Comprehensive Study Guide for Electrical and Transient Circuits

Introduction to Electrical Circuits

• Definition of an Electrical Circuit: An electrical circuit is an interconnection of electrical components or a model of such an interconnection, consisting of specific electrical elements.

• Core Electrical Elements: The primary components involved in these interconnections include:

  • Resistors ($R$)

  • Capacitors ($C$)

  • Inductors ($L$)

Types of Electric Current

• Direct Current (DC):

  • Defined as current that does not change in time but remains constant over its duration.

  • Represented graphically as a flat horizontal line (figure a in the transcript).

  • At Direct Current, the frequency is zero ($f = 0\,\text{Hz}$).

• Alternating Current (AC):

  • Defined as a time-varying current (figure b in the transcript).

  • Characterized by frequency ($f$), which is measured in the unit of Hertz ($\text{Hz}$).

Types of Flow Notation

• Conventional Flow: This notation assumes electric charge moves from the positive side of the battery (or power source) toward the negative side.

• Electron Flow: This notation follows the actual physical movement of electrons, which move from the negative side of the battery toward the positive side.

Fundamental Concepts of Resistance

• Definition: The flow of electric current is subject to friction. This friction or opposition to the flow of current is defined as Resistance ($R$).

• Physical Factors Determining Resistance: The resistance of an electrical conductor depends on four distinct factors:

  • (a) The length of the conductor ($L$).

  • (b) The cross-sectional area of the conductor ($A$).

  • (c) The type of material (represented by resistivity, $\rho$).

  • (d) The temperature of the material.

• Mathematical Formula for Resistance:     \n    R = \rho \frac{L}{A}\n         Where:

  • $R$ is the resistance in Ohms ($\Omega$).

  • $L$ is the length in meters ($m$).

  • $A$ is the cross-sectional area in square meters ($m^2$).

  • $\rho$ is the resistivity in Ohm-meters ($\Omega \cdot m$).

Resistor Coding Systems

Color Coding Table

• Standard resistors use color bands to indicate value, multiplier, and tolerance.

• General Rules:

  • For a 4-band resistor: 1st band is the 1st digit, 2nd band is the 2nd digit, 3rd band is the multiplier, and 4th band is the tolerance.

  • Special Note: If there are 5 colors, the first three bands represent digits (denoted as "if 5 colors 1st na tatlo ay digits"). This is noted as likely appearing in ECT examinations.

BS 1852 Letter Coding for Resistors

• This system uses the code "mRn" where m and n are positive integers.

• Common Examples:

  • $0.47\,\Omega$ = $R47$ or $0R47$

  • $1.00\,\Omega$ = $1R0$

  • $4.70\,\Omega$ = $4R7$

  • $47.0\,\Omega$ = $47R$

  • $470\,\Omega$ = $470R$ or $0K47$

  • $1.0\,\text{k}\Omega$ = $1K0$

  • $4.7\,\text{k}\Omega$ = $4K7$

  • $47\,\text{k}\Omega$ = $47K$

  • $470\,\text{k}\Omega$ = $470K$ or $0M47$

  • $1\,\text{M}\Omega$ = $1M0$

  • $6.7\,\text{k}\Omega$ = $6k7$

  • $6.7\,\text{M}\Omega$ = $6M7$

Resistor Temperature Coefficient

• The resistance at a final temperature can be calculated using the initial resistance and the temperature change:     \n    R_2 = R_1 (1 + \alpha (T_2 - T_1))\n         Where:

  • $R_2$ = final resistance

  • $R_1$ = initial resistance

  • $T_2$ = final temperature

  • $T_1$ = initial temperature

  • $\alpha$ = temperature coefficient (alpha reference)

Common Types of Resistors

1. Carbon Composition: Made out of carbon material.

2. Wirewound: Made out of wires; typically used for high power applications. (Classified as Fixed/Variable).

3. Potentiometer: A variable resistor featuring 3 terminals.

4. Rheostat: A variable resistor featuring 2 terminals.

Network Theorems and Circuit Properties

• Parallel Circuit Rules:

  • In a parallel circuit with 3 branches, the total resistance will be lower than the value of the smallest resistor in the branches.

