Electricity

CIRCUIT THEORY 1

Fundamentals of Electricity

Topics Introduced:

• Elements:

  • Fundamental building blocks of matter (approximately 118 on the periodic table)

• Compounds:

  • Chemical combinations of two or more elements

• Molecules:

  • Smallest parts of a compound retaining original properties; e.g., H₂O₂ = H₂ + O₂

• Atoms:

  • Composed of three subatomic particles:

    • Protons: Positively charged

    • Neutrons: Uncharged

    • Electrons: Negatively charged, orbit the nucleus in shells; shell configuration influences chemical properties

Electrical Charge

• Atoms generally neutral; electric charge produced through gain/loss of electrons

• Ionization: Process leading to charged atoms (ions) - negatively charged when gaining electrons, positively charged otherwise

• Attraction and Repulsion: Charged objects exert non-contact forces based on their charge polarity

Electrical Current and SI Units

• Current (I):

  • The flow of charge measured in Amperes (A); defined as I = Q/t

• Electrical Charge (Q):

  • Measured in Coulombs (C); 1C = approximately 6.25 x 10^18 electrons

• Electrical Energy:

  • Measured in Joules (J); potential difference (V) as work done per unit charge

Resistance and Electromotive Force

• Resistance (R):

  • Opposition to current flow, measured in Ohms (Ω)

• Electromotive Force (EMF):

  • Characteristic of an energy source that drives current in a circuit; differentiated from voltage which measures energy usage

Electrical Power

• Power (P):

  • Rate of energy conversion, measured in Watts (W); calculated from P = E/t

• Relationship between voltage and current in devices:

  • Calculated through V = IR

CIRCUIT THEORY 2

Power and Energy Recap

• Definition: Power is the rate at which energy is transferred or converted.

  • Formula:

    • P = E/t

  • Units:

    • Power (P): Watts (W)

    • Energy (E): Joules (J)

    • Time (t): seconds (s)

SI Units and Common Prefixes

Electrical Power

• Power can also be calculated using current and voltage:

  • P = VI

• Ohm's Law relationship:

  • P = I²R or P = V²/R

• When current or voltage is unknown, use:

  • If current is unknown: P = V²/R

  • If voltage is unknown: P = I²R

Direct Current (DC) vs. Alternating Current (AC)

• DC:

  • Flows in one direction with consistent voltage levels.

  • Common applications: batteries, electronic devices.

• AC:

  • Voltage and current direction change periodically.

  • Typically used in power distribution (homes, offices).

  • Devices such as TVs convert AC to DC for operation.

Conventional Current

• Definition: Assumes current flows from positive to negative terminal of a power source.

• Note: Electron flow is in the opposite direction (from negative to positive).

Resistors

• Definition: Components that restrict electrical charge flow.

Types of Resistors

• Fixed Resistors: Constant resistance value.

• Variable Resistors: Resistance can be adjusted.

Functions of Resistors

• Reduce current and protect circuits from overcurrent by converting kinetic energy into thermal energy.

  • Power dissipated in resistor: P = I²R

Factors Affecting Resistance

• Variations due to length, cross-sectional area, and material type.

Practical Considerations

• Fixed resistors have slight resistance variation due to temperature changes.

• Common resistance values: 100Ω, 10kΩ, 100kΩ, etc.

• Cost: Fixed resistors are more expensive due to the need for individual units for varied resistance values.

Resistor Value Classification

• “Preferred values” are common resistance values with color-coded bands for identification.

  • Resistor colors indicate value and tolerance.

Resistor Tolerance

• Measure of variation from specified resistance—typically from 1% to 10%.

• Precision Resistors: Tolerances lower than 2% are more costly.

Ohm's Law

• Relationship between voltage (V), current (I), and resistance (R):

  • Formulas:

    • V = IR

    • I = V/R

    • R = V/I

• Developed by George Ohm based on experiments with current and electricity.

Short Circuit

• Definition: Occurs when current flows through an unintended path with low or no resistance, potentially damaging components and generating heat.

