Understanding Parallel Circuits

In parallel circuits, each component is connected across the same voltage source, creating multiple paths for current flow. The total impedance of the circuit is determined by the combined effect of all branch impedances.

R-L Parallel Circuit

An R-L parallel circuit consists of a resistor (R) and an inductor (L) connected in parallel.

Key Characteristics:

• The current splits into two paths: through the resistor and the inductor.
• The current through the resistor (IR) is in phase with the voltage.
• The current through the inductor (IL) lags the voltage by 90°.

Impedance (Z):

The total impedance is calculated as:

1 / Z = 1 / R + 1 / jXL, where XL = ωL.

Current Relationship:

I = √(IR2 + IL2)

R-C Parallel Circuit

An R-C parallel circuit includes a resistor (R) and a capacitor (C) connected in parallel.

Key Characteristics:

• The current splits into two paths: through the resistor and the capacitor.
• The current through the resistor (IR) is in phase with the voltage.
• The current through the capacitor (IC) leads the voltage by 90°.

Impedance (Z):

1 / Z = 1 / R + jωC

Current Relationship:

I = √(IR2 + IC2)

R-L-C Parallel Circuit

An R-L-C parallel circuit contains a resistor (R), inductor (L), and capacitor (C) connected in parallel.

Key Characteristics:

• The impedance is determined by the combination of all branch impedances.
• At resonance, the inductive and capacitive reactances cancel each other out.

Impedance (Z):

1 / Z = 1 / R + 1 / jXL + jωC

Resonance:

• Resonance occurs when XL = XC, resulting in minimum circuit impedance.

Voltage and Current Relationships

• In parallel circuits, the voltage across each branch is the same.
• Total current is the vector sum of branch currents.

Phase Relationship:

The total current phase depends on the dominant reactive component:

• Inductive dominance: Current lags voltage.
• Capacitive dominance: Current leads voltage.

Applications of Parallel Circuits

• Power Distribution Systems: To supply equal voltage across devices.
• Filters: Band-pass and band-stop filters in communication systems.
• Power Factor Correction: Parallel capacitors improve power factor in AC systems.

Problem-Solving Techniques

1. Calculate individual branch currents using Ohm’s Law:
   • IR = V / R, IL = V / XL, IC = V / XC
2. Determine total current:
   • I = √(IR2 + IL/C2)
3. Compute total impedance:
   • 1 / Z = 1 / R + 1 / jXL + jωC

Advantages and Challenges

Advantages:

• Independent current paths for each component.
• Lower total impedance than individual branches.

Challenges:

• Managing complex phasor calculations.
• High currents in branches may lead to overheating.

Conclusion

Parallel circuits offer unique advantages for distributing current and maintaining constant voltage. This lesson explained the behavior of R-L, R-C, and R-L-C parallel circuits, focusing on impedance, resonance, and applications. The next lesson will discuss power in single-phase AC circuits.