Circuit Theory Final

Elements – building blocks of matter. Compounds are the chemical combination of two or more elements.

Molecule are the smallest part of the compound. It is the chemical combination of two or more atoms.

The number of protons, neutrons and electrons an atom has tells us the material. Some materials hold these particles more tightly than others.

Electrons orbit the nucleus in shells, each shell has a different energy level, increasing the further it gets from the nucleus. The most outer shell is called the valence. 2,8, etc.

The electrons in the shells closest to the nucleus have a strong force of attraction to the protons. Sometimes the electrons in an atom's outermost shells do not have a strong force of attraction to the protons.

These electrons can be pushed out of their orbits, applying a force can make them shift from one atom to another – these are electricity.Insulators prevent the flow of electrons or electricity. These materials have 5 or more electrons in the valence shell.

Semiconductors:

Materials that can be altered to function as either a conductor or insulator. They have 4 electrons in the valence shell. Used extensively in electronic circuits.

Forming a circuit -->

Looking inside a copper cable, the electrons of atoms move from one atom to another, randomly and freely.

If the slice of a cable were in a closed circuit, the battery voltage forces free electrons to move and flow in the same direction.

Atoms generally have a neutral state. They are at a "ground" state. With enough energy we can change the atoms charge by causing it to gain or lose electrons. They are electrically charged if there are too many or few electrons.

This process is called ionization. Can happen through collision with other atoms. Ionized atoms = ions.

If it gains electrons --> negatively charged.

If it loses electrons --> positively charged.

In the neutral state an atom has little attraction. Ions have strongly attraction, as they are highly chemically reactive.

Ions tend to discharge and return to their neutral state over time.

Non-contact forces depend on the charge of two objects. Protons and electrons create electric fields, which exert the Coulomb force – radiates outward in all direction.

Electrical Charge:

All charged objects have an electric field around them, which shows how they will interact with other charged particles.

Electricity is the movement of electrons between atoms. We harness electrons to do "work".

Movement of electrons through a circuit is electric current.

Wires are made of metal and these always have loose electrons moving throughout them. If you can make them move in an organized way – is a current.

Electrical Charge (Q) measured in Coulombs (C)

1 Coulomb is 6250000000000000 electrons.

1 electron is 1.6 x 10^-19

Electrical Current (I) measured in Amps (A)

Q = I t (time – seconds)

Energy (E) is often measured in Joules (J).

1J = amount of energy transferred to an object when the force of 1N is applied to that object over a distance of 1m.

Can be potential or kinetic. Voltage = PD is a measure of electrical potential energy between two points. Voltage – measured in volts = 1V = 1J/1C

Electrical charge or energy is converted into other forms of energy – often to a component called a "Load". Electrical potential is needed to move these charges. Circuits go from high potential to low.

How much current is flowing through a component of a circuit depends of the resistance and PD (V) across the component.

Voltage is the force that moves electrons across a circuit, and produces current flow.

E = QV. When charge moves through a PD, electrical work is done and energy is transferred.

V = E/Q

Electrical grounding is the process of removing excess charge, by providing a path for the current to return to the battery, or physically into the ground.

Resistance is the opposition to the flow of electrons, is measured in Ohms. Every material/component has some element of resistance. Conductors have a low resistance, inductors have a high resistance to current flow.

Electromotive Force:

The characteristic of any energy source capable of driving electric charge around a circuit. The force within a voltage that drives the current around a circuit.

EMF is the measure of energy supply to each coulomb of charge.

Voltage is the energy use by one coulomb of charge to move from one point to another.

EMF = E or Vs.

EMF is measured between the end points of the source. Voltage is measured between any two points of the closed circuits.

Devices – transducers provide an EMF by converting other forms of energy to electrical.

Includes batteries (chemical) or generators (mechanical)

Power is the rate of energy use of conversion and is measured in Joules Per second (Watts)

E = Fs

P= E/T

P= QV/T

Electrical components are given a power rating, that indicated the maximum rate of conversion.

A source of energy delivers power to a load which absorbs it. Light bulbs for example, convert this electrical power.

If energy is the ability to do work, power is the rate of "doing work". P = QV/t and Q= It and P = IV.

Amps usually A, mA, μA

Volts usually kV, V, mV

Resistance usually MΩ, KΩ, mΩ

Capacitance is μF, pF

Inductance is mH, μH

Active Power is MW, kW, mW

The electrical energy transferred each second is also found by multiplying voltage by current.

If we don't know the current or voltage, can use Ohm's Law.

When current is unknown P = v^2 / R

When voltage is unknown P = I^2 R

Direct Current is the flow of electricity in one direction. DC power is far more consistent in terms of voltage delivery, meaning most electronics use DC power sources – such as batteries.