Circuit Theory Session Summary

• Topics Covered in Week 2:

  • Behavior of resistors in series and parallel, potential dividers, assignments, and reading materials.

• Weekly Questions: Available on Blackboard; solutions provided through support sessions.

Resistors in Series and Parallel

Resistors in Series

• The current remains the same; voltage drops across each resistor sum to the source voltage.

  • Formula:

    • R_total = R1 + R2 + R3

Resistors in Parallel

• The voltage across resistors is the same; supply current is the sum of branch currents.

  • Formula:

    • 1/R_total = 1/R1 + 1/R2 + 1/R3

Voltage and Current Dividers

Voltage Divider

• Uses resistors in series to divide voltage.

  • Formula:

    • V_out = V_in * (R2 / (R1 + R2))

Current Divider

• Splits supply current in parallel circuits.

  • For 2 resistors:

    • I1 = (R2 / (R1 + R2)) * I_total

    • I2 = (R1 / (R1 + R2)) * I_total

Conclusion

• Understanding concepts in circuit theory is essential for handling electrical systems, with practical applications in designing and analyzing circuits.

CIRCUIT THEORY 3

EMF (Electromotive Force)

Definition: Fundamental part of a voltage source that represents its potential difference when no current flows.

Characteristics:

• Depends on the source of potential difference.

• Influenced by internal resistance (r), which affects output voltage under load. Higher internal resistance typically results in a lower terminal voltage under load conditions

Terminal Voltage

Definition: Voltage measured between battery terminals.Conditions:

• No current: EMF = Terminal Voltage

• With current: EMF > Terminal Voltage, indicating that some voltage is lost due to internal resistance during current flow.Formula:

• V = EMF - I × r

  Where I is the current (positive when flowing away from the positive terminal)

  and r is internal resistance.

Series Connection: In series, both EMFs and internal resistances of voltage sources add, leading to an overall increase in EMF while total resistance increases as well. This is important for calculating the total voltage and current in a series circuit.

Maximum Power Transfer

Definition: In electricity, the power delivered to a load is maximized when load resistance (R_L) equals the source resistance.

Application: Represents how circuit loads can be optimized for efficiency. This principle is crucial in designing audio systems, RF systems, and communication circuits to ensure that maximum power is delivered to the intended load.

Measurement Devices

Types of Instruments:

• Traditional Analogue Moving Coil Galvanometer

• Digital Voltmeter (DVM) / Multimeter (MM)

Advantages of Digital Instruments:

• Higher precision and accuracy

• Robustness, less susceptibility to damage

• Easier to read with clear numerical displays

Components of a MultimeterParts:

• Voltmeter: Measures voltage

• Ohmmeter: Measures resistance

• Ammeter: Measures current

  • Display: Typically 3-4 digits with a negative sign indication for polarity.Selection Dial: Chooses measurement type.

  • Ports:

    • COM: Common, connected to circuit ground (-)

    • mAVΩ: Measures current (up to 200mA), voltage, and resistance

    • 10A Port: For high current measurements (>200mA) to prevent damage to the multimeter.

Measuring Voltage with a Voltmeter

Connection: Must connect in parallel across the point of interest.Effect: Connection introduces voltmeter resistance, changing potential difference.Ideal Property for Voltmeter: Should ideally have infinite resistance to avoid affecting the circuit, ensuring that it accurately measures the potential difference without drawing current.

Measuring Current with an Ammeter

Connection: Connected in series, changing circuit current due to added resistance of the meter. It is crucial to ensure that the meter is within its rated current to prevent damage.Fuses: Protect against excessive current by blowing and stopping measurement until replaced, which is an important safety feature.

Wheatstone Bridge

Use: Measures unknown resistance using two known resistors and a variable resistor for balance.

Balance

Condition: Occurs when the voltage reading across the bridge is zero; enables calculation of unknown resistance based on the known values and the properties of the bridge circuit. This method is highly accurate for resistance measurement.

Kirchhoff’s Laws

• Current Law (KCL): Total current entering a junction equals the total current leaving.

