negative to positive, and we can say that we have a directed flow of charge (electrons). This directed flow of electrons is called **electric** **current**.

**Figure 2** Directed electron motion in copper.

Let’s look at what happens on a larger scale when electron motion is directed by an outside force. In **Figure 3**, the negative potential directs electron flow (current) toward the positive potential. The current passes through the lamp, causing it to produce light and heat. The more intense the current (meaning the greater its value), the greater the light and heat produced by the bulb.

**Figure 3** Current through a basic lamp circuit.

Electric Current is represented in formulas by the letter* I* (For intensity). The intensity of current is determined by the amount of charge flowing per second. The greater the flow of charge per second, the more intense the current.

**Coulombs and Amperes **

The change on a single electron is not sufficient to provide in a practical unit of measure for charge. Therefore, the **Coulomb (C)** is used as the basic unit of charge One coulomb equals the total charge on 6.25 × 10^{18} electrons. When, one coulomb of charge passes a point in one second, we have one **Ampere (A)** of electric current. In other words,

$\begin{matrix}1\text{ ampere=1 Coulomb per second} \\Or \\1\text{ A=1 C/s} \\\end{matrix}$

The total current passing a point (in amperes) can be found by dividing the total charge ( in coulombs) by the time ( in seconds) . By Formula

\[\begin{matrix}I=\frac{Q}{t} & {} & \left( 1 \right) \\\end{matrix}\]

Where

I= the intensity of electric current in amperes

Q= the total charge, in coulombs

T= the time it takes the charge to pass a point, in seconds

This relationship is illustrated in **Example 1**.

**Example 1**

Three coulombs of charge pass through a copper wire every second. What is the value of electric current?

**Solution**

Using equation 1, the current is found as

**\[I=\frac{Q}{t}=\frac{3C}{1s}=3{}^{C}/{}_{s}=3A\]**

**Example 1** is included here to help you understand the relationship between amperes, coulombs, and seconds. In practice, electric current is not calculated using **Equation 1** because you cannot directly measure coulombs in charge. As you will learn, there are far more practical ways to calculate current.

**Two Theories: Conventional Current and Electron Flow. **

There are two theories that describe electric current, and you will come across both in practice.

The **Conventional Current** theory defines current as the *flow of charge from positive to negative. *This theory is called “conventional current” because it is the older of the two approaches to current, and for many years was the only one taught outside of military and trade schools.

**Electron Flow** is the newer of the two current theories. Electron flow theory defines current as *the flow of electrons from negative to positive*.

The two electric current theories are contrasted in **Figure 4**. Each circuit contains a battery and a lamp. **Conventional current** begins at the positive battery terminal, passes through the lap, and returns to the battery through its negative terminal. **Electron flow** is in the opposite direction: It begins at the negative terminal, passes through the lamp, and returns to the battery through its positive terminal.

**Figure 4** Conventional current and electron flow.

It is worth nothing that the two circuits in **Figure 4** are identical. The only difference between the two is how we describe the electric current. In practice, how you view current does not affect any circuit calculations, measurements, or test procedures. Even so you should get comfortable with both viewpoints, since both are used by many engineers technicians, and technical publications. In this text, we take the **electron flow **approach to current. That is, we will assume current is the flow of electrons from negative to positive.

**Direct Current (DC) Versus Alternating Current (AC) **

Current is generally classified as being either Direct Current (DC) or Alternating Current (AC). The differences between direct current and alternating current are illustrated in **Figure 5**.

**Figure 5** Direct current (DC) and alternating current (AC).

**Direct current** is unidirectional. That is, the flow of charge is always in the same direction. The term direct current usually implies that the current has a fixed value. For example, the graph in **Figure 5a** shows that the current has a constant value of 1A. While a fixed value is implied, direct current can change in value. However, the direction of current does not change.

**Alternating Current** is Bidirectional. That is, the direction of current changed continually. For example, in **figure 5b**, the graph shows that the current builds to a peak value in one direction and then builds to a peak value in the other direction. Note that the alternating current represented by the graph not only changes direction but is constantly changing in value.

**Electric Current Produces Heat **

Whenever electric current is generated through a component or circuit, heat is produced. The amount of heat varies with the level of current: The greater the current, the more heat it produces. This is why many high-current components, like motors, get hot when they are operated. Some High-current circuits get so hot that they have to be cooled.

The heat produced by electric current is sometimes a desirable thing. Toasters, electric stoves, and heat lamps are common items that take advantage of the heat produced by current.

**Figure 6** High current causes a stove heating element burner to glow red.

**Putting it all together**

Free electrons are generated in copper at room temperature. When undirected, the motion of these free electrons is random, and the net flow of electrons in any one direction is zero.

When directed by an outside force, free electrons are forced to move in a uniform direction. This directed flow of charge is referred to as electric current.

Electric Current is represented by the letter I, which stands for intensity. The intensity of current depends on the amount of charge moved and the time required to move it.

Electric Current is measured in amperes (A). When one coulomb of current passes a point every second, you have one ampere of current.

There are **two current theories**. The electron flow theory describes current as the flow of charge (electrons) from negative to positive. The conventional current theory describes current as the flow of charge from positive to negative. Both approaches are widely followed. The way you view current does not affect the outcome of any circuit calculations, measurements, or test procedures.

Most electrical and electronic systems contain both direct current (DC) and alternating current (AC) circuits. In DC circuits, the current is always in the same direction. In AC circuits, current continually changes direction.

**Review Questions**

**How are free electrons generated in a conductor at room temperature?**

The thermal energy (heat) present at room temperature is enough to generate free electrons.

**What is electric current? What factors affect the intensity of electric current?**

Current is the directed flow of electrons in a material. The intensity of current depends on the amount of charge moved and the time required to move it.

**What is a coulomb?**

One coulomb equals the total charge on 6.25 × 10^{18} electrons.

**What is he basic unit of electric current?**

The ampere is the basic unit of electric current. It is defined as 1 coulomb per second or 1 A = 1 C/s.

**Contrast the electron flow and convention current theories.**

Conventional current theory defines current as the flow of charge from positive to negative. Electron flow is the flow of charge from negative to positive.

Voltage can be described as a force that generates the flow of electrons (current) through a circuit. In this section, we take a detailed look at voltage and how it generates current.

**Generating Current with a Battery**

Battery in **Figure 7a** has two terminals. The positive (+) terminal has an excess of positive ions and is described as having a positive potential. The negative (-) terminal has an excess of electrons and is described as having a negative potential.

**Figure 7** A difference of potential and a resulting current.

Thus there is a Difference of Potential, or voltage (V), between the two terminals.

If we connect the two terminals of the battery with the copper wire and lamp (**Figure7b)**, a current is produced as the electrons are drawn to the positive terminal of the battery. In other words, there is a directed flow of electrons from the negative (-) terminal to the positive (+) terminal of the battery.

There are **several important points** that need to be made:

**1.** Voltage is a force that moves electrons, for this reason, it is often referred to as Electrical Force (E) or Electromotive Force (EMF).