Alternating current describes the flow of charge that changes direction periodically. The voltage level also reverses along with the current. AC is used to power houses and buildings.

Some applications like TV's convert the AC that goes into the TV into DC for the electronic circuit.

Conventional current assumes that the current flows out of the positive terminal, through the circuit and into the negative terminal.

However what actually happens is Electron Flow, electrons flow out of the negative terminal, through the circuit and into the positive terminal of the source.

Resistors are an electrical component that restricts the flow of electrical charge.

These can be fixed value or variable.

These play a major role in reducing the current in circuits and protect circuits from damage, which can result from overdraw of current – by dissipating the kinetic energy of electrons in current as thermal energy.

Heat dissipation within a resistor is simply the power dissipated across that resistor since power represents energy per time put into a system.

Resistance variations exist due to length, cross sectional area, and material.

An ideal fixed resistor provides a constant resistance under all environments, however the resistance of the fixed resistors varies slightly with increase of temperature.

The cost of fixed resistors is high when compared to variable resistors, as each time resistance needs to be changed, a new resistor needs to be bought.

As it is impractical to have resistors of every single value possible, they are manufactured in "preferred values.

Ohm's Law:

The relationship between current, voltage and resistance.

Includes 3 main formulas:

V = IR

I = V/R

R = V/I

Short Circuit occurs when an electrical current though an unintended path with very little (or zero) resistance.

This usually happens when two points in a circuit that should not be directly connected come into contact – faulty wiring, damaged insulation, wrong connections, allowing current to bypass the intended load.

Often leads to excessive current flow, which can generate hear, damage components, pose safety risks, or in extreme cases, cause fires.

When resistors are in series, the current through them stays the same. The voltage across each resistor is different – this is known as resistor voltage drop, a calculation of the voltage value across a resistor. - KVL

These must add up to the voltage source = Vs = V1 + V2 + V3

With Ohm's Law --> (V = IR)

I Rs = I R1 + I R2 + I R3

Resistors in Parallel --> Rp

The voltage drop across each resistor is the same, but the current is different. When resistors are connected in parallel, the supply current is equal to the sum of the currents through each resistor.

The currents in the branches of a parallel circuit add up to the supply current.

A potential divider, aka Voltage Divider, is a simple circuit which takes advantage of the way voltages

drop across resistors in series.

The general idea is that by using two resistors in series, it is possible to divide a voltage and create a different voltage between them.

It is usually used to make a larger voltage into a smaller one.

The current running through each resistor is the same - as is in series.

This is an open circuit, there is no load on it yet, so there is 0A flowing out.

Therefore, I = V/(R1 + R2), as (I = V/R1) = (I = V/R2)

The general equation is :

V out = V in * R2/ (R1+R2)

Current Divider:

Are parallel circuits in which the source divides into several parallel paths.

In a parallel connected circuit, all the components have their terminals connected, sharing the same two end nodes.

Different paths and branches = different amounts of current flowing across.

The voltage drop across these parallel resistors is the same as that of the power supply. This is because the resistors have common potential points shared between them, so the voltage will be the same, but the current is different.

With the same voltage across each resistor, the greatest current will flow through the lowest value resistor.

Therefore, the currents through each resistor will be in inverse proportion to the ratio of the resistances.

As the source or total currents equals the sum of the individual branch currents, then the total current is flowing in the circuit is given by Kirchhoff's current law as being =

It = IR1 + IR2.

EMF:

All voltage sources have two fundamental parts – a source of electrical energy that has a characteristic electromotive force (EMF), and an internal resistance r.

The emf is the potential difference of a source when no current is flowing.

The numerical value of the emf depends on the source of potential difference.

The internal resistance r of a voltage source affects the output voltage when a current flows.

Terminal Voltage:

Is the voltage (potential difference) measured between the terminals (positive and negative terminals) of a battery.

When no current is flowing --> emf = terminal voltage.

When current is flowing --> emf > terminal voltage.

V = EMF – Ir

(I = electric current and is positive and r is internal resistance.)

(When multiple voltage sources are in series, their internal resistances add and their emfs add algebraically.)

Maximum Power Transfer:

A "load" in electrical terms is when we add a component such as a resistor to a circuit. We can represent the load with a resistor having resistance of Rl ohms.

The maximum power transfer theorem states that the DC voltage source will deliver maximum power to the variable load resistor only when the load resistance is equal to the source resistance.

Measurement Devices:

I, V and R can be measured by two types of instruments:

Instruments based on a traditional analogue moving coil galvanometer.