• Voltage Law (KVL): In closed loops, the sum of voltage drops equals the source EMF. These laws are fundamental for circuit analysis and design, ensuring conservation of charge and energy in electrical circuits.

CIRCUIT THEORY 4

Multiple Power Sources

Series Connection:

• If voltage sources are in series, voltages add algebraically considering polarity.

  • Example: 5V + 10V in the same direction = 15V.

  • If opposite polarities, subtract smaller from larger (e.g., 5V - 10V = -5V, indicating reversed polarity).

• Same current flows through all components, maintaining constant flow despite individual voltage contributions.

Parallel Connection:

• Voltage sources must have the same voltage levels to avoid short circuits or damage to components (e.g., two 12V batteries in parallel).

• Total current is divided between parallel sources based on their internal resistances, adhering to ( I_{total} = I_1 + I_2 + ... )

• Polarity of voltage sources affects current direction and voltage drops, impacting the behavior of connected loads.

Branch Current Method (BCM)

• Assumes directions of currents in a circuit network and requires writing equations relating them using Kirchhoff's and Ohm’s Laws.

• Essential for breaking down complex circuits into manageable calculations.

Diagramming Current Directions

• Properly diagramming directions of travel in circuits enhances clarity in analysis.

BCM Step 1

• Choose a node (wire junction) as a reference point for unknown currents, establishing a foundation for applying KCL.

BCM Step 2

• Make an educated guess on the direction of wires' currents (label as I1, I2, I3).

• Directions are speculative; adjustments may be needed based on results.

BCM Step 3

• Label voltage drop polarities across resistors according to assumed current directions.

  • Positive where current enters, negative where it exits, ensuring all polarities are consistently noted for subsequent calculations.

BCM Step 4: Kirchhoff’s Voltage Law (KVL)

• States that the algebraic sum of all voltages in a loop equals zero.

  • Example for Loop 1: -28 + VR2 + VR1 = 0.

• Applying Ohm’s Law: -28 + 2I2 + 4I1 = 0 helps in solving for unknown currents.

Loop Equations

• For Loop 2: -VR3 + 7 - VR2 = 0

• Apply Ohm’s Law: -1I3 + 7 - 2I2 = 0, creating a system of equations for further analysis.

System of Equations

• Mathematical system with:

  • KCL: I2 = I1 + I3

  • I1 - I2 + I3 = 0

  • KVL Equations:

    • -1I3 + 7 - 2I2 = 0

    • -28 + 2I2 + 4I1 = 0

Understanding Negative Current

• “Negative” current indicates the real direction of the current is opposite to the assumed direction, aiding in clarifying current flow in complex networks.

Multi-Loop Circuits

• Circuit with three EMFs demonstrated for advanced understanding.

• Label current directions and apply KCL for current equations to reinforce analytical skills.

• Use KVL for loop equations ensuring that every branch is accounted for accurately during analysis.

Example Circuit Analysis

• Given: E1 = 19V, E2 = 6V, E3 = 2V, with their respective resistances.

• Illustrate assumed current flows with arrows to visualize circuit behavior.

KCL Application

• Use KCL at junction (a): I1 = I2 + I3.

• Different equations derived when applying KCL at nodes highlight versatility in circuit analysis.

Polarity Changes

• Current transitions through batteries affect potential changes directly correlating with battery EMF, necessitating accurate notation of battery connections to prevent confusion in analysis.

Writing Loop Equations

• Loop 1: (E1) - (I1R1) - (I3R4) - (E3) - (I1R2) = 0

• Loop 2: (I2R3) - (E2) + (I3R4) + (E3) = 0, ensuring all components are represented.

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Detailed Loop Analysis for Loop 1

• Expand and simplify the loop equation to isolate unknowns:

  • 19V - 6I1 - I3 - 2 - 4I1 = 0

  • Rearranged to find values of currents effectively.

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Detailed Loop Analysis for Loop 2

• Expand Loop 2 equation and solve:

  • 4 + 4I2 - I3 = 0

  • Rearranging facilitates the identification of the relationship between I2 and I3.

