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Practice MCQ

Unit 1 - Notes

ECE131

Unit 1: Fundamentals of D.C. Circuits

1. Basic Circuit Variables and Elements

Voltage (Potential Difference)

Definition: The work done (energy required) to move a unit positive charge from one point to another in an electric field.
Symbol: $V$ or $v$
Unit: Volts ( $V$ ) or Joules/Coulomb ( $J/C$ ).
Formula: $V = \frac{dW}{dq}$

Current

Definition: The rate of flow of electric charge through a conductor's cross-sectional area.
Symbol: $I$ or $i$
Unit: Amperes ( $A$ ) or Coulombs/second ( $C/s$ ).
Formula: $I = \frac{dq}{dt}$
Direction:
- Conventional Current: Flows from positive to negative potential (direction of positive charge flow).
- Electron Flow: Flows from negative to positive potential (actual flow of electrons).

Power and Energy

Power ( $P$ ): The time rate of expending or absorbing energy.
- Formula: $P = \frac{dW}{dt} = \frac{dW}{dq} \cdot \frac{dq}{dt} = V \cdot I$
- Unit: Watts ( $W$ ).
- Sign Convention: If current enters the positive terminal of an element, power is absorbed (Load). If current leaves the positive terminal, power is delivered (Source).
Energy ( $W$ or $E$ ): The capacity to do work.
- Formula: $W = \int P \, dt$
- Unit: Joules ( $J$ ) or Kilowatt-hour ( $kWh$ ).
- Relation: $1 \, kWh = 3.6 \times 10^6 \, J$ .

Passive Circuit Parameters

1. Resistance ( $R$ )

Definition: The property of a material that opposes the flow of current.
Unit: Ohm ( $\Omega$ ).
Physical Formula:
- $\rho$ : Resistivity ( $\Omega \cdot m$ )
- $l$ : Length ( $m$ )
- $A$ : Cross-sectional area ( $m^2$ )
Conductance ( $G$ ): Reciprocal of resistance ( $G = 1/R$ ). Unit: Siemens ( $S$ ) or Mho ( $\mho$ ).

2. Inductance ( $L$ )

Definition: The property of a circuit element to store energy in the form of a magnetic field. It opposes changes in current.
Unit: Henry ( $H$ ).
Voltage-Current Relation: $v = L \frac{di}{dt}$
DC Behavior: In a steady DC circuit (where current is constant), $\frac{di}{dt} = 0$ , so voltage across an inductor is 0. It acts as a Short Circuit.
Energy Stored: $E = \frac{1}{2} L I^2$

3. Capacitance ( $C$ )

Definition: The property of a circuit element to store energy in the form of an electric field. It opposes changes in voltage.
Unit: Farad ( $F$ ).
Current-Voltage Relation: $i = C \frac{dv}{dt}$
DC Behavior: In a steady DC circuit (voltage is constant), $\frac{dv}{dt} = 0$ , so current through a capacitor is 0. It acts as an Open Circuit.
Energy Stored: $E = \frac{1}{2} C V^2$

2. Fundamental Laws

Ohm’s Law

Statement: The current flowing through a metallic conductor is directly proportional to the potential difference across its ends, provided physical conditions (temperature, strain, etc.) remain constant.
Formula: $V = I \cdot R$
Limitations: Not applicable to non-linear devices (diodes, transistors) or electrolytes.

Kirchhoff’s Laws

1. Kirchhoff’s Current Law (KCL)

Principle: Law of Conservation of Charge.
Statement: The algebraic sum of currents meeting at a junction (node) is zero.
Mathematically: $\sum I_{in} = \sum I_{out}$ or $\sum I = 0$ .

2. Kirchhoff’s Voltage Law (KVL)

Principle: Law of Conservation of Energy.
Statement: The algebraic sum of all voltages (voltage drops and voltage rises) around a closed loop or mesh is zero.
Mathematically: $\sum V_{rises} = \sum V_{drops}$ or $\sum V = 0$ .
Sign Convention (Standard):
- Moving from $-$ to $+$ is a Rise ( $+V$ ).
- Moving from $+$ to $-$ is a Drop ( $-V$ ).

3. Circuit Sources

Independent Sources

Does not depend on any other voltage or current in the circuit.
Ideal Voltage Source: Maintains constant terminal voltage regardless of load current. Internal resistance $R_{int} = 0$ .
Ideal Current Source: Maintains constant output current regardless of load voltage. Internal resistance $R_{int} = \infty$ .

Dependent (Controlled) Sources

Magnitude depends on a voltage or current elsewhere in the circuit. Denoted by a diamond shape.
1. Voltage Controlled Voltage Source (VCVS): $V = \mu V_x$
2. Current Controlled Voltage Source (CCVS): $V = r I_x$
3. Voltage Controlled Current Source (VCCS): $I = g V_x$
4. Current Controlled Current Source (CCCS): $I = \beta I_x$

4. Intuitive Methods of Circuit Analysis

Series Circuits

Components are connected end-to-end.
Current: Same through all components ( $I_{total} = I_1 = I_2$ ).
Voltage: Adds up ( $V_{total} = V_1 + V_2$ ).
Equivalent Resistance: $R_{eq} = R_1 + R_2 + \dots + R_n$

