# The Source- Free RC Circuit

To enlarge the image, please click the picture.
The source- free RC circuit takes place when the source of a direct current is disconnected abruptly.

Dealing with this kind of circuit, you must consider the ff:

• The initial voltage v(0)= Vo across the element capacitor
• The time constant t

resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors controlled by a voltage or .current sources.

A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit.

Please click on the image for enlargement

I want to show you a video which helps you to understand more, what is all about this source-free RC circuit.

Another example of this circuit is shown in this figure.

Sample Problem:

click to enlarge this picture

The solution will be like this,

# First- Order Circuits

The circuits or networks which carry energy storage elements are solved and calculated using differential equations. You must know how to solve problems using differentiation.

The “order” of such circuit is stipulated by the order of the differential equations that solved it.

• The “purely resistive” circuit is a zero order circuit. Obviously, this circuit has zero energy storage elements.
You can use equations which are zero-order differential equations, such as purely algebraic equations.
• The circuit with an (irreducible) energy storage element is what we call first order circuit”
You can use equations which are first order differential equations.

So here guys, let’s consider a circuit with just only one capacitor or only one inductor, which is an example of a first order circuit.

Now, look at the figure shown:

Viewing this circuit from the perspective of the energy storage element, we can apply on these networks, the Thevenin’s and Norton’s Theorems. We can always reduce a first order circuit to one of these:

For the capacitor voltage or the inductor current, we will have  to find, then solve the equation shown below on each of these circuits. These equations merely came from applying KVL (to the circuit with the capacitor) and KCL (to the circuit with the inductor).

I got what you are thinking right now! Is it, how will you solve these equations?

Note that they all have the same mathematical form!

The form is a homogeneous, (i.e. one variable), linear, (i.e. first order), differential equation. The solution to these differential equations, are functions of time:  v(t) or i(t)

Rewriting both equations above in a general way as follows:

The letters you can find on this equation are defines as follows;

• x(t) is either the capacitor voltage,
• v(t) or the inductor current,
• i(t) (both are functions of time)
• t is the constant RC or L/R (note: C might be Ceq , L might be Leq  and R = RT)
• y(t) is the forcing function (related to the voltage source or current source)

# DC CIRCUITS

This diagram shows the elements, quantities, laws, and components which comprise the dc circuit.

Some terms that you have seen in the diagram will be discuss, in order for you to familiarize these things , and to fully understand the meaning and applications of these, but there are some electrical terms, that will be discuss in the next chapters.

Moving Coil Meters

The design of a voltmeter, ammeter or ohmmeter begins with a current-sensitive element. Though most modern meters have solid state digital readouts, the physics is more readily demonstrated with a moving coil current detector called a galvanometer. Since the modifications of the current sensor are compact, it is practical to have all three functions in a single instrument with multiple ranges of sensitivity. “Multimeter” might be designed as illustrated.

A voltmeter measures the change in voltage between two points in an electric circuit and therefore must be connected in parallel with the portion of the circuit on which the measurement is made.

An ammeter is an instrument for measuring the electric current in amperes in a branch of an electric circuit. It must be placed in series with the measured branch.

The standard way to measure resistance in ohms is to supply a constant voltage to the resistance and measure the current through it, It is by the use of ohmmeter.

Direct current (DC) circuits basically consist of a loop of conducting wire (like copper) through which an electric current flows. An electric current consists of a flow of electric charges, analogous to the flow of water (water molecules) in a river. In addition to the copper wire in a circuit, there usually are components such as resistors which restrict the flow of electric charge, similar to the way rocks and debris in a river restrict the flow of the river water.

In the figure shown,

Circuit has:

•  battery
•  bulb/ lamp
•  switch(control)
•  wires connecting it all up

The connections must be made in such a way as to allow the energy to flow from the source, through the load and back into the source to form a loop.

Simple circuits: is a closed loop of conductor through which charges can flow.

Direct current (dc) is used in such equipment as automobiles, mine locomotives, metal refining, computers, communication transmitters and receivers. Direct current is sometimes used in long distance transmission of large amounts of high voltage power through it is converted back to AC/ alternating current before being distributed to the power company’s customers.