**2.** Current and voltage is not the same thing. Current is the directed flow of electrons from negative to positive. Voltage is the electrical force that generates current. In other words, current occurs as a result of an applied voltage (electric force).

The volt (V) is the unit of measure for voltage. Technically defined, one volt is the amount of electrical force that uses one joule (J) of energy to move one coulomb (C) of charge that is,

\[\begin{align}& \text{1 volt= 1 joule per coulomb} \\& \text{or} \\& \text{1 V=1 }{}^{J}/{}_{C} \\\end{align}\]

**Review Questions**

**What is voltage?**

Voltage is the force that generates current in a circuit.

**How does voltage generate a current through a wire?**

A voltage source has an excess of electrons (negative charge) on one terminal and an excess of positive ions on the other. This is referred to as a potential difference. The excess electrons at the negative terminal are attracted by the positive ions on the positive terminal. This results in the flow of charge in any wire that connects the two terminals of the voltage source.

**What is the unit of measure for voltage? How is it defined?**

The unit of measure for voltage is the volt. One volt is the amount of electrical force that uses one joule (J) of energy to move one coulomb (C) of charge. 1 V = 1 J/C.

**How would you define a coulomb in terms of voltage and energy?**

1 coulomb equals 1 joule per volt, 1 C = 1 J/V

**How would you define a joule in terms of voltage and charge?**

1 joule equals 1 V times 1 coulomb, 1 J = 1 V × 1 C

All elements provide some opposition to current. This opposition to current is called resistance. The higher the resistance of an element, component, or circuit, the lower the current produced by the given voltage.

Resistance (R) is measured in **Ohms**. Ohms are represented using the Greek letter omega (Ω). Technically defined, one ohm is the amount of resistance that limits current to one ampere when one volt of electrical force is applied. This definition is illustrated in **Figure 8**.

**Figure 8** A basic electric circuit.

The schematic diagram in **Figure 8** shows a battery that is connected to a resistor. A resistor is a component that provides a specific amount of resistance. As shown in the figure, a resistance of 1Ω limits the current to 1A when 1V is applied. Note that the long end-bar on the battery schematic symbol represents the battery’s positive terminal and the short end bar represents its Negative terminal.

**Putting It All Together**

We have now defined charge, current, voltage and resistance. For convenience, these electrical properties are summarized in **Table 1**.

**Table 1**: Basic Electrical Properties

Many of the properties listed in **Table 1** can be defined in terms of the others. For example, in our discussion on resistance, we said that one ohm is the amount of resistance that limits current to one amp when one volt of electrical force is applied. By the same token, we can redefine the ampere and the volt as follows:

**1.** One ampere is the amount of current that is generated when one volt of electrical force is applied to one ohm of resistance.

**2.** One volt is the amount of electrical force required to generate one amp of current through one

ohm of resistance.

**Review Question**

**What is resistance?**

Resistance is the opposition to current.

**What is the basic unit of resistance and how is it defined?**

The unit of resistance is the Ohm (Ω). One ohm is the amount of resistance that limits current to 1 ampere when one volt is applied. 1 Ω = 1 A/V.

**Define each of the following values in terms of the other two: current, voltage, and resistance.**

1 V is the force required to cause 1 ampere of current through 1 ohm of resistance.

1 A is the current that results from when 1 volt is applied to 1 ohm of resistance.

1 Ω is the resistance that limits current to 1 ampere when 1 volt is applied.

The post Electric Current, Voltage, and Resistance Overview | Three Basic Electrical Quantities appeared first on Electrical A2Z.

]]>The post Electrical Components and Symbols appeared first on Electrical A2Z.

]]>While the number of components used and their arrangement varies from one circuit to another, the types of components used are almost universal. In this article, we will take a brief look at the basic types of components, the symbols used to represent them in electrical diagrams, and the units that are used to express their values.

Circuit:A group of components that performs a specific function.

**Resistors**

We have all heard the term current as it applies to a river or stream: it is the flow of water from one point to another. In electrical circuits, the term current is used to describe the flow of electricity from one point in a circuit to another. A **Resistor** is a component that is used to restrict the current in a circuit, just as a valve can be used to restrict the flow of water through a pipe. Some resistors are shown in **Figure 1a**.

**Figure 1a **Resistors

The resistor symbols shown in **Figure 1b** are used in **Schematic Diagrams, **which show you how the components in a circuit are interconnected. The schematic symbol for a resistor indicates whether its value is fixed or variable.

Resistors:A component that is used to restrict (limit) current.

**Figure 1b **Resistors Symbols: Variable and Fixed Resistor Symbols

**Capacitors**

A **Capacitor **is a component that stores energy in an electric field. Capacitors are used in a wide variety of electric and electronic circuits. Several-capacitors and the common capacitor schematic symbols are shown in **Figure 2**.

**Figure 2:** Capacitors and their Symbols: Variable and Fixed Capacitors Symbols

Capacitors:A component that stores energy in an electric field, also called a condenser.

**Inductors**

An **Inductor **is a component that stores energy in a magnetic field. Though both inductors and capacitors store energy, they are very different in terms of their construction and characteristics. Several types of inductors and the basic inductor schematic symbols are shown in **Figure 3**. Inductors are often referred as Coils or Chokes**. **

**Figure 3: (a) & (b):** Inductors: Variable and Fixed Inductors symbols

Inductors:A component that stores energy in a magnetic field, also called a coil or a choke.

**Transformers**

A **Transformer** is a component that contains one or more inductors that are wrapped around a single physical structure. Transformers are used to:

• Change one voltage level to another.

• Transfer electrical energy from one circuit to another.

Transformers come in all types and sizes and are used in most electrical and electronic systems. A **Step-Down Transformer** decreases its input voltage to a lower level. A **Step-Up** **Transformer** increases its input voltage to a higher level. Some smaller transformers are shown in **Figure 4**.

**Figure 4 **Transformers.

**Switches**

A **Switch** is a component that is used to make or break an electrical connection. Several common types of switches are shown in **Figure 5**.

**Figure 5 **Switches

**Fuses and Circuit Breakers**

A **Fuse** is a component that automatically breaks an electrical connection. If the current increases beyond a certain value, if some condition, like an electrical short causes the current in a circuit to suddenly increase, the circuit fuse “burns open” to break the current path and protect the circuit. Some typical fuses are shown in **Figure 6**.

**Figure 6** Fuses and circuit breakers

A fuse can only protect a circuit once. If a fuse burns opens, it must be replaced. Unlike a fuse, a **Circuit Breaker** can be reset (closed) and used again. Two circuit breakers are included in **Figure 6**. Note the mini-breaker (center-blue) that can be inserted in a fuse box.

The post Electrical Components and Symbols appeared first on Electrical A2Z.

]]>The post Resistor Types and Color Code appeared first on Electrical A2Z.

]]>

**Figure 1.** Group of carbon composition resistors and the fixed resistor symbol.

The chemical makeup that causes resistance is accurately controlled in the resistor manufacturing process. Resistor values can be purchased in a range of values from less than 1 ohm to over 22 mega-ohms.