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Simultaneous Equations Formation

• Rearranged relationships between currents derive a system of simultaneous equations:

  • I1 = I2 + I3

  • 11I1 - I2 = 17

  • This ultimately leads to the resolution of I1, I2, and I3.

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Solution: Finding I1

• Solve simultaneous equations to find:

  • I1 = 1.5 A

  • Clear verification process enables accurate results.

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Finding I2 and I3

• Substitute I1 into related equations to ascertain:

  • I2 = -0.5 A (indicates opposite direction, hence the importance of checking assumptions)

  • I3 is calculated from values obtained, demonstrating interdependencies in the circuit.

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Voltage Labeling Method

• To verify calculations, systematically label voltages using a chosen reference point (usually set to 0V) ensuring clarity in directionality and polarity.

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Practice Questions Overview

• Details on ideal voltmeter and ammeter characteristics crucial in circuit testing.

• Review Wheatstone Bridge equations relating to balanced conditions to develop understanding of practical circuit applications.

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Circuit Analysis using Kirchhoff's Laws

• Example with dual supplies and internal resistances offers insights into real-world applications.

• Multiple parts addressing current, power loss, and Thevenin’s theorem derivations deepen conceptual comprehension.

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Advanced Circuit Questioning

• Focus on KCL, KVL, loop equations, and Thevenin's equivalent circuits fosters critical thinking skills essential in advanced electronic design.

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Learning Outcomes

• Focus on understanding and applying the Superposition Theorem, reinforcing theoretical knowledge with practical application.

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Superposition Theorem Definition

• The theorem states that the response of a multi-source circuit is the sum of responses from each source independently, simplifying complex circuit analysis.

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Strategy of Elimination

• Eliminate all but one source to analyze circuits, applying KCL/KVL for voltage drops or currents independently, streamlining problem-solving.

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Circuit Analysis with Superposition

• Calculate current before splits utilizing the total resistance to find the current using Ohm's Law, enhancing accuracy in circuit behavior predictions.

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Voltage Drop Calculations

• Calculate the voltage drop across R1 and use KCL to find remaining current splits, ensuring that all aspects of the circuit are adequately addressed.

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KCL Application with Splitting Currents

• Ensure current relationship from earlier established equations, utilizing KCL to substantiate the flow of currents and voltage drops.

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Loop Calculations with Current Splits

• Apply KCL to derive necessary equations linking R2 and R3, which clarifies the interactions between parallel components.

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Finalizing Equations from Superposition

• Rearranging final equations to isolate current variables, providing clarity and focus in problem resolution.

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Parallel Resistance Identification

• Analysis of two parallel resistances contributing to voltage divider effects illustrates practical implications in circuit simplifications, enhancing design efficiency.

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Final Voltage Drops Confirmation

• Confirm voltage drops across resistors in the modified circuit setup, ensuring compliance with established theoretical predictions.

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Testing with Alternate Circuit Form

• Practical exercise to reinforce understanding of superposition enhances student engagement and integrates learning with real-world applications.

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Superimposing Results

• Combine voltage/current impacts to provide total circuit conditions, essential in overall circuit functionality evaluations.

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Detailed Voltage Encoding

• Care for polarity and direction when adding voltage drops algebraically informs the analysis, vital in accurate calculations.

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Summary of Calculation Results

• Resulting circuit after applying superimposed voltages and voltage drops confirmed via rigorous checks and balances, ensuring reliability of outcomes.

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Currents Superimposing

• Currents also combine algebraically similar to voltages, integrating resistances in the final evaluations to develop a sophisticated understanding of circuit behavior.

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Final Example Verification

• Review of outputs and voltages confirm circuit equilibrium, crucial in validating theoretical predictions against practical findings.

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Practical Problem-Solving

• Questions focusing on circuit analysis through Kirchhoff's Laws and the Superposition theorem facilitate honing analytical skills necessary for future coursework.

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Practical Application of Superposition

• Challenge to apply superposition principles across various circuit scenarios, enhancing understanding through applied methodologies.

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Reflections and Insights

• Expected competencies from the session include demonstrating both the Branch Current Method and the Superposition Theorem, enabling nuanced grasp of electronic circuit functionality.

