19.3 Parallel Circuits

Parallel circuits offer multiple paths for current flow, keeping voltage constant across branches while dividing current. This setup allows for safer connections of components with different voltage ratings and enables flexible circuit designs.

Calculating equivalent resistance in parallel circuits involves reciprocals, unlike series circuits. ###Ohm's_Law_0### applies to each branch independently, making it easier to analyze current and power distribution across the circuit's components.

Parallel Circuits

Analysis of parallel circuit diagrams

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• Parallel circuits provide multiple paths for current to flow through
  • Each path in a parallel circuit is called a branch, allowing current to split and flow through different components simultaneously
  • Voltage remains constant across each branch in a parallel circuit, ensuring that all components connected in parallel experience the same potential difference
• Current divides at each junction in a parallel circuit and recombines at the other end
  • Total current in a parallel circuit is calculated by summing the currents flowing through each individual branch: $I_{total} = I_1 + I_2 + ... + I_n$
• Resistors connected in parallel have the same voltage drop across them
  • Voltage across each resistor in a parallel circuit is equal to the source voltage, regardless of the resistor's value: $V_1 = V_2 = ... = V_n = V_{source}$
  • This property allows components with different voltage ratings to be safely connected in parallel (LED arrays)

Equivalent resistance in circuit combinations

• Equivalent resistance ([object Object],[object Object]) represents the single resistance value that can replace a combination of resistors while maintaining the same overall current and voltage characteristics
• For resistors connected in parallel, the reciprocal of the equivalent resistance is equal to the sum of the reciprocals of the individual resistances: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$
  • Simplified formula for calculating the equivalent resistance of two resistors in parallel: $R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2}$
  • Connecting resistors in parallel decreases the overall resistance, allowing more current to flow (parallel battery configurations)
• For resistors connected in series, the equivalent resistance is calculated by summing the individual resistances: $R_{eq} = R_1 + R_2 + ... + R_n$
• Complex circuits with combinations of series and parallel resistors can be simplified by calculating the equivalent resistance of each parallel or series section, then combining the resulting equivalent resistances
  • This process is repeated until the circuit is reduced to a single equivalent resistance (voltage divider circuits)
• The inverse of resistance, conductance, is often used in parallel circuit calculations to simplify the process of finding equivalent resistance

Ohm's law for parallel circuits

• Ohm's law describes the relationship between voltage, current, and resistance in a circuit: [object Object],[object Object]
  • $V$ represents voltage measured in volts (V)
  • $I$ represents current measured in amperes (A)
  • $R$ represents resistance measured in ohms ($\Omega$)
• In parallel circuits, Ohm's law can be applied to each branch independently
  • Current flowing through each branch is calculated using the formula: $I_n = \frac{V}{R_n}$, where $V$ is the source voltage and $R_n$ is the resistance of the specific branch
• Total current in a parallel circuit is determined by summing the currents flowing through each branch: $I_{total} = \frac{V}{R_1} + \frac{V}{R_2} + ... + \frac{V}{R_n}$
• Power dissipated by each resistor in a parallel circuit is calculated using the formula: $P_n = \frac{V^2}{R_n}$
  • Total power dissipated in a parallel circuit is the sum of the power dissipated by each individual resistor: $P_{total} = \frac{V^2}{R_1} + \frac{V^2}{R_2} + ... + \frac{V^2}{R_n}$
  • Power calculations are essential for determining the proper wattage ratings for components in a circuit (resistors, light bulbs)

Circuit Analysis Techniques

• Nodal analysis is used to solve for voltages at specific points in a circuit by applying Kirchhoff's Current Law
• Mesh analysis involves solving for currents in closed loops of a circuit using Kirchhoff's Voltage Law
• Thévenin's theorem and Norton's theorem are powerful tools for simplifying complex circuits into equivalent circuits, making analysis easier