The physical size and material used for resistance is rated in watts. A resistor’s wattage rating refers to the resistor’s ability to safely dissipate heat. Heat is generated by electrons flowing through the resistor. Common wattage sizes range from 1/4 watt to 25 watts. Resistors are grouped by ohms and watt sizes. See **Figure 2**.

**Figure 2.** The physical size of a resistor can vary according to wattage rating. The higher the wattage, the larger the resistor. The largest shown is 20 watts, and the smallest is 1/8 watt.

When purchasing a resistor, the desired resistance and wattage rating must be specified. For example: 1000 ohms and the watt size, 1/4 watt, 1/2 watt, or 2 watts, etc. In each watt size, the resistance value would be the same. See **Figure 3** to examine the construction of a **molded composition resistor**.

**Figure 3**. Cutaway of a carbon composition resistor. (Allen-Bradley)

**Thin Film Resistor**

Another type of small wattage, fixed value resistor is the **thin film resistor**. The thin film resistor is similar to the molded composition resistor in appearance and function. However, the thin film resistor is made by depositing a resistance material on a glass or ceramic tube. A photographic process is used to deposit this film. Leads with caps are fitted over each end of the tube to make the body of the resistor. Thin film resistors are usually color coded.

The term “film resistor” is generally used to classify very compact resistors used in micro-electronics or on very small-scale electronic circuit boards. Film resistors can also be referred to as **surface-mounted resistors (SMRs)**. The demand for smaller and smaller electronic devices, such as cell phones, created the need for small discrete components such as resistors, to be manufactured in a more compact method.

Thick film and thin film are two general classifications based on how the film resistor is manufactured. Thin film deposits resistive material on an insulated substrate. Then the undesired portion is etched away leaving the desired pattern of resistive material. See **Figure 4**.

**Thick film** deposits a special resistive paste directly on the insulated substrate by using a stencil or silk screen process. As a result, thick film is typically a thicker deposit of resistive material as compared to thin film. The advantage of thick film is the resistor can support higher currents and wattage than the thin film. The advantage of thin film is smaller components requiring less height can be made.

**Figure 4.** This cutaway drawing displays how a film resistor is constructed.

The **substrate** is made from glass, ceramic, or silicon, and it is used as an insulator base for the resistor. A layer of resistor material is deposited on the substrate in a zigzag pattern precisely engineered to produce the desired resistance value.

The **resistor materials** are made from metals or carbon mixed in a precise proprietary formula. The result is a thin film, which is only a few micrometers thick or thick film, which is 10 to 50 micrometers thick.

A **protective coating** is used to cover the resistor material deposited on the substrate. The ends of the film resistor serve as connection terminals and are made of metal such as nickel or silver, or they can be made from an alloy.

**Solder** is used to mount and connect the film resistor to the circuit board. See **Figure 5**.The process is very similar to the same process used to manufacture integrated circuits (ICs).

**Figure 5.** You can see film resistors mounted on a circuit board.

The number stenciled directly on surface-mounted resistors indicates the resistance value. When a three-digit number is used, the first two digits represent the first two digits of the resistance value and the third digit represents the number of zeros to add to the first two digits. **For example**, the 202 in **Figure 5** represents the resistance value 2000 or 2k, + or – 5% tolerance (meaning it is allowed to vary by 5%).

When **four digits are used**, a tolerance of + or – 1% exists. For four-digit numbers, the same rule applies that the last digit represents the number of zeros. **For example**, 5002 would represent 50,000 or 50 k ohms, + or – 1% tolerance. For resistance values that require a decimal, the letter “R” is used to indicate the decimal point. For example, 7R5 would represent 7.5 ohms.

Film technology is not restricted to single resistors but may also be designed to produce resistor networks on a single chip. A resistor network consists of two or more independent resistors contained in one single surface- mounted package.

**Wire Wound Resistor**

For higher current uses, resistors are **wire wound**. A thin wire is wound on a ceramic core. The wire has a specific fixed-value resistance. The entire component is insulated by a coat of vitreous (opaque) enamel. These resistors are shown in **Figure 6**.

Wire wound resistors are commonly manufactured in sizes from 5 to 200 watts. The wattage chosen depends on the heat dissipation required during operation. Metal oxide resistors are also used for high voltage and wattage requirements.

**Figure 6**. Wire wound resistors.

**Adjustable Resistor **

Another type of wire wound resistor is the adjustable resistor. Unlike the standard wire wound resistor, the adjustable resistor is not entirely covered by enamel material. Instead, a portion of one side of the wire is exposed.

An adjustable sliding tap is attached to move across the exposed surface. This allows the resistance value to be varied. Adjustable resistors may have two or more taps for providing various resistance values in the same circuit. An adjustable resistor and symbol are shown in **Figure 7**.

**Figure 7.** This adjustable resistor provides a sliding tap for voltage divider uses. On the right is the schematic symbol for an adjustable resistor.

**Potentiometers**

Most electronic equipment requires the use of variable resistance parts. A potentiometer is a very common type of variable resistor found in electronic projects. The potentiometer has a rotary knob that varies the resistance value as it is turned.

The variation in resistance is provided by a contact that is attached to a ring of resistive material inside the device. This device is similar to the wire wound resistor.

Many potentiometers are constructed with thin wire inside as the source of resistance. Various styles of potentiometers are illustrated in **Figure 7a**. Also shown is the potentiometer’s schematic symbol.

**Figure 7a.** Potentiometers are used in electronic circuitry for fixed and variable resistance. Notice that the schematic symbol for the potentiometer is the same as for the adjustable resistor.

**Thermistors**

A special type of resistor is called a thermistor. In comparison to other types of resistors, the thermistor is unusual due to its ability to change resistance value rapidly as its temperature changes. It is commonly used to prevent high inrush currents in electrical circuits.

**An example** of a thermistor use can be seen in a **blow dryer**. A common blow dryer has heating elements composed of tungsten wire. The tungsten wire has a very low resistance value when cold, and a high resistance value when red hot. The thermistor is placed in series with the heating elements to prevent a high current value when the dryer is first turned on. As the blow dryer heats up, the resistance value goes down. The result is a fairly consistent current value as the dryer’s heating element changes from low resistance (cold) to high resistance (hot).

Early blow dryer models caused a dimming and flickering of lights and other electronic equipment in the home because of inconsistent current draw. The thermistor eliminates this problem.

Larger resistors are usually marked with their numerical resistance value printed directly on the side of the resistor. However, this type of labeling is not always practical, especially on small resistors. The resistor color code system was developed for this purpose.

The color code marking system has been adopted by the Electronics Industries Association (EIA) and the United States Armed Forces. This system of color coding is recognized throughout the world. Refer to **Figure 8**. Note how the color codes are printed, or banded, around the entire body of the resistor. This method of coding permits the value of the resistor to be read regardless of the mounting position. To see how to read the color coded bands refer to **Figure 9**.

F**igure 8. **Color coded bands encircle the resistor.

**Figure 9.** Standard color code for resistors.

Resistors commonly have three or four (and sometimes five) bands. Each band has a unique meaning.

