Unit 1: Fundamentals of D.C. circuits - Subjective Questions

Resistance ():

• It is the property of a material that opposes the flow of electric current.
• Unit: Ohm ().
• V-I relation: (Ohm's law).
• It dissipates energy as heat.

Inductance ():

• It is the property of a coil that opposes a change in current by storing energy in a magnetic field.
• Unit: Henry (H).
• V-I relation: .
• Energy stored: .

Capacitance ():

• It is the property of a device to store electric charge in an electric field.
• Unit: Farad (F).
• V-I relation: .
• Energy stored: .

In a pure DC steady state, an inductor behaves as a short circuit and a capacitor behaves as an open circuit.

Voltage ():

• The work done per unit charge to move charge between two points.
• ; Unit: Volt (V).

Current ():

• The rate of flow of electric charge.
• ; Unit: Ampere (A).

Power ():

• The rate at which electrical energy is consumed or delivered.
• ; Unit: Watt (W).

Energy ():

• The total work done over a period of time.
• ; Unit: Joule (J) or kWh (commercial unit).

Interrelationship:

• Power is the product of voltage and current.
• Energy is the integral of power over time: .

Statement:
At constant temperature, the current () flowing through a conductor is directly proportional to the potential difference () across its ends.

where is the resistance (constant of proportionality).

Explanation:

• If voltage is doubled, current also doubles, provided temperature remains constant.
• The ratio gives the resistance .

Limitations:

• Valid only when temperature and other physical conditions remain constant.
• Not applicable to non-linear devices (e.g., diodes, transistors).
• Not applicable to unilateral elements that behave differently in forward and reverse directions.
• Not applicable to arc lamps and gaseous conductors.

Kirchhoff's Current Law (KCL):

• The algebraic sum of currents meeting at any node (junction) is zero.
• Based on the conservation of charge.
• Example: If and enter a node and leaves, then .

Kirchhoff's Voltage Law (KVL):

• The algebraic sum of all voltages around any closed loop is zero.
• Based on the conservation of energy.
• Example: In a loop with source and drops across , : .

Sign Conventions:

• Current entering a node is taken positive, leaving negative (or vice versa).
• Rise in potential is positive, drop is negative when traversing a loop.

Series Connection:

• Same current flows through all resistors.
• By KVL:

Parallel Connection:

• Same voltage appears across all resistors.
• By KCL:

For two resistors in parallel: .

Voltage Division Rule (VDR):
For two resistors and in series carrying current with supply :

• Voltage across :
• Voltage across :

Current Division Rule (CDR):
For two resistors and in parallel with total current and voltage :

• Current through :
• Current through :

Note that in CDR, the current through one resistor is proportional to the opposite resistor.

Introduction:
Star-Delta transformation is used to simplify networks that cannot be reduced using simple series-parallel combinations (e.g., bridge networks).

Delta () to Star () Conversion:
Let the delta resistances be , , and . The equivalent star resistances , , are obtained by equating terminal resistances.

Rule: Each star resistance = (product of two adjacent delta resistances) / (sum of all three delta resistances).

Star () to Delta () Conversion:

Similarly for (divide by ) and (divide by ).

Rule: Each delta resistance = (sum of products of star resistances taken two at a time) / (opposite star resistance).

Independent Sources:

• Provide a fixed voltage or current that does not depend on any other circuit variable.
• Types:
  • Independent Voltage Source: maintains constant voltage regardless of current.
  • Independent Current Source: maintains constant current regardless of voltage.
• Example: Battery, DC generator.

Dependent (Controlled) Sources:

• The source value depends on a voltage or current elsewhere in the circuit.
• Four types:
  • VCVS (Voltage Controlled Voltage Source):
  • CCVS (Current Controlled Voltage Source):
  • VCCS (Voltage Controlled Current Source):
  • CCCS (Current Controlled Current Source):

Dependent sources are widely used to model electronic devices such as transistors and op-amps.

Mesh Analysis is a systematic method based on KVL used to find unknown currents in a planar circuit by assigning loop (mesh) currents.

Procedure:

1. Identify all independent meshes (loops that do not contain any other loop).
2. Assign a mesh current to each mesh (usually clockwise).
3. Apply KVL to each mesh, expressing voltages in terms of mesh currents.
4. For a resistor shared between two meshes, the net current is the difference of the two mesh currents.
5. Form the simultaneous equations.
6. Solve for the mesh currents using substitution, elimination, or matrix methods.

General Matrix Form:

For a 2-mesh circuit:

where , are self-resistances and , are mutual resistances.

Advantages:

• Fewer equations compared to branch analysis.
• Best suited for circuits with many voltage sources.

Nodal Analysis is a systematic method based on KCL used to find unknown node voltages in a circuit.

Procedure:

1. Identify all the nodes in the circuit.
2. Select one node as the reference node (ground, 0 V).
3. Assign voltage variables to the remaining nodes.
4. Apply KCL at each non-reference node, expressing branch currents in terms of node voltages using Ohm's law.
5. Form the simultaneous equations.
6. Solve for the node voltages.