In the figure shown,

It is the schematic diagram of the DC circuit. It shows how each resistors / wire in the installation or equipment is connected electrically to the system.

Real Life Application:

Anywhere, anytime,in most buildings, including our homes, offices and schools, we are surrounded by many devices and equipment that internally operates on Direct Current (DC). These devices are plug into a typical Alternating Current (AC) outlet, and then converted from AC to DC to operate the equipment and devices.

Such examples that use dc are television remote control and cellphones. Whether watching TV, power heating and cooling systems, or charging a cell phone, we rely on our home’s electrical system to provide us with power when and where we need it. By understanding the basics of how electricity is distributed around your home, you can keep this important system properly maintained and in safe working condition. These things worked through batteries and as well as all the factors contributing to DC circuits.

FUNDAMENTAL QUANTITIES OF ELECTRICITY

The electrical quantities commonly involved charge, electric current, voltage, power, and energy. The movement of charge is called current. This current needs pressure to overcome resistance and to flow in an electric circuit.

This explains the relationship between the factors of voltage (pressure), current (flow), power and others in a circuit. For you to make calculations and solutions in a circuit, it is very necessary to know how these factors or components are related to each other and the units in which these factors are being measured and expressed.

a.)            CHARGES

In an electrical state of protons and electrons, the words positive and negative are used. We all know that the negatively charged particles are called electrons. These particles repel each other and attracted by the protons, which are the positively charged particles that repel each other and attract electrons. These two were indicated by these signs (-) and (+) respectively. The physical law regarding charges stated,” LIKE CHARGES REPEL EACH OTHER; UNLIKE CHARGES ATTRACT EACH OTHER.”

To come up with a possible actual current flow, charge must be delivered. A charge is created when there’s a surplus of electron/ proton in a given place. If positively charged, the area is subject to a flow of electrons into it, otherwise, when it comes to negatively charged, the area is subject to a flow of electrons out of it.

In the figure, the atom has five electrons and six protons. Since it has more positive protons than negative electrons, it is said to be positively charged.

In the figure, the atom has seven electrons and six protons. Since it has more negative electrons than positive protons, it is said to be negatively charged.

Charge is measured in Coulomb(c) which is equal to 6.24 x 10 ^18 electrons. The coulomb with its unit symbol: c is the SI derived unit of the electric charge with a symbol Q or q. It is meant as the charge transported by a constant current of one ampere in one second. It comes from the name of Charles Coulomb who experimentally established the fundamental law of electric force between two stationary particles. An important characteristic of charge is that electric charge is always conserved.

Real Life Application:

After running a plastic comb through your hair, you will see and find that the comb attracts bits of paper. The attractive force is often strong enough to suspend the paper from the comb, defying the gravitational pull of the entire Earth. The same effect occurs with other rubbed materials, such as hard rubber and glass. Another example is to rub an inflated balloon against wool (or across your hair). You can observe that the rubbed balloon will then stick to the wall of the room. These materials become electrically charged. Another one is when you give your body an electric charge by vigorously rubbing your shoes on a wool rug or by sliding across a car seat. By this, we can conclude that like charges repel one another and unlike charges attract one another.

b.)            CURRENT

When the bundle or mass of displaced electrons (sometimes called “free” electrons) moves along a conductor, we say that current is flowing in the conductor. It is an electrical movement measured in AMPERES (A) and can be defined as the time rate of change of the charge.

Andre-Marie Ampere is a French mathematician and physicist who invented the science of electromagnetism. He was the first person to come up with techniques for measuring the electricity. The ampere which is the SI unit of measurement of electric current is named after him.

AMPERE: Current is measured in amperes abbreviated as A. This term is used in honor of Andre- Marie Ampere, who pioneered in the science of electricity. The letter symbol for the current in amperes is or i. When a charge of 6,280,000,000,000 electrons passes a given point in a conductor in one second, we say that one ampere is flowing. For the smaller units of measurement, the terms milliampere and microampere are used.

From the figure shown, You can see the relationship between the current (i), the charge (q), and the time (t).

A current of 1 A is flowing in a circuit if a charge of 1 coulomb passes any point in the circuit every second.