- The
**first band**represents the value of the first digit of the resistance value. - The
**second band**represents the second digit of the resistance value. - The
**third band**is called the multiplier. The multiplier gives the factor of ten that the first two digits should be multiplied by. - The
**fourth band**represents the tolerance of the resistor.

Resistor tolerance is a reflection of the precision of resistor’s value. If a 20 ohm resistor has a 10% tolerance, the resistor’s value can vary by ± 2 ohms. In this case the resistor can have a true resistance value of 18 to 22 ohms. A fifth band is sometimes used to indicate resistor reliability or expected failure rate.

See **Figure 10**. In the first sketch, the **first bar** is red and the second bar is violet. Checking the color code chart shows the first two digits in the resistor’s value to be 2 and 7 (or 27). This number is then multiplied by the **third band**. The third band is brown (× 10). This indicates that the value 27 is multiplied by 10, for a final value of 270 ohms. This is what the resistor value would be if the resistor was perfect. **However**, the **fourth band** is silver. This indicates that this resistor has a tolerance of 10 percent. **In this example**, the tolerance is a ±27 ohms (270 × 0.1 = 27). Thus the value of this resistor is somewhere between 243 ohms and 297 ohms.

Work out the stated value, maximum value, and minimum value for the other two sketches in **Figure 10** on your own.

**Figure 10**. Examples of standard color-coded resistors.

The post Resistor Types and Color Code appeared first on Electrical A2Z.

]]>The post Electrical Circuit Devices appeared first on Electrical A2Z.

]]>**Switches**

Switches are installed in circuits to control the flow of electrons through the circuit. They can be categorized by their actuator and electrical switching path. The actuator is the mechanical device that causes the circuit to open and close.

Various types of switches are illustrated in **Figure 1**. The schematic symbol associated with each type of switch is also included.

**Figure 1.** Typical switch types. Notice how many have the same electrical symbol but different actuators.

Some of the most common actuators are the slide, toggle, rotary, and **push button**. As you can see, the name of the switch indicates the type of actuator used to turn the circuit on and off.

The electrical circuit inside the switch is described in terms of poles and throws. The simplest type of switch is the **single-pole single-throw switch**, which is abbreviated SPST. The term single-pole means that the switch provides one path for the electron flow and that it can be turned on or off. The term single-throw means that the switch controls only one circuit.

A **single-pole double-throw switch (SPDT)** has one common connection point and can complete a circuit path to two different circuits. However, only one circuit can be completed at a time. There are many possibilities and combinations for a switch of this type.

A useful application of the single-pole double-throw switch is its ability to control a load, such as a lamp, from two different locations. In residential wiring systems, the single-pole double-throw switch is referred to as a **three-way switch**.

In **Figure 2**, two single-pole double-throw switches are connected to a lamp. Either switch is capable of turning the lamp on or off.

**Figure 2.** Two single-pole double-throw switches (SPDT) can be used to control a lamp from two different locations.

A double-pole double-throw switch (DPDT) has two common connection points and can provide two circuit paths simultaneously. The DPDT switch is like having two SPST switches connected in parallel.

Switches are rated for **ampacity and voltage**.

The

ampacityrating of a switch is an indication of how much current it can safely handle.The

voltage ratingis the maximum voltage for which a switch is designed.

Exceeding the maximum voltage rating will cause the electrical-mechanical circuitry inside the switch to fail. **For example**, if a toggle switch is rated as one amp and 24 volts, a current in excess of one amp will burn out the switching circuitry inside the switch. If the 24-volt switch circuit sufficiently to stop the flow of electrons. This action will result in a dangerous situation that can melt the switch’s insulation and short circuit the switch.

**Connectors**

There are many types of connectors used with electrical conductors. The type of connection used varies according to the type and size of the conductor, the purpose served by the connection, and the type of device to be connected.

Look at **Figure 3**. You will see many common types of connectors. One general classification is solder less connectors. A solder less connector does not require the use of solder to make the connection. These connectors generally require a crimping tool. The crimping tool squeezes the connector to the conductor. **Figure 3** shows common wire crimps on terminals and splices.

**Figure 3**. Some wire connectors are made to be crimped on the end of stranded conductors. After the connector is crimped on the wire, the wire can be easily secured under a termination block screw.

Some types of connectors use screws and bolts to form the mechanical connection to conductors. These connectors are used primarily for larger conductors. See **Figure 4**.

**Figure 4.** Two types of solder less connectors are the wire nut connector and the split-bolt connector. The wire nut is used extensively in residential and commercial wiring. The split-bolt connector is used mainly on large diameter conductors.

Common circuit protection devices are fuses and circuit breakers.

**Fuse**

Fuses, such as those shown in **Figure 5**, are constructed from small, fine wire. This wire is engineered to burn if certain amperages are exceeded. Fuses are sized by their voltage and current capacity, primarily current.

**Figure 5.** A typical fuse and the schematic symbol that represents the fuse.

**For example, **a three-amp fuse is designed to burn and open the circuit when the current exceeds three amps. A load that draws three amps or greater will generate sufficient heat in the fuse to melt the fuse link inside the glass tube. The **time required** to melt the fuse link is inversely proportional to the amount of overload. This means that the higher the overload current, the faster the melting action occurs. When a fuse melts, it must be replaced.

**Circuit breaker**

A **circuit breaker**, sometimes called a reset, is another device used to protect a circuit from overload and short circuit conditions. See **Figure 6**.

The main advantage of a circuit breaker over the fuse link is that the circuit breaker need not be replaced after tripping. It can be reset by moving the handle to the off position and then returning it to the on position.

Some circuit breakers have an actuator similar to a push button switch. These breakers are pushed in to reset after tripping. Circuit breakers and resets require a waiting period to allow the internal trip mechanism to cool down.

Most homes today use circuit breakers as the safety device to prevent overloads. Overloads could result in house fires.

**Figure 6. **A typical circuit breaker.

Circuit breakers are produced with two different tripping methods.

**One method** uses bimetallic strips. A bimetallic strip is a metal strip made of two different types of metal. Different metals expand at different rates. Heat generated from the overload condition causes the bimetal trip to expand. The different metals expand at different rates. This causes the breaker’s trip mechanism to bend and break contact. Some trip mechanisms are adjustable to allow for a more precise trip current.

A **second tripping mechanism** uses magnetism to operate. The circuit current runs through a coil. As the current increases through the coil, the amount of magnet- ism in the coil increases. When a predetermined point is reached, the tripping mechanism operates and opens the circuit.

The magnetic circuit breaker is much faster and more accurate than the bimetallic circuit breaker. The magnetic circuit breaker, however, is more expensive than the bimetallic type.

The post Electrical Circuit Devices appeared first on Electrical A2Z.

]]>The post Low Pass and High Pass Filter Bode Plot appeared first on Electrical A2Z.

