Chapter 21: CIRCUITS AND DC INSTRUMENTS

Resistors in Series

• Definition: Resistors are connected one after another, sharing the same current throughout the circuit. The arrangement affects how voltage is distributed among the components.

• Formula: The total or equivalent resistance (Rs) is calculated as: $R</em>s = R<em>1 + R</em>2 + R_3 + …$

• Key Characteristics:

  1. Series resistances add: Total resistance is the sum of all individual resistances, leading to an increase in the overall resistance of the circuit.

  2. Same current flows through each resistor: The same amount of current (I) passes through all components in a series circuit, which is crucial for current calculations.

  3. Voltage division: Each resistor receives a fraction of the total voltage. The voltage drop across each resistor can be calculated using Ohm's law (V = IR), where V is the voltage across the resistor, I is the current, and R is the resistance.

Resistors in Parallel

• Definition: Resistors are connected across the same voltage source, allowing multiple paths for current to flow, which impacts the overall behavior of the circuit.

• Formula for Parallel Resistance: The total or equivalent resistance (Rp) is calculated as: $\frac{1}{R</em>p} = \frac{1}{R<em>1} + \frac{1}{R</em>2} + \frac{1}{R_3} + …$

• Key Characteristics:

  1. Smaller equivalent resistance: The total parallel resistance is always less than the smallest resistance in the network, providing a pathway for greater current flow.

  2. Same voltage across each resistor: Each resistor experiences the same full voltage from the source (V), which is essential to ensuring equal distribution of voltage.

  3. Current division: Resistors do not share the total current equally; instead, current divides among them in proportion to their resistances, calculated using the formula ( In = \frac{V}{Rn} ).

Combinations of Series and Parallel

• In complex circuits, resistors can be connected in both series and parallel configurations.

• Method for Finding Equivalent Resistance: Break down complex networks into simpler series and parallel parts, calculate their individual equivalents, and then combine results step by step, often by starting with the parallel groups first.

Electromotive Force (emf) and Internal Resistance

• Definition: emf is the voltage provided by a source when no current is flowing, representing the source's potential energy per unit charge.

• Internal Resistance: Represented as r, it affects terminal voltage and the current delivered to the load, especially significant in series connections.

  • If a load resistance (Rload) is connected, the effective voltage is reduced due to internal resistance. The equation for the terminal voltage (Vterminal) can be expressed as ( V_{terminal} = emf - I \times r ).

Kirchhoff's Laws

Kirchhoff’s Junction Rule (Current Law)

• Statement: The total current entering a junction must equal the total current leaving the junction, ensuring conservation of charge.

• Example Calculation:

  • If incoming current is I1 and outgoing currents are I2 and I3, then: $I</em>1 = I<em>2 + I</em>3$

  • If I1 = 11 A and we have I2 = 7 A, then:
    $I_3 = 11 A - 7 A = 4 A$

Kirchhoff’s Loop Rule (Voltage Law)

• Statement: The sum of the potential differences around a closed loop must equal zero, reflecting the conservation of energy in electrical circuits.

• Example Calculation:

  • Consider a loop with a voltage supply of 18 V and several resistors:

  • If V_internal = 1 V, and total drop across resistors is 12 V + 5 V, then the equation should satisfy:
    $18 V = 1 V + 12 V + 5 V$

Problem-Solving Strategies for Kirchhoff's Rules

1. Circuit Diagram: Draw and label all known and unknown elements in the circuit, including voltage sources and orientation of components.

2. Apply Junction Rule: Use this at junctions to write equations involving currents flowing in and out of each junction effectively.

3. Apply Loop Rule: Create equations for loops in the circuit using potential changes for all components involved.

4. Solve Equations: Use algebraic techniques to solve simultaneous equations for unknown currents and voltages.

5. Check Answers: Make sure results are reasonable and satisfy both Kirchhoff’s laws to ensure accuracy in your calculations.

Measuring Voltage and Current

• Voltmeter: Measures potential differences, connected in parallel across components or the whole circuit to avoid altering the circuit functionality.

• Ammeter: Measures current, connected in series with the circuit; the current reading remains consistent regardless of the positioning of the ammeter in series through the circuit, ensuring accuracy in measurement.