1 Amp = 1 Coulomb per second. We can write this formula as:  Current (I) = Charge (Q) / Time (t) or Charge (Q) = Current (I) x Time (t)

So now, let’s begin solving the simplest part among all the topics.

Sample Problems:

1. A charge of 12 C passes through the filament of a car headlamp bulb in 4 s. What is the current?

Answer: Current = Charge/time = 12/4 = 3 A

2. A current of 0,5 A flows for 20 s through a small electric motor. How much charge has passed?

Answer: Charge = Current x time = 0.5 x 20 = 10C

3. A current of 200 mA flows for 2 minutes. How much charge has passed?

Charge = 0.200 x 120 = 24 C

Answer: (Current in amps, time in seconds)

The charge displacement between the original time t0 and t is acquired by simply integrating both sides of the i=dq/dt. Resulting to this equation:

Sample Problem:

_________________________________________________________________________

From the two figures shown, these are the common types of current. From your left side, it is the alternating current (AC) which is dissimilar and changes sinusoidally with the time. In your left side, however, is the Direct current (DC) which remains constant in time.

Real Life Application:

You use current in your household, like operating your electric appliances. One example is refrigerator. Our refrigerators are one of the most valuable equipments found in our home today. This is very necessary for us, humans. Almost every household needs something to store their foods, vegetables, meats, and others. It is to prevent them from spoiling. This equipment which is craftily made turns on every five minutes and keeps everything cold and fresh. Without it, there will be enormous amount of food that will go to be on the garbage every day. Surely, such invention is great that it affects almost every people on earth regarding whatever their walks of life are. Natural preservatives used in the past are through salt and ice. This will relatively lengthen and preserve the foods, but they are not that easy to do, it makes your lives more complicated, and the preparations are delicate. Refrigerators today are not only helping us on preserving our foods, but also it provides us a luxurious kind of life. It keeps our beverages cool, whenever we wanted them to, and it has freezer, which make a fine delicious desert that all of us are enjoying today, especially when preparing foods for occasions, celebrations, and holidays. There’s a lot of advancement happens in our world today due to the technology with the aid of current, and that includes the advancement in the features of a refrigerator. The first model of the refrigerator consumes only a small amount of power consumption, but today, it consumes more energy than usual.

c.)            VOLTAGE

An atom that has lost some of its electrons has a surplus of protons and is positively charged. Protons are locked in the nucleus or core and cannot be displaced by ordinary means. Having a surplus of protons, the atom is in need of negative electrons to neutralize the positive charge. This results in a force of attraction for negatively charged electrons to neutralize the strain. Because other nearby electrons exert a force of repulsion against them, a two-way force is exerted on the displaced electrons. They are repelled toward the positively charged atom and are also attracted by it. The total force exerted on the electron is called electromotive force or simply called as “voltage”.

The Vab voltage between the two points a and b in an electric circuit is what we call the energy or work needed to move a unit charge from a to b; stated mathematically as:

Where Vab or simply v is measured in volts, w measured in joules, which is the energy, and the q, measured in coulombs, which is the charge.

1 volt is equal to 1 joule per coulomb, expressed as 1 newton-meter/ coulomb.

History

The Italian physicist Alessandro Volta (1745-1827) invented the electric battery, or “voltaic pile,” thus providing for the first time a sustained source of current electricity.

Alessandro Volta was born on Feb. 18, 1745, in Como. He resisted pressure from his family to enter the priesthood and developed instead an intense curiosity about natural phenomena, in particular, electricity. In 1769 he published his first paper on electricity. It contained no new discoveries but is of some interest as the most speculative of all Volta’s papers, his subsequent ones being devoted almost exclusively to the presentation of specific experimental discoveries.

Early Investigations and Inventions of Volta:

In 1774 Volta was appointed professor of physics at the gymnasium in Como, and that same year he made his first important contribution to the science of electricity, the invention of the electrophorus, a device which provided a source of electric potential utilizing the principle of electrostatic induction. Unlike earlier source of electric potential, such as the Leyden jar, the electrophorus provided a sustained, easily replenishable source of static electricity. In 1782 Volta announced the application of the electrophorus to the detection of minute electrical charges. His invention of the so-called condensing electroscope culminated his efforts to improve the sensitivity of earlier electrometers.