]]>\[\begin{matrix}{{\left| \frac{{{A}_{0}}}{{{A}_{i}}} \right|}_{dB}}=20{{\log }_{10}}\left| \frac{{{A}_{0}}}{{{A}_{i}}} \right|=20{{\log }_{10}}\left| \frac{{{A}_{0}}}{{{A}_{i}}} \right| & {} & (1) \\\end{matrix}\]

While logarithmic plots may at first seem a daunting complication, they have two significant advantages:

1. The product of terms in a frequency response function becomes a sum of terms because log(*ab/c*) = log(*a*) + log(*b*) − log(*c*). The advantage here is that Bode (logarithmic) plots can be constructed from the sum of individual plots of individual terms. Moreover, there are only four distinct types of terms present in any frequency response function:

a. A constant *K*.

b. Poles or zeros “at the origin”(*jω*).

c. Simple poles or zeros (1 + *jωτ*) or (1 + *jω/ω _{o}*).

d. Quadratic poles or zeros [1+ *jωτ* +(*jω/ω _{n}*)

2. The individual Bode plots of these four distinct terms are all well approximated by linear segments, which are readily summed to form the overall Bode plot of more complicated frequency response functions.

Consider the RC low-pass filter. The frequency response function is:

\[\frac{{{V}_{0}}}{{{V}_{i}}}(j\omega )=\frac{1}{1+j\omega /{{\omega }_{0}}}=\frac{1}{\sqrt{1+{{(\omega /{{\omega }_{0}})}^{2}}}}\angle -{{\tan }^{-1}}\left( \frac{\omega }{{{\omega }_{0}}} \right)\begin{matrix}{} & (2) \\\end{matrix}\]

where the circuit time constant is τ = *RC* = 1/*ω _{0}* and

**Figure 1** shows the Bode magnitude and phase plots for the filter.

**Figure 1** Bode plots for a low-pass *RC* filter; the frequency variable is normalized to *ω*/*ω*0. (a) Magnitude response; (b) phase angle response

The normalized frequency on the horizontal axis is *ωτ*. The magnitude plot is obtained from the logarithmic form of the absolute value of the frequency response function.

\[{{\left| \frac{{{V}_{0}}}{{{V}_{i}}} \right|}_{dB}}=20{{\log }_{10}}\frac{\left| K \right|}{\left| 1+j\omega \tau \right|}=20{{\log }_{10}}\frac{\left| K \right|}{\left| 1+j\omega /{{\omega }_{0}} \right|}\begin{matrix}{} & (3) \\\end{matrix}\]

When *ω ≪ ω _{0}*, the imaginary part of the simple pole is much smaller than its real part, such that |1 +

\[\begin{matrix}{{\left| \frac{{{V}_{0}}}{{{V}_{i}}} \right|}_{dB}}\approx 20{{\log }_{10}}K-20{{\log }_{10}}1=20{{\log }_{10}}K & (dB) & (4) \\\end{matrix}\]

Thus, at very low frequencies (*ω ≪ ω _{0}*),

When *ω ≫ ω*_{0}, the imaginary part of the simple pole is much larger than its real part, such that |1 + *jω/ω _{0}| ≈ | jω/ω_{0}*| = (

\[\begin{matrix}{{\left| \frac{{{V}_{0}}}{{{V}_{i}}} \right|}_{dB}}\approx 20{{\log }_{10}}K-20{{\log }_{10}}\frac{\omega }{{{\omega }_{0}}} & {} & {} \\{} & {} & (5) \\\approx 20{{\log }_{10}}K-20{{\log }_{10}}\omega +20{{\log }_{10}}{{\omega }_{0}} & {} & {} \\\end{matrix}\]

Thus, at very high frequencies (*ω ≫ ω _{0}*),

Finally, when *ω = ω _{0}*, the real and imaginary parts of the simple pole are equal, such that |1 +

\[20{{\log }_{10}}\frac{\left| K \right|}{\left| 1+j\omega /{{\omega }_{0}} \right|}=20{{\log }_{10}}K-20\log \sqrt{2}=20{{\log }_{10}}K-3dB\begin{matrix}{} & (6) \\\end{matrix}\]

Thus, the Bode magnitude plot of a **first-order low-pass filter** is approximated by two straight lines intersecting at *ω _{0}*.

The phase angle of the frequency response function $\angle \left( \frac{{{V}_{o}}}{{{V}_{i}}} \right)=-{{\tan }^{-1}}\left( \frac{\omega }{{{\omega }_{o}}} \right)$ has the following properties:

As a first approximation, the phase angle can be represented by three straight lines:

- For ω < 0.1ω
_{o}, ∠ (V_{o}/V_{i}) ≈ 0. - For 0.1 ω
_{o}and 10ω_{o}, ∠ (V_{o}/V_{i}) ≈ – (π/4) log (10ω/ω_{o}). - For ω > 10ω
_{o}, ∠ (V_{o}/V_{i}) ≈ –pi/2.

These straight-line approximations are illustrated in **Figure 1(b).**

**Table 1** lists the differences between the actual and approximate Bode magnitude and phase plots. Note that the maximum difference in magnitude is 3 dB at the cutoff frequency; thus, the cutoff is often called the **3-dB frequency** or the *half-power frequency*.

**Table 1 **Correction factors for asymptotic approximation of first-order filter

ω/ω_{0} |
Magnitude response error, (dB) |
Phase response error (deg) |

0.1 | 0 | −5.7 |

0.5 | −1 | 4.9 |

1 | −3 | 0 |

2 | −1 | −4.9 |

10 | 0 | +5.7 |

The case of an ** RC high-pass filter** is analyzed in the same manner as was done for the

\[\begin{matrix}\frac{{{V}_{0}}}{{{V}_{0}}}=\frac{j\omega CR}{1+j\omega CR}=\frac{j(\omega /{{\omega }_{0}})}{1+j(\omega /{{\omega }_{0}})} & {} & {} \\=\frac{(\omega /{{\omega }_{0}})\angle (\pi /2)}{\sqrt{1+{{(\omega /{{\omega }_{0}})}^{2}}}\angle \arctan (\omega /{{\omega }_{0}})} & {} & (7) \\=\frac{\omega /{{\omega }_{0}}}{\sqrt{1+{{(\omega /{{\omega }_{0}})}^{2}}}}\angle \left( \frac{\pi }{2}-\arctan \frac{\omega }{{{\omega }_{0}}} \right) & {} & {} \\\end{matrix}\]

**Figure 2** depicts the Bode plots for **equation 7**, where the horizontal axis indicates the normalized frequency *ω/ω _{0}*. Straight-line asymptotic approximations may again be determined easily at low and high frequencies. The results are very similar to those for the first-order low-pass filter.