Parallel Circuits

Components are connected across the same two nodes.
Voltage: Same across all components ( $V_{total} = V_1 = V_2$ ).
Current: Adds up ( $I_{total} = I_1 + I_2$ ).
Equivalent Resistance:
- Special case for two resistors: $R_{eq} = \frac{R_1 R_2}{R_1 + R_2}$

Voltage Division Rule (VDR)

Used for series circuits to find voltage across a specific resistor without calculating current first.
Voltage across $R_x$ :
$V_x = V_{total} \times \frac{R_x}{R_{total}}$

Current Division Rule (CDR)

Used for parallel circuits to find current through a specific branch.
General Formula: Current through branch $x$ :
$I_x = I_{total} \times \frac{R_{equivalent}}{R_x}$
Two Resistor Specific Formula: Current through $R_1$ :
$I_1 = I_{total} \times \frac{R_2}{R_1 + R_2}$

Star-Delta ( $\text{Y}-\Delta$ ) Transformation

Used to simplify bridge circuits or networks that are neither series nor parallel.

Delta ( $\Delta$ or $\pi$ ) to Star ( $\text{Y}$ or $\text{T}$ )

To find resistance $R_A$ in the Star network (connected to node A):

R_A = \frac{R_{AB} \cdot R_{AC}}{R_{AB} + R_{BC} + R_{AC}}

Mnemonic: Product of adjacent arms divided by sum of all arms.

Star ( $\text{Y}$ or $\text{T}$ ) to Delta ( $\Delta$ or $\pi$ )

To find resistance $R_{AB}$ in the Delta network (between nodes A and B):

R_{AB} = \frac{R_A R_B + R_B R_C + R_C R_A}{R_C}

Mnemonic: Sum of products of all pairs divided by the opposite arm.

5. Systematic Circuit Analysis

Mesh Analysis

Based on: KVL.
Applicability: Only for Planar circuits (circuits that can be drawn without crossing lines).
Steps:
1. Identify meshes (loops) and assign mesh currents (usually clockwise: $I_1, I_2$ ).
2. Apply KVL to each mesh to generate linear equations.
3. Solve the simultaneous equations to find mesh currents.
Supermesh: Occurs when a current source (independent or dependent) is shared between two meshes. Combine the meshes and write a constraint equation for the current source.

Nodal Analysis

Based on: KCL.
Applicability: Planar and Non-planar circuits.
Steps:
1. Identify principal nodes. Select one as the Reference/Ground Node ( $V=0$ ).
2. Assign voltages ( $V_1, V_2$ ) to remaining nodes relative to ground.
3. Apply KCL at each non-reference node (Sum of currents leaving = 0).
4. Solve simultaneous equations for node voltages.
Supernode: Occurs when a voltage source is connected between two non-reference nodes. Treat the two nodes as one generalized node for KCL and write a constraint equation ( $V_1 - V_2 = V_{source}$ ).

6. Network Theorems

Superposition Theorem

Statement: In a linear, bilateral network containing multiple sources, the response (current or voltage) in any element is the algebraic sum of the responses caused by each source acting alone.
Procedure:
1. Activate one source at a time.
2. Deactivate other sources:
  - Voltage Sources $\rightarrow$ Short Circuit ( $0V$ ).
  - Current Sources $\rightarrow$ Open Circuit ( $0A$ ).
3. Calculate the response for the active source.
4. Sum individual responses.
Note: Cannot be used to calculate Power directly (because $P=I^2R$ is non-linear).

Thevenin’s Theorem

Statement: Any linear, bilateral, two-terminal network can be replaced by an equivalent circuit consisting of a single voltage source ( $V_{th}$ ) in series with a single resistor ( $R_{th}$ ).
Procedure:
1. Remove the load resistor $R_L$ .
2. Find $V_{th}$ : Calculate the Open Circuit Voltage across the terminals.
3. Find $R_{th}$ : Turn off all independent sources (Voltage $\rightarrow$ Short, Current $\rightarrow$ Open) and calculate equivalent resistance looking into the terminals.
4. Draw the equivalent circuit and reconnect $R_L$ .
5. Load Current: $I_L = \frac{V_{th}}{R_{th} + R_L}$

Norton’s Theorem

Statement: Any linear, bilateral, two-terminal network can be replaced by an equivalent circuit consisting of a single current source ( $I_N$ ) in parallel with a single resistor ( $R_N$ ).
Procedure:
1. Remove the load resistor $R_L$ .
2. Find $I_N$ : Place a short circuit across the terminals and calculate the Short Circuit Current ( $I_{sc}$ ).
3. Find $R_N$ : Same procedure as $R_{th}$ (Norton Resistance = Thevenin Resistance).
4. Draw equivalent circuit with $R_L$ in parallel.
5. Load Current (via CDR): $I_L = I_N \times \frac{R_N}{R_N + R_L}$

Maximum Power Transfer Theorem

Statement: A DC source delivers maximum power to a variable load resistor when the load resistance equals the internal resistance of the source (Thevenin Resistance).
Condition: $R_L = R_{th}$ (or $R_S$ ).
Maximum Power Formula:
$P_{max} = \frac{V_{th}^2}{4 R_{th}}$
Efficiency: At maximum power transfer, the efficiency of the circuit is exactly 50%, because equal power is dissipated in the source resistance and the load resistance.