  • If 3 resistors of the same value are in parallel, divide that common value by 3 to find the equivalent resistance.

• Millman's Theorem: Also known as the parallel generator theorem. It states that any number of parallel voltage sources can be reduced to a single equivalent voltage source.

  • \n        R_{eq} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}}\n        

• Maximum Power Transfer: A load will receive maximum power from a linear bilateral DC network when its total resistive value is exactly equal to the Thvenin resistance ($R_{Th}$) of the network as "seen" by the load.

  • Condition: $R_L = R_{Th}$

  • Efficiency at maximum power transfer is $50\%$.

• Reciprocity Theorem: The current at one point in a circuit due to a voltage at a second point is the same as the current at the second point due to the same voltage at the first point.

• Tellegen's Theorem: The sum of the instantaneous powers in all branches of any network is zero at any given time ($\sum P = \sum V \cdot I = 0$).

  • This implies total power supplied by sources equals total power absorbed by loads.

Transient Circuits Analysis

• Definition: The study of terminal characteristics (current, potential drop, power, energy) across various load parameters when energized by a DC or AC source through the activation of a switch.

• Response of L and C to a DC Source (DISCO):

  • At $t = 0$ (Transient State):

    • Inductor ($L$) acts as an OPEN circuit.

    • Capacitor ($C$) acts as a SHORT circuit.

  • At $t \rightarrow \infty$ (Steady State):

    • Inductor ($L$) acts as a SHORT circuit.

    • Capacitor ($C$) acts as an OPEN circuit.

• Time Constant ($\tau$): Defined as the time taken for a transient to reach its final state if the initial rate of change is maintained. In control systems, it is the time required to reach $63\%$ of the final value.

  • For RC Circuits: $\tau = RC$

  • For RL Circuits: $\tau = \frac{L}{R}$

Mathematical Curves in Transients

• Exponentially Rising Curve:     \n    y = y_0(1 - e^{-\frac{t}{\tau}})\n    

  • Examples include voltage or current charging/storing.

• Exponentially Falling Curve:     \n    y = y_0(e^{-\frac{t}{\tau}})\n    

  • Examples include voltage or current decay/discharging.

RL and RC Specific Transient Formulas

RL Transient Circuit

• Storage Cycle (Position 1):

  • Current: $i_L = \frac{E}{R} (1 - e^{-\frac{Rt}{L}})$

  • Resistor Voltage: $V_R = E (1 - e^{-\frac{Rt}{L}})$

  • Inductor Voltage: $V_L = E e^{-\frac{Rt}{L}}$

• Decay Cycle (Position 2):

  • Current: $i_L = \frac{E}{R_T} (e^{-\frac{R_T t}{L}})$ where $R_T = R_1 + R_2$.

RC Transient Circuit

• Charging Phase (Position 1):

  • Charge ($q$): $q = EC + (q_0 - EC)e^{-\frac{t}{RC}}$. If starting at zero, $q = EC(1 - e^{-\frac{t}{RC}})$.

  • Current: $i = \frac{E}{R} e^{-\frac{t}{RC}}$

  • Capacitor Voltage: $V_C = E (1 - e^{-\frac{t}{RC}})$

  • Resistor Voltage: $V_R = E e^{-\frac{t}{RC}}$

• Discharging Phase (Position 2):

  • Discharge Voltage: $V_d = E e^{-\frac{t}{RC}}$

Summary: 3 Steps to Success in Transient Circuits

1. DISCO at infinity: Determine the steady-state behavior of Inductors and Capacitors.

2. Determine Trend: Decide whether the unknown quantity is RISING or FALLING.

3. Apply Formulas: Use the specific equations for initial value ($y_0$) and time constant ($\tau$).