General Matrix Form:

For a 2-node circuit:

where , are self-conductances and , are mutual conductances.

Advantages:

• Fewer equations in circuits with many parallel branches.
• Best suited for circuits with many current sources.

When to prefer:

• Use Mesh Analysis when the number of meshes is small and voltage sources dominate.
• Use Nodal Analysis when the number of nodes is small and current sources dominate.

The choice is made to minimize the number of simultaneous equations to be solved.

Statement:
Any linear, bilateral, two-terminal network containing sources and resistances can be replaced by an equivalent circuit consisting of a single voltage source in series with a single resistance .

Thevenin Equivalent:

Procedure:

1. Remove the load resistance from the terminals A-B.
2. Find the open-circuit voltage across A-B.
3. Deactivate all independent sources (short voltage sources, open current sources).
4. Compute the equivalent resistance looking into terminals A-B.
5. Draw the Thevenin equivalent: in series with .
6. Reconnect and find the load current:
   

Note: For circuits with dependent sources, is found by applying a test source at the terminals.

Advantage: Simplifies analysis when load varies, since the equivalent needs to be computed only once.

Statement:
Any linear, bilateral, two-terminal network can be replaced by an equivalent circuit consisting of a single current source in parallel with a single resistance .

Norton Equivalent:

Procedure:

1. Remove the load resistance from terminals A-B.
2. Short the terminals A-B and find the short-circuit current .
3. Deactivate all independent sources and find looking into A-B.
4. Draw the Norton equivalent: in parallel with .
5. Reconnect and find load current using CDR:
   

Relation with Thevenin:

Norton's and Thevenin's equivalents are duals of each other and are interconvertible by source transformation.

Statement:
Maximum power is transferred from a source to a load when the load resistance equals the internal (Thevenin) resistance of the source network.

Proof:
Consider a source with internal resistance connected to load .

Load current:

Power delivered to load:

For maximum power, differentiate w.r.t. and set to zero:

Simplifying the numerator:

Maximum Power:

Efficiency at maximum power transfer is only 50%, so this condition is important in communication/electronics but not in power systems.

Statement:
In a linear circuit containing more than one independent source, the response (current or voltage) in any element is the algebraic sum of the responses produced by each independent source acting alone, with all other independent sources deactivated.

Deactivation Rules:

• Replace independent voltage sources by a short circuit.
• Replace independent current sources by an open circuit.
• Dependent sources are left unchanged.

Steps to Apply:

1. Consider one independent source at a time.
2. Deactivate all other independent sources.
3. Compute the response (current/voltage) due to the active source.
4. Repeat for every independent source.
5. Add all individual responses algebraically to get the total response.

Limitations:

• Applicable only to linear circuits.
• Cannot be used to calculate power directly, since power is a non-linear () quantity.
• Not applicable to circuits with only dependent sources.

Basic (Systematic) Method:

• Uses systematic formulation of equations based on KVL and KCL.
• Includes mesh analysis and nodal analysis.
• Involves writing and solving a set of simultaneous equations.
• Works for any circuit, including complex ones that cannot be simplified.
• More rigorous but computationally heavier.

Intuitive Method:

• Simplifies the circuit step-by-step using series-parallel reduction.
• Uses shortcuts like the Voltage Division Rule (VDR) and Current Division Rule (CDR).
• Faster and requires less computation.
• Applicable only to circuits that can be reduced by series/parallel combinations.
• Not suitable for bridge/complex networks without transformations.

Summary:

Given: , ,

Equivalent Resistance (parallel):

Total Current:

Branch Currents (CDR / Ohm's law):

Check: ✓

Power Dissipated:

Total Power: ✓

Source Transformation is a technique to convert a practical voltage source into an equivalent current source and vice versa, keeping the terminal behavior unchanged. It simplifies circuit analysis.

Voltage Source to Current Source:
A voltage source in series with resistance can be replaced by a current source in parallel with the same resistance , where:

Current Source to Voltage Source:
A current source in parallel with resistance can be replaced by a voltage source in series with the same resistance , where:

Key Points:

• The resistance remains the same in both representations.
• The direction of the current source arrow points toward the positive terminal of the voltage source.
• Only valid for sources that have an associated series/parallel resistance (practical sources).
• Dependent sources can also be transformed, but the controlling variable must be retained.

Application: It helps combine sources and reduce circuits, and establishes the relation between Thevenin and Norton equivalents.

Given: , , ,

Total Resistance (series):

Circuit Current:

Voltage Across Each Resistor (VDR):

Check: ✓ (confirms KVL)

Energy Stored in an Inductor:
The voltage across an inductor is .

Instantaneous power:

Energy stored:

Energy Stored in a Capacitor:
The current through a capacitor is .

Instantaneous power:

Energy stored:

DC Steady-State Behaviour:

• Inductor: In steady state, current is constant, so . It behaves as a short circuit.
• Capacitor: In steady state, voltage is constant, so . It behaves as an open circuit.