CIRCUIT THEORY 5

The Capacitor

• Definition: A capacitor stores energy in an electric field.

• Functionality Comparison: Capacitors operate similarly to batteries, but they can charge and release energy much more quickly and typically have a shorter energy output duration.

• Uses:

  • Acts as a storage bank of electrical charge, smoothing out variations in voltage levels across circuit components.

  • Removes electrical noise, helping to stabilize circuit performance, particularly in LED circuits.

  • Applications include medical devices such as MRI machines, telecommunications filtering, audio oscillators, and radio receivers.

Types of Capacitors

• Capacitors vary greatly in size, from tiny components (like disc capacitors) to large high-voltage capacitors used in power transmission.

• Structure: Comprises two conductive plates separated by a dielectric (non-conducting) material.

• Types: Often categorized by the type of dielectric material used, including ceramic, electrolytic, tantalum, and film capacitors.

Capacitor Dielectrics

• Definition: Insulating materials that exhibit polarization when an electric field is applied.

• Function: When connected to battery terminals, positive and negative charges (+Q and -Q) separate across the plates, maintaining the electric field.

• Examples of Dielectrics: Common materials include air, ceramic, mica, and polyethylene.

Energy Storage in Capacitors

• Capacitors accumulate energy by keeping opposite charges separate, creating potential energy due to their mutual attraction, which is released when the circuit is closed.

• Design: A parallel-plate capacitor is characterized by its two conductive plates with a defined gap filled with a dielectric material, where equal and opposite charges are stored.

Electric Field in a Capacitor

• The electric field lines extend straight across the gap between the plates, directed from the positively charged plate to the negatively charged plate.

• When fully charged, the capacitor holds a charge Q that corresponds to the power supply's voltage.

Capacitor Discharge

• After disconnection from the power source, the stored charge can remain in the capacitor until it is connected to a circuit load (like a light bulb), enabling a rapid discharge of energy.

Maximum Voltage in Capacitors

• Limitations: Each capacitor has a specified maximum voltage (known as the voltage rating), and exceeding this can lead to failure or explosion.

• Safety Tip: Always check a capacitor's voltage rating before use and avoid working with live circuits to ensure safety.

Capacitance

• Definition: Capacitance is the ability of a capacitor to store charge, represented as C and measured in Farads (F).

• Relationship to Charge and Voltage: 1 Farad equals the storage of 1 coulomb of charge at 1 volt.

• Common subunits include Microfarads (μF) and Picofarads (pF).

Charge and Voltage Relationship

• The charge (Q) stored in a capacitor is directly proportional to the voltage (V) across it.

• Formula: C = Q/V or Q = CV.

• Examples:

  • To find the voltage across a 4μF capacitor charged with 5mC:

    • V = Q/C = 5mC / 4μF = 1.25 kV.

  • For a 50pF capacitor at 2kV:

    • Q = CV = 50pF x 2000 = 0.1 μC.

Energy Stored in a Capacitor

• Formulas: The energy stored can be calculated as follows:

  • E = 1/2 QV

  • E = 1/2 CV²

  • E = 1/2 Q²/C

• The stored energy represents electric potential energy accrued during the charging process, often averaging at V/2.

Practical Uses of Capacitors

• In devices like defibrillators, capacitors deliver rapid charges to restore heart rhythms.

• Example: A defibrillator that discharges 400 J at 10,000 V requires precise calculations to determine capacitance for effective treatment.

Capacitor Safety Practices

• Treat all capacitors as if they are always charged, even after disconnection.

• Precautions:

  • Never touch both leads of a capacitor simultaneously.

  • Ensure capacitors are discharged automatically through a resistor when the power supply is switched off to prevent accidental electric shock.

Capacitors in Series

• Configuration: Capacitors in series share the same current but can store differing voltages across them.

• Charge Equality: Each capacitor stores the same charge Q in this arrangement.

• Voltage Drop: The total voltage is divided among the capacitors according to capacitance values, with larger capacitors having a lower voltage drop.