For *ω < ω _{0}*, the Bode magnitude approximation intercepts the origin (

**Figure 2** Bode plots for RC high-pass filter. (a) Magnitude response; (b) phase response

The straight- line approximations of the Bode phase plot are:

- For
*ω*< 0.1*ω*, ∠ (V_{o}_{o}/V) ≈ π/2._{i} - 2. For 0.1ω
_{o}and 10ω_{o}, ∠ (V_{o}/V≈ − (π/4) log (10_{i)}*ω*/*ω*)._{o} - For
*ω > 10ω*, ∠ (V_{o}/V_{o}) ≈ 0._{i}

These straight-line approximations are illustrated in **Figure 2(b).**

Bode plots of high-order systems may be obtained by combining Bode plots of factors of the higher-order frequency response function. Let, for example,

\[H (j\omega )={{H }_{1}}(j\omega ){{H }_{2}}(j\omega ){{H }_{3}}(j\omega )\begin{matrix}{} & {} & ( \\\end{matrix}8)\]

which can be expressed, in logarithmic form, as

\[{{\left| H (j\omega ) \right|}_{dB}}={{\left| {{H }_{1}}(j\omega ) \right|}_{dB}}+{{\left| {{H }_{2}}(j\omega ) \right|}_{dB}}+{{\left| {{H }_{3}}(j\omega ) \right|}_{dB}}\begin{matrix}{} & {} & ( \\\end{matrix}9)\]

And

\[\angle H (j\omega )=\angle {{H }_{1}}(j\omega )+\angle {{H }_{2}}(j\omega )+\angle {{H }_{3}}(j\omega )\begin{matrix}{} & {} & ( \\\end{matrix}10)\]

Consider as an example the frequency response function

\[H (j\omega )=\frac{j\omega +5}{(j\omega +10)(j\omega +100)}\begin{matrix}{} & {} & (11) \\\end{matrix}\]

The first step in computing the asymptotic approximation consists of factoring each term in the expression so that it appears in the form a_{i} ( jω/ω_{i} +1), where the frequency ω* _{i}* corresponds to the appropriate 3-dB frequency, ω

\[\begin{matrix}H (j\omega )=\frac{5(j\omega /5+1)}{10(j\omega /10+1)100(j\omega /100+1)} & {} & {} \\{} & {} & (12) \\\frac{0.005(j\omega /5+1)}{10(j\omega /10+1)100(j\omega /100+1)}=\frac{K(j\omega /{{\omega }_{1}}+1)}{(j\omega /{{\omega }_{2}}+1)(j\omega /{{\omega }_{3}}+1)} & {} & {} \\\end{matrix}\]

**Equation 12** can now be expressed in logarithmic form:

\[\begin{matrix}H (j\omega ){{\left| _{dB}=\left| 0.005 \right| \right.}_{dB}}+\left| \frac{j\omega }{5}+1 \right|-\left| \frac{j\omega }{10}+1 \right|-\left| \frac{j\omega }{100}+1 \right| & {} & {} \\{} & {} & (13) \\\angle H (j\omega )=\angle 0.005+\angle \left( \frac{j\omega }{5}+1 \right)-\angle \left( \frac{j\omega }{10}+1 \right)-\angle \left( \frac{j\omega }{5}+1 \right) & {} & {} \\\end{matrix}\]

Each of the terms in the logarithmic **magnitude expression** can be plotted individually.

The constant corresponds to the value −46 dB, plotted in **Figure 3(a)** as a line of zero slope.

The numerator term, with a 3-dB frequency *ω _{1}* = 5, is expressed in the form of the first-order Bode plot of

You see that the individual factors are very easy to plot by inspection once the frequency response function has been normalized in the form of **equation 9.**

If we now consider the **phase response portion** of **equation 13,** we recognize that the first term, the phase angle of the constant, is always zero.

The numerator first-order term, on the other hand, can be approximated, that is, by drawing a straight line starting at 0.1ω_{1} =0.5, with slope +π/4rad/decade *(positive because this is a numerator factor)* and ending at 10ω_{1} = 50, where the asymptote +π/2 is reached.

The two denominator terms have similar behavior, except for the fact that the slope is −π/4 and that the straight line with slope −π/4 rad/decade extends between the frequencies 0.1ω_{2 }and 10ω_{2}, and 0.1ω_{3} and 10ω_{3}, respectively.

**Figure 3** depicts the asymptotic approximations of the individual factors in **equation 13**, with the magnitude factors shown in **Figure 3(a)** and the phase factors in **Figure 3(b).** When all the asymptotic approximations are combined, the complete frequency response approximation is obtained.

**Figure 4** depicts the results of the asymptotic Bode approximation when compared with the actual frequency response functions.

**Figure 3** Bode plot approximation for a second-order frequency response function. (a) Straight-line approximation of magnitude response; (b) straight-line approximation of phase angle response

**Figure 4** Comparison of Bode plot approximation with the actual frequency response function. (a) Magnitude response of second-order frequency response function; (b) phase angle response of second-order frequency response function.

You can see that once a frequency response function is factored into the appropriate form, it is relatively easy to sketch a good approximation of the Bode plot, even for higher-order frequency response functions.

This section illustrates the Bode plot asymptotic approximation construction procedure. The method assumes that there are no complex conjugate factors in the response and that both the numerator and denominator can be factored into first-order terms with real roots.

1. Express the frequency response function in factored form, resulting in an expression similar to the following:

\[H\left( j\omega \right)=\frac{K\left( {j\omega }/{{{\omega }_{1}}+1}\; \right)\cdots \left( {j\omega }/{{{\omega }_{m}}+1}\; \right)}{\left( {j\omega }/{{{\omega }_{m+1}}+1}\; \right)\cdots \left( {j\omega }/{{{\omega }_{n}}+1}\; \right)}\]

2. Select the appropriate frequency range for the semi logarithmic plot, extending at least a decade below the lowest 3-dB frequency and a decade above the highest 3-dB frequency.

3. Sketch the magnitude and phase response asymptotic approximations for each of the first-order factors, using the techniques illustrated in **Figures 1 to 4**.

4. Add, graphically, the individual terms to obtain a composite response.

5. If desired, apply the correction factors of **Table 1**.

The post Low Pass and High Pass Filter Bode Plot appeared first on Electrical A2Z.

]]>The post Band Pass Filter Frequency Response appeared first on Electrical A2Z.

]]>\[H (j\omega )=\frac{{{V}_{0}}}{{{V}_{i}}}(j\omega )\]

**Figure 1** *RLC* bandpass filter.

Apply voltage division to find:

\[\begin{matrix}{{V}_{0}}(j\omega )={{V}_{i}}(j\omega )\frac{R}{R+1/j\omega C+j\omega L} & {} & {} \\{} & {} & (1) \\={{V}_{i}}(j\omega )\frac{j\omega CR}{1+j\omega CR+{{(j\omega )}^{2}}LC} & {} & {} \\\end{matrix}\]

Thus, the frequency response function is:

\[\frac{{{V}_{0}}}{{{V}_{i}}}(j\omega )=\frac{j\omega CR}{1+j\omega CR+{{(j\omega )}^{2}}LC}\begin{matrix}{} & (2) \\\end{matrix}\]

**Equation 2** be factored into the form

\[\frac{{{V}_{0}}}{{{V}_{i}}}(j\omega )=\frac{jA\omega }{(j\omega /{{\omega }_{1}}+1)+(j\omega /{{\omega }_{2}}+1)}\begin{matrix}{} & (3) \\\end{matrix}\]

Where *ω _{1}* and

An immediate observation we can make is that if the signal frequency *ω* is zero, the response of the filter is equal to zero since at *ω* = 0 the impedance of the capacitor 1/*jωC* **becomes infinite**. Thus, the capacitor acts as an **open-circuit**, and the output voltage equals zero.