The figure shown shows the voltage across an element (represented by a rectangle) which is connected to points a and b. To define the reference direction or the polarity of the voltage, we need to use the plus and minus signs, represented as this (+_ and (-).

The vabcan be analyzed by these two processes: 1.) the point a is at the potential vab  volts higher than the point b.

2.) the potential at point a with respect to point b is vab.

It is logically stated as: vab.= -vab.

The Volt:

Voltage is measured in volts (abbreviated as V). The term volt is used in honor of Alessandro Volta, an Italian physicist who invented an electric cell. The letter symbol for voltage is V. Voltage is also referred to as potential, potential difference (pd), or electromotive force (emf). Voltage is measured with a voltmeter. Smaller units of measurement are the millivolt(mV) which is 1/1000 of a volt, and microvolt; however the larger unit for this is the kilovolt, which is 1000 V.

d.)        POWER

Electric Power is measured in watts (abbreviated as W). The name watt is used in honor of James Watt, a Scottish engineer and inventor, who originated the term horsepower and defined it as a unit of mechanical power. The letter symbol for electric power is P. Power may be measured with a wattmeter. A larger unit of measurement of electric power is the kilowatt, abbreviates as kw, which is equal to 1000 w.

Electric power

Electric power is the amount of work done by an electric current in a unit time.

When a current flows in a circuit with resistance, it does work. Devices can be made that convert this work into heat (electric heaters), light (light bulbs and neon lamps), or motion.

Where P is the electric power in watt (W).

E is the energy consumption in joule (J).

t is the time in seconds (s).

Sample Problem:

Find the electric power of an electric circuitthat consumes 120 joules for 20 seconds.

Solution:

E = 120J

t = 20s

answer: P = E / t = 120J / 20s = 6W

Electric power calculation

In the figure a, it shows the relationship of power, energy and time.

Simply written mathematically as

P = · Iwhich you can find in figure b.

or P = I · R or P = V 2 R

Where P is the electric power in watt (W).

V is the voltage in volts (V).

I is the current in amps (A).

R is the resistance in ohms (Ω).

James Watt is the man who discovered the steam engine which steered the industrial revolution. The unit of measurement which is used in the measurement of electrical and mechanical power ‘the watt’ was named in his honour.

e.)            ENERGY

Watts are used to measure the rate at which electric power is being used in a given amount of time. This measurement does not indicate how much electric energy has been used. Time is a factor that must be considered in determining the amount of energy used in a given period. Usually, this is done by multiplying watts by hours. The result is watthours, abbreviated as Wh. If power is measured in kilowatts and multiplied by hours, the result is kilowatthours, abbreviated as kWh. The unit kilowatthour is used to measure a definite amount of electric energy.

This is an energy made available by the flow of electric charge through a conductor; “they built a car that runs on electricity”. It is a form of energy that is absorbed or delivered by an electric circuit. It is also described as energy that results from the conversion of electric potential energy which is the potential energy in a device with electric fields that change with time.

Real Life Application:

We see the lights everywhere. We see different things around our surroundings, which are being operated by the aid of electricity, and this all happened, as we all know that they exist and it is by the help of what we call “electric energy”. From computer, grinder, television, dryer, radio, toaster, loudspeaker, laptop, light bulbs, everything runs on the electrical/ electric energy. So now,let’s see what is all about electrical energy, and some of its uses.

The electrical energy is nothing, but it is defined as the flow of electric charge through the conductor. The flow of tiny charged particles called electrons takes place, due to the potential difference or voltage developed between the conductors. Here, you can find some examples which use electrical energy and you can go through it:

The first or the most common approach use of electrical energy is the use of batteries. Regardless on their sizes, they are used to store electrical energy as chemical energy and later using it for our needs especially when there is a brownout, as flash light or bulb light.

Another one that best illustrates electrical energy is the lightning. This includes charge separation, where the positively charged particles get separated from the negatively charged particles. There develops a charge potential when the separation occurs. When the separation is high enough discharge takes place that leads to flow of electric current.