Capacitors in Parallel

• Configuration: All capacitors in parallel retain the same voltage supplied across them.

• Total Capacitance Calculation: The total capacitance is the sum of the individual capacitances: C_T = C1 + C2 + C3.

Exam Review Questions

• Example questions could involve calculations for total charge or capacitance under different circuit configurations, reinforcing theoretical knowledge and practical application of the concepts learned.

CIRCUIT THEORY 6

Capacitors in Parallel - Recap

• For the circuit shown:

  • a) Calculate total capacitance using the formula:

    • ( C_Total = C_1 + C_2 + C_3 + ... )

  • b) Determine charge on each capacitor, considering ( Q = CV ) for each.

  • c) Find total charge held by the circuit as ( Q_T = Q_1 + Q_2 + Q_3 + ... )

  • d) Calculate total energy stored by the circuit using ( E_{total} = \frac{1}{2} C_{total} V^2 ).

Charging Process of Capacitors and Resistors in Series

• Charge on a capacitor cannot change instantaneously due to its ability to store energy.

• Current relationship:( I = \frac{dQ}{dt} )

  • As time approaches zero, any change in charge approaches zero.

• Steady-state current into a capacitor is zero which indicates charge builds up to a point where the voltage across the capacitor equals that of the voltage supply:( V_c = V ).

Step 1 Charging Process

When the switch (S) is closed:

• Apply Kirchhoff’s Voltage Law (KVL):

  • ( E = V_c + V_r )

    • Where ( E ) is the voltage from the supply, ( V_c ) is the voltage across the capacitor, and ( V_r ) is the voltage across the resistor.

Voltage Relationships in Charging

• The battery voltage (E) remains constant during the charging process.

• Capacitor voltage can be expressed as:( V_c = \frac{Q}{C} )

• The voltage drop across the resistor can be calculated:( V_r = IR )

• KVL yields:( E = \frac{Q}{C} + IR ).

Step 2 - Initial Condition

• At the moment the switch is closed:

• Initial charge is ( Q = 0 ) leading to ( V_c = 0 )

• Initial current through the circuit is given by:( I = \frac{E}{R} ).

Step 3 - After Initial Charge

• Short time after S is closed (say t1 seconds):

• Charge partly rises to ( Q_1 ) coulombs.

• Capacitor voltage is now:( V_{C1} = \frac{Q_1}{C} )

• Voltage drop across the resistor is now given by ( V_r = I_{t1}R ).

• Applying KVL at t1:( E = V_{C1} + V_r ).

Step 4 - Continued Charging

• The charge increases to ( Q_2 ) coulombs at time ( t2 ).

• Capacitor voltage becomes ( V_C = \frac{Q_2}{C} ).

• The voltage drop is given by:( V_r = I_{t2}R ).

• As time progresses, ( V_C ) continuously increases while current & voltage across R decrease due to the charging nature of capacitors.

Step 5 - Steady State

• After a few seconds, the capacitor becomes fully charged:

• Current reduces to zero ( ( I = 0 ) ), while the capacitor maintains voltage PMAX, equal to the supply voltage:( V_C = E ).

• Steady-state established: KVL becomes effective in the form ( E = \frac{Q}{C} + IR ).

Summary of Charging Process Steps

• Key equations identified:

  • Initial condition: ( V_C = 0 ) + contributions from R.

  • Continuous rise in capacitor voltage ( ( V_C ) ) while current ( ( I ) ) decreases.

  • Steady state occurs when ( V_C = E ) and current ceases.

Voltage-Time Curve for Capacitors

• Charging follows an exponential growth curve for voltage over time after initiation:

• Current and voltage across R follow an exponential decay curve, demonstrating the relationship between voltage, charge, and time.

Time Constant in C-R Circuits

• With constant DC voltage, the time constant is denoted ( ( \tau = RC ) ) measured in seconds.

• Example calculation: ( R = 1000 \Omega ) and ( C = 10 \mu F ) gives a time constant of ( \tau = 0.0099s ).