Further, we note that the filter output in response to an input signal at sinusoidal frequency approaching infinity is again equal to zero. This result can be verified by considering that as ω approaches infinity, the impedance of the **inductor becomes infinite**, that is, an **open-circuit**. Thus, the filter cannot pass signals at very high frequencies.

In an intermediate band of frequencies, the bandpass filter circuit will provide a variable attenuation of the input signal, dependent on the frequency of the excitation. This may be verified by taking a closer look at **equation 1:**

\[\begin{matrix}H (j\omega )=\frac{{{V}_{0}}}{{{V}_{i}}}(j\omega )=\frac{jA\omega }{(j\omega /{{\omega }_{1}}+1)+(j\omega /{{\omega }_{2}}+1)} & {} & {} \\=\frac{A\omega {{e}^{j\pi /2}}}{\sqrt{1+{{(\omega /{{\omega }_{1}})}^{2}}}\sqrt{1+{{(\omega /{{\omega }_{2}})}^{2}}}{{e}^{j\arctan (\omega /{{\omega }_{1}})}}{{e}^{j\arctan (\omega /{{\omega }_{2}})}}} & {} & \left( 4 \right) \\=\frac{A\omega }{\sqrt{\left[ 1+{{(\omega /{{\omega }_{1}})}^{2}} \right]\left[ 1+{{(\omega /{{\omega }_{2}})}^{2}} \right]}}{{e}^{j\left[ \pi /2-\arctan (\omega /{{\omega }_{1}})-\arctan (\omega /{{\omega }_{2}}) \right]}} & {} & {} \\\end{matrix}\]

**Equation 2** is of the form **H** (*jω*) =|H|e^{j∠H}, with

\[\begin{matrix} \begin{align} & \left| H (j\omega ) \right|=\frac{A\omega }{\sqrt{\left[ 1+{{(\omega /{{\omega }_{1}})}^{2}} \right]\left[ 1+{{(\omega /{{\omega }_{2}})}^{2}} \right]}} \\ & \angle H (j\omega )=\frac{\pi }{2}-\arctan \frac{\omega }{{{\omega }_{1}}}-\arctan \frac{\omega }{{{\omega }_{2}}} \\\end{align} & {} & (5) \\\end{matrix}\]

The magnitude and phase plots for the frequency response of the bandpass filter of **Figure 1** are shown in **Figure 2**. These plots have been normalized to have the filter passband centered at the frequency *ω* = 1 rad/s.

The frequency response plots of **Figure 2** suggest that, in some sense, the bandpass filter acts as a combination of a high-pass and a low-pass filter. As illustrated in the previous cases, it should be evident that one can adjust the filter response as desired simply by selecting appropriate values for *L, C*, and *R*.

**Figure 2** Frequency response of RLC bandpass filter

The response of second-order filters can be explained more generally by rewriting the frequency response function of the second-order bandpass filter of **Figure 1** in the following forms:

\[\begin{matrix}\frac{{{V}_{0}}}{{{V}_{_{i}}}}(j\omega )=\frac{j\omega CR}{LC{{(j\omega )}^{2}}+j\omega CR+1} & {} & {} \\=\frac{(2\zeta /{{\omega }_{n}})j\omega }{{{(j\omega /{{\omega }_{n}})}^{2}}+(2\zeta /{{\omega }_{n}})j\omega +1} & {} & (6) \\=\frac{(1/Q{{\omega }_{n}})j\omega }{{{(j\omega /{{\omega }_{n}})}^{2}}+(1/Q{{\omega }_{n}})j\omega +1} & {} & {} \\\end{matrix}\]

with the following definitions:

\[\begin{matrix}{{\omega }_{n}}=\sqrt{\frac{1}{LC}}=natural\begin{matrix}or\begin{matrix}resonant\begin{matrix}frequency \\\end{matrix} \\\end{matrix} \\\end{matrix} & {} & {} \\Q=\frac{1}{2\zeta }=\frac{1}{{{\omega }_{n}}CR}={{\omega }_{n}}\frac{L}{R}=\frac{1}{R}\sqrt{\frac{L}{C}}=quality\text{ }factor & {} & (7) \\\zeta =\frac{1}{2Q}=\frac{R}{2}\sqrt{\frac{C}{L}}=damping\text{ }ratio & {} & {} \\\end{matrix}\]

**Figure 3** depicts the normalized frequency response (magnitude and phase) of the second-order band pass filter for *ω _{n}* = 1 and various values of Q (and ζ). The peak displayed in the frequency response around the frequency

Note that as the **quality factor** *Q* increases, the sharpness of the resonance increases and the filter becomes increasingly ** selective** (i.e., it has the ability to filter out most frequency components of the input signals except for a narrow band around the resonant frequency).

One measure of the selectivity of a bandpass filter is its **bandwidth**. The concept of bandwidth can be easily visualized in the plot of **Figure 3(a**) by drawing a horizontal line across the plot (we have chosen to draw it at the amplitude ratio value of 0.707 for reasons that will be explained shortly). The frequency range between (magnitude) frequency response points intersecting this horizontal line is defined as the **half-power bandwidth** of the filter.

The name ** half-power** stems from the fact that when the amplitude response is equal to 0.707 (or ${}^{1}/{}_{\sqrt{2}}$ ), the voltage (or current) at the output of the filter has decreased by the same factor, relative to the maximum value (at the resonant frequency). Since power in an electric signal is proportional to the square of the voltage or current, a drop by a factor ${}^{1}/{}_{\sqrt{2}}$ in the output voltage or current corresponds to the power being reduced by a factor of ½.

Thus, we term the frequencies at which the intersection of the 0.707 line with the frequency response occurs the **half-power frequencies**. Another useful definition of bandwidth *B* is as follows. Note that a high-*Q* filter has a narrow bandwidth and a low-* Q* filter has a wide band with.

\[B=\frac{{{\omega }_{n}}}{Q}\begin{matrix}{} & Ban{{d}_{{}}}width & \begin{matrix}{} & (8) \\\end{matrix} \\\end{matrix}\]

**Figure 3(a)** Normalized magnitude response of second-order bandpass filter; **(b)** normalized phase response of second-order bandpass filter

The post Band Pass Filter Frequency Response appeared first on Electrical A2Z.

]]>The post Low Pass and High Pass Filter Frequency Response appeared first on Electrical A2Z.