Instruments based on a digital voltmeter / multimeter.

Nowadays, digital instruments are more common as they are more precise, robust and easier to read.

A multimeter is a measuring device that can measure multiple electrical properties.

This includes – voltmeter – ohmmeter – ammeter

A multimeter has a display, selection dial and ports. You choose which measurement to record.

There are two ports – COM (common) and is connected to the ground or the – of a circuit. It is conventionally black.

The mAVΩ port allows the measurement of current up to 200mA, voltage V and resistance Ω. The 10A is a special port used for measuring large currents – greater than 200mA.

You can use a multimeter to check resistors. You can pick out a random resistor and set the multimeter to 20KΩ setting. Then you hold the probes against the resistor legs.

Here it can return 0.00 if the setting is too high, 1 or OL if it is overloaded.

They can also have discrepancies of 500Ω. Results are never perfect.

The ammeter can measure electric current, and must be connected in series. Adding an ammeter adds a resistance. This changes the current – but negligibly.

If an ammeter is accidently connected across a substantial voltage – it will blow it's fuse.

Fuses work by having an easily meltable wires – are usually 3, 5 or 13 A.

Voltmeters are added to circuits in parallel, around the part of the circuit you want to measure the voltage drop of. They also add resistance – and the P.D will change.

To measure resistance – you can measure V and I and calculate it.

Measuring resistance in a circuit can be inaccurate, and the circuit should not be on if you must measure it on the circuit.

Breadboards – are used for rapid prototyping of circuits – and operate with low voltages.

A Wheatstone is an alternative method of measuring resistance (when a resistance is unknown) and is a bridge that comprises two potential dividers.

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)

One consisting of two precisely known fixed resistors R1 and R2. The other consists of the unknown resistor Rx in series with accurately calibrated variable resistor Rs

You adjust Rs to find the unknown resistor by balancing two legs of the bridge circuit.

It is usually used with transducers – a device that converts energy from one form to another – to measure physical quantities like temperature, pressure and strain.

Wheatstone Bridges are used in application where small changes in resistance are to be measured in sensors. This is used to convert a change in resistance to a change in voltage of a transducer.

The variable resistor Rs denoted by R4 is adjusted until the voltmeter reads 0 volts. At this point the bridge is balanced.

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](data:image/jpeg;base64,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For a balanced condition, the voltage across R1 and R2 is the same – voltage across parallel resistors is the same. This also applies to the voltage across R3 and R4.

Rx = Rs x R2/R4

Kirchoff's Laws:

Are two simple laws reflecting the way current and voltage behave.

KCL – Action of currents at a point or junction.

The total current entering a circuit is exactly equal to the total current leaving the same junction.

KVL – Sources of EMF and voltage drops around a closed loop.

In a closed loop, the algebraic sum of the product of current and resistance in each part of the circuit is equal to the resultant emf in the loop.

The sum of the voltage drops around a closed loop is equal to the resultant emf.

E = I R1 + I R2 Volts

Branch Current Method

You can use loops to determine the voltage drops

1. Lable currents.

2. Solve as equation.

Multiple Power Sources:

When connected in series, voltage sources add algebraically, if two voltage sources of 5V and 10V are in series, in the same direction, the total is 15V.

If they have opposite polarities, subtract the smaller from the larger.

The same current flows through all components.

When connected in parallel, voltage sources must have the same voltage to avoid short circuits, if they are not equal, current may flow between them and cause excessive current draw.

The total current supplied is divided between the parallel sources based on their internal resistances.

Voltage source polarity affects the direction of current and voltage drops in the circuit. Incorrect polarity - especially in series - can lead to opposing voltages.

The BCM method assumes directions of currents in a network, requires Kirchhoff’s and Ohm’s laws. First, you choose a node - a junction.Then you make a guess which way the currents go into and out the junction, and label them 1, 2 and 3. After, you label the polarities with the resistors, positive where the current enters and negative where it exits.

As all voltage should = 0, we can use V = IR and follow round the current to make 3 simultaneous equations to solve:

I1 - I2 + I3 = 0, and two voltage equations.

If you get a negative current, it shows the assumed direction for a current was the opposite of it’s real direction. For multi-loop circuits: For a circuit with 3 EMfs:

Label the current and current direction - can just pick.

Use KCL for current equations for each junction.

Use KVL for loop equations to include each branch at least once. 1. Draw arrows for assumed current flow

2. Write KCL equation for junction a —> I1 = I2 + I3

3. Note polarity changes - current passes through - to +, so is a positive change.

4. Write the two loop equations:

+(E1) - (I1R1) - (I3R4) - (E3) - ((I3+I2)R2) = 0