Transient Behavior of C-R Circuits

• Voltage and current response in capacitors exhibit exponential changes:

  • Voltage across capacitor: ( V_C = E(1 - e^{-t/\tau}) ).

  • Voltage drop across resistor: ( V_R = V e^{-t/\tau} ).

  • Current: ( I = I_0 e^{-t/\tau} ) where ( I_0 ) is the initial current.

  • Relationship for resistor current: ( I = \frac{E - V_C}{R} ).

Example Calculation

• For a 20μF capacitor connected to a 50K resistor with a 20V DC supply:

  • Find the initial current, time constant, current after 1s, voltage across the capacitor after 2s, and time required to achieve 15V across the capacitor.

Further Example Calculation

• Consider a 100K Ohm resistor with a 5μF capacitor, and a 50V DC supply:

  • Tasks: Calculate capacitor voltage after 1s, current after 2s, and time to reach a voltage of 35V.

Formula Booklet Summarized Equations

• Capacitor relationships are as follows:

  • Voltage across the capacitor: ( V_C = \frac{Q}{C} )

  • Energy stored in the capacitor: ( E_{cap} = \frac{1}{2} C V^2 )

  • Total charge in the system: ( Q_T = C_pV )

    • Where ( C_p = C_1 + C_2 + C_3 ) for capacitors in parallel.

Charging and Discharging Formulas

• Charging formula:

  • ( V_C(t) = E(1 - e^{-t/\tau}) )( I(t) = I_0 e^{-t/\tau} )

• Discharging formula:

  • ( V_C(t) = V_0 e^{-t/\tau} )( I(t) = -\frac{V_C}{R} )

Charging Process Explained

• The current flow caused by electrons in the capacitor leads to voltage balance between the capacitor and the resistor, illustrating the relationship:( V_R = I R ).

Voltage and Current Charging Curve

• Graphical representations of capacitor voltage and charging current against time demonstrate expected behavior during charging.

Fully Charged Capacitor Explanation

• After 5 time constants, the voltage across the capacitor nearly equals the applied voltage (E).

• Voltage reaches approximately 63.2% of the voltage difference from its initial value in each time constant period.

Discharging Process

• Upon switching to position B, the capacitor discharges through a resistor, initially producing maximum current flow in the opposite direction with associated exponential voltage decay.

Discharging Current Graph

• Illustrative representations of discharging voltage and current over time reveal the expected behavior during the discharge phase.

Discharging Equations

• During discharge, the capacitor voltage decreases according to:( V_C(t) = V_0 e^{-t/\tau} )

  • This assumes full discharge occurs approximately after 5 time constants.

Discharging Calculations Example

• For a 100V charged capacitor discharging through a 50K resistor, calculate:

  • Capacitor value, time to reach 20V, current after 0.5s, and resistor voltage after 1s.

Practical Application Example

• For a 20mF capacitor charged to 120V:

  • Calculate time constants and expected voltages after 10 seconds.

Past Paper Question 3 Summary

• Analyze circuit charging and discharging under given specifications.

  • Investigate maximum voltage during charge and potential reasons for differing discharging durations.

Past Paper Question Insight

• Detailed solution outlines provided insights on time constants and specific voltage changes during charging and discharging phases.

Past Paper 2021-22

• Series connection involving specific criteria to evaluate current flow and time constants.

CIRCUIT THEORY 7

Inductors

Definition and Function

An inductor is a passive circuit element that resists changes in current. Acting as a magnetic energy storage device, it plays a crucial role in the management of electrical energy in circuits. Inductors are commonly referred to as coils or chokes, indicated in circuit diagrams by the letter ‘L’.

Behavior of Inductors

Inductors work by storing kinetic energy from moving electrons in a magnetic field. Unlike resistors, which dissipate energy as heat, inductors conserve energy. Typically, they are made of insulated wire coiled around a core to amplify the magnetic field's strength. Their operation adheres to Faraday’s law of induction, which describes how a change in magnetic field can induce voltage, and Lenz’s law, which dictates the direction of induced voltage based on the change in current.