Next, the hydroelectric dam that acts as a medium. It converts the kinetic energy of the falling water into what we call electrical energy.

Electric energy is a basic part of our nature and it is one of our most widely used forms of energy nowadays. We get electricity, which is a secondary energy source, from the conversion of other sources of energy, like coal, natural gas, oil, nuclear power and other natural sources, which are called primary sources.

SAMPLE PROBLEM:

f.)              CIRCUIT ELEMENTS

Color Code:

The ohm is the unit of resistance. One symbol for ohm Ω (Greek letter omega). Resistance values are indicated by standard color code adopted by manufacturers.

This code involves the use of color bands on the body of the resistor. The colors and their numerical values are given in the resistor color chart found in the ff. figure.

Common resistors are marked with a standard color code from which you can determine the resistance and tolerance. It is accompanied in the following table by the rude and politically incorrect mnemonic sentence which has been used for generations.

CLICK ON THE IMAGES TO ENLARGE:

A variation on the color code is used for precision resistors which may have five colored bands. In that case the first three bands indicate the first three digits of the resistance value and the fourth band indicates the number of zeros. In the five band code the fifth  band is gold for 1% resistors and silver for 2%.

There is another scheme for resistors which have the values stamped on them. Since a decimal point is easy to miss, this code uses R instead of a decimal point. For values over 100 W four numbers are used. The tolerance is indicated by a letter.

Examples:

8R2K = 8.2 W +/- 10%

2202F = 22000 W +/- 1%

• K, A MULTIPLIER WHICH STANDS FOR 1,000
• M, A MULTIPLIER WHICH STANDS FOR 1,000,000

33 kilo ohms (33kΩ) stands for 33,000Ω; 1.2 mega ohms(1.2 MΩ) stands for 1,200,000Ω.

Transformer

A transformer makes use of Faraday’s law and the ferromagnetic properties of an iron core to efficiently raise or lower AC voltages. It of course cannot increase power so that if the voltage is raised, the current is proportionally lowered and vice versa.

Inductors

Inductance is typified by the behavior of a coil of wire in resisting any change of electric current through the coil.

Arising from Faraday’s law, the inductance L may be defined in terms of the emf generated to oppose a given change in current:

Capacitors

Capacitance is typified by a parallel plate arrangement and is defined in terms of charge storage:

where

Q = magnitude of charge stored on each plate.

V = voltage applied to the plates.

Capacitor

Independent Source Elements

Use independent source element statements to specify DC, AC, transient, and mixed independent voltage and current sources. Some types of analysis use the associated analysis sources. For example, in a DC analysis, if both DC and AC sources are specified in one independent source element statement, the AC source is taken out of the circuit for the DC analysis. If an independent source is specified for an AC, transient, and DC analysis, transient sources are removed for the AC analysis and DC sources are removed after the performance of the operating point. Initial transient value always overrides the DC value.

The figure shown are the independent voltage sources. In a.) It is for the time- varying or constant voltage, while in b.) it is used for the constant voltage.

The figure shown is a symbol for the independent current source.

Dependent Sources:

It is a voltage source or a current source whose value and equivalent depend on a voltage or current somewhere else in the circuit. The values of these sources are proportional to some other voltage or current in the circuit.

Sample Problem:

Lessons I’ve learned: (Basic Concept)

• An electrical circuit is a path in which electrons from a voltage or current source flow. Electric current flows in a closed path called an electric circuit
•  Electric charge have the ff: properties.
• Unlike charges attract one another and like charges repel one another
• Electric charge is always conserved.
• Charge comes in discrete packets that are integral multipliers of basic electric charge.
•  Electric current is an electrical movement measured in AMPERES(A) and can be obtained by the time rate of change of charge.
•  Voltage is a measure of the difference in electric potential between two points in space, a material, or an electric circuit, expressed in volts.
•  Electrical energy is the energy carried by moving electrons in an electric conductor. It cannot be seen, but it is one of our most useful forms of energy.
• A circuit element is basically just a component that makes up a complete electrical circuit. They are like building blocks that can be combined to create interesting circuits and model real world electronics. Some examples include conductors, voltage sources, current sources, and resistors.