]]>**Figure 1** depicts a simple **RC filter** and denotes its input and output voltages, respectively, by *V _{i}* and

**Figure 1** RC Low-pass filter

The frequency response for the filter may be obtained by considering the function

\[H (j\omega )=\frac{{{V}_{0}}}{{{V}_{i}}}\left( j\omega \right)\begin{matrix}{} & (1) \\\end{matrix}\]

and noting that the output voltage may be expressed as a function of the input voltage by means of a voltage divider, as follows

\[{{V}_{0}}(j\omega )={{V}_{i}}(j\omega )\frac{1/j\omega C}{R+1/j\omega C}={{V}_{i}}(j\omega )\frac{1}{1+j\omega RC}\begin{matrix}{} & (2) \\\end{matrix}\]

Thus, the frequency response of the *RC* filter is

\[\frac{{{V}_{0}}}{{{V}_{i}}}(j\omega )=\frac{1}{1+j\omega CR}\begin{matrix}{} & (3) \\\end{matrix}\]

An immediate observation upon studying this frequency response is that if the signal frequency ω is zero, the value of the frequency response function is 1. That is, the filter is passing all the input. Why? To answer this question, we note that at ω = 0, the impedance of the capacitor, 1*/jωC*, becomes infinite. Thus, the capacitor acts as an open-circuit, and the output voltage equals the input:

\[{{V}_{0}}(j\omega =0)={{V}_{i}}(j\omega =0)\begin{matrix}{} & (4) \\\end{matrix}\]

Since a signal at sinusoidal frequency equal to zero is a DC signal, this filter circuit does not in any way affect DC voltages and currents. As the signal frequency increases, the magnitude of the frequency response decreases since the denominator increases with *ω*. More precisely, **equations 5 to 8** describe the magnitude and phase of the frequency response of the RC filter:

\[\begin{matrix}H (j\omega )=\frac{{{V}_{0}}}{{{V}_{i}}}(j\omega )=\frac{1}{1+j\omega CR} & {} & {} \\=\frac{1}{\sqrt{1+{{(\omega CR)}^{2}}}}=\frac{{{e}^{j0}}}{{{e}^{-j\arctan (\omega CR/1)}}} & {} & \left( 5 \right) \\=\frac{1}{\sqrt{1+{{(\omega CR)}^{2}}}}{{e}^{-j\arctan (\omega CR)}} & {} & {} \\\end{matrix}\]

OR

\[H (j\omega )=\left| H (j\omega ) \right|{{e}^{j\angle H(j\omega )}}\begin{matrix}{} & (6) \\\end{matrix}\]

WITH

\[\left| H(j\omega ) \right|=\frac{1}{\sqrt{1+{{(\omega CR)}^{2}}}}=\frac{1}{\sqrt{1+{{(\omega /{{\omega }_{0}})}^{2}}}}\begin{matrix}{} & (7) \\\end{matrix}\]

AND

\[\angle H (j\omega )=-\arctan (\omega CR)=-\arctan \frac{\omega }{{{\omega }_{0}}}\begin{matrix}{} & (8) \\\end{matrix}\]

WITH

\[{{\omega }_{0}}=\frac{1}{RC}\begin{matrix}{} & (9) \\\end{matrix}\]

The simplest way to envision the effect of the filter is to think of the phasor voltage scaled by a factor of |**H**| and shifted by a phase angle ∠**H** by the filter at each frequency, so that the resultant output is given by the phasor with

\[\begin{matrix}{{V}_{0}}=\left| H \right|.{{V}_{i}} & {} & {} \\{} & {} & (10) \\{{\phi }_{0}}=\angle H +{{\phi }_{i}} & {} & {} \\\end{matrix}\]

and where |**H**| and ∠**H** are functions of frequency. The frequency *ω _{0}* is called the

It is customary to represent **H **(*jω*) in two separate plots, representing |**H**| and ∠**H** as functions of *ω*. These are shown in **Figure 2** in normalized form, that is, with |**H**| and ∠**H** plotted versus *ω/ω _{o}*, corresponding to a cutoff frequency

Note that, in the plot, the frequency axis has been scaled logarithmically. This is a common practice in electrical engineering because it enables viewing a very broad range of frequencies on the same plot without excessively compressing the low- frequency end of the plot.

The frequency response plots of **Figure 2** are commonly employed to describe the frequency response of a circuit since they can provide a clear idea at a glance of the effect of a filter on an excitation signal. The cutoff frequency *ω* = 1/*RC* has a special significance in that it represents approximately the point where the filter begins to filter out the higher-frequency signals. The value of |**H** (jω)| at the cutoff frequency is ${}^{1}/{}_{\sqrt{2}}=0.707$ . Note how the cutoff frequency depends exclusively on the values of *R* and *C*. Therefore, one can adjust the filter response as desired simply by selecting appropriate values for *C* and *R*, and therefore one can choose the desired filtering characteristics.

**Figure 2** Frequency response of an RC low-pass filter

Just as a low-pass filter preserves low-frequency signals and attenuates those at higher frequencies, a **high-pass filter** attenuates low-frequency signals and preserves those at frequencies above a cutoff frequency. Consider the high-pass filter circuit shown in **Figure 3**.

**Figure 3** RC High-pass filter

The frequency response is defined as:

\[H (j\omega )=\frac{{{V}_{0}}}{{{V}_{i}}}(j\omega )\]

Voltage division yields:

\[{{V}_{0}}(j\omega )={{V}_{i}}(j\omega )\frac{R}{R+1/j\omega C}={{V}_{i}}(j\omega )\frac{j\omega CR}{1+j\omega CR}\begin{matrix}{} & (11) \\\end{matrix}\]

Thus, the frequency response of the filter is:

\[\frac{{{V}_{0}}}{{{V}_{i}}}(j\omega )=\frac{j\omega CR}{1+j\omega CR}\begin{matrix}{} & (12) \\\end{matrix}\]

Which can be expressed in magnitude-and-phase form by

\[\begin{align}& H (j\omega )=\frac{{{V}_{0}}}{{{V}_{i}}}(j\omega )=\frac{j\omega CR}{1+j\omega CR}=\frac{\omega CR{{e}^{j\pi /2}}}{\sqrt{1+{{(\omega CR)}^{2}}}{{e}^{j\arctan (\omega CR/1)}}} \\& =\frac{\omega CR}{\sqrt{1+{{(\omega CR)}^{2}}}}.{{e}^{j\left[ \pi /2-\arctan (\omega CR) \right]}} \\\end{align}\]

Or

\[\begin{matrix}H (j\omega )=\left| H \right|{{e}^{j\angle H }} & {} & \left( 13 \right) \\\end{matrix}\]

with

\[\begin{matrix}\left| H (j\omega ) \right|=\frac{\omega CR}{\sqrt{1+{{(\omega CR)}^{2}}}} & {} & {} \\{} & {} & (14) \\\angle H (j\omega )={{90}^{o}}-\arctan (\omega CR) & {} & {} \\\end{matrix}\]

You can verify by inspection that the amplitude response of the high-pass filter will be zero at *ω* = 0 and will asymptotically approach 1 as *ω* approaches infinity while the phase shift is *π*/2 at *ω* = 0 and tends to zero for increasing ω.

Amplitude and phase response curves for the high-pass filter are shown in **Figure 4.** These plots have been normalized to have the filter cutoff frequency *ω*_{0} = 1 rad/s. Note that, once again, it is possible to define a cutoff frequency at *ω _{0}* = 1/

**Figure 4** Frequency response of an RC high-pass filter

The post Low Pass and High Pass Filter Frequency Response appeared first on Electrical A2Z.

]]>