Voltage Characteristics

• Increasing Current: When current flows through an inductor, it generates a back electromotive force (EMF) opposing the current's direction, a phenomenon referred to as charging. This back EMF prevents the current from instantaneously reaching its maximum value.

• Decreasing Current: Conversely, if the current decreases, the inductor generates a voltage that aids the current flow, effectively acting as a power source, a process termed discharging. The energy stored in the magnetic field is released back into the circuit.

Inductance

Definition

Inductance measures the induced EMF in response to a change in current. Self-inductance occurs when the induced EMF is within the same circuit.

Measurement

Inductance is measured in Henrys (H), where 1 Henry induces a voltage of 1 Volt when the current changes at a rate of 1 Ampere per second. Typical inductance values range from 1 µH (microhenry, 10^-6 H) to 20 H (Henrys).

Energy Stored in Inductors

The energy stored in an inductor can be calculated using the equation:E_L = (1/2) L I²For instance, an 8H inductor carrying a current of 3A stores energy as follows:E = 1/2 * 8 * 3² = 36 Joules.

Factors Affecting Inductance

• Number of Turns: The more turns in the wire coil, the greater the inductance due to increased magnetic field strength.

• Cross-sectional Area: A larger cross-sectional area of the coil enhances the inductance by allowing more magnetic field lines to pass through.

• Magnetic Core: Utilizing a magnetic core, like iron, concentrates the magnetic field, thereby increasing inductance significantly.

• Coil Arrangement: The geometrical arrangement and physical dimensions of the coil play critical roles in determining inductance. A short, thick coil possesses higher inductance than a long, thin one.

Applications of Inductors

Inductors are employed for various purposes, including:

• Filtering and smoothing high-frequency noise in electric circuits

• Storing and transferring energy in power converters (DC-AC or AC-DC)

• Creating tuned oscillators or LC circuits for frequency modulation

• Matching impedances in RF applications to maximize power transfer

Combining Inductors

Inductors can be connected in series or parallel:

• In Series: The total inductance (Ls) is the sum of individual inductances: Ls = L1 + L2 + L3…

• In Parallel: The total inductance (Lp) is calculated using the reciprocal formula: Lp = 1/(1/L1 + 1/L2 + 1/L3)Care should be taken to shield inductors during calculations to account for interactions between their magnetic fields.

Mutual Inductance

Mutual inductance occurs when two inductors are placed close to each other, allowing one to induce EMF in the other due to the shared magnetic field generated by their respective current flow.

Transients in L-R Circuits

Transients occur when the circuit undergoes a change, such as switching from one DC configuration to another.

Current Growth in L-R Circuit

When the circuit switch is closed, the initial current begins to increase towards its final value, determined by Kirchhoff's Voltage Law (KVL). The inductor actively counters this change through back EMF, leading to a gradual rise in current.

Current Decay in L-R Circuit

When the switch is opened, current rapidly decreases, leading to a collapsing magnetic field that induces a back EMF, which momentarily continues to drive current. The voltage across both the resistor and the inductor decays exponentially to zero over time.

Time Constant for Inductors

The time constant (τ) is critical in defining how quickly current reaches its steady state after a switch is closed. τ is calculated as the ratio of inductance (L) to resistance (R) in the circuit and is often approximated to five time constants for the current to stabilize fully.

Thevenin's Theorem

Thevenin's theorem is used to simplify complex circuit analysis by transforming a complicated circuit into a single equivalent voltage source in series with a resistance. This approach facilitates easier calculations, particularly in power or battery systems.

Steps to Use Thevenin's Theorem

1. Remove the load resistance from the circuit.

2. Replace EMF sources with their internal resistances and calculate the Thevenin resistance (Rth).

3. Find the open-circuit voltage (Vth) at the terminals.

4. Draw the equivalent circuit and calculate other currents/voltages easily from there.

Practice Problems

1. Calculate the resistance of a coil with an inductance of 6H connected in series with a 10 Ohm resistor to a 120 V DC supply.

2. Determine the current flow immediately after a short circuit occurs.

3. Find the time required for the current to fall to 10% of its initial value after opening the switch.