Calculate RL Parallel Circuit

Calculator and formulas for calculating the parameters of an RL parallel circuit

RL Parallel Circuit Calculator

RL Parallel Circuit

The calculator calculates current, power, apparent and reactive resistance in the parallel connection of a resistor and an inductor. Enter the values for inductor, resistor, frequency and voltage.

Results
Reactance XL:
Total impedance Z:
Resistor current IR:
Inductor current IL:
Total current I:
Real power P:
Reactive power Q:
Apparent power S:
Phase angle φ:

Circuit Diagram & Parameters

RL Parallel Circuit
Parameter Legend
ITotal current
IRCurrent through the resistor
ILCurrent through the inductor
XLInductive reactance
ZImpedance (total resistance)
YAdmittance (apparent conductance)
PReal power
QReactive power
SApparent power
φPhase angle

Example Calculations

Practical Calculation Examples

Example 1: Low Frequency

Given: L = 100 mH, R = 100 Ω, f = 50 Hz, U = 230 V

\[X_L = 2\pi f L = 2\pi \cdot 50 \cdot 0.1 = 31.4 \text{ Ω}\]
\[Z = \frac{R \cdot X_L}{\sqrt{R^2 + X_L^2}} = \frac{100 \cdot 31.4}{\sqrt{100^2 + 31.4^2}} = 29.8 \text{ Ω}\]
\[I = \frac{U}{Z} = \frac{230}{29.8} = 7.72 \text{ A}\]
Total current: 7.72 A
Example 2: Audio Frequency

Given: L = 10 mH, R = 8 Ω, f = 1 kHz, U = 12 V

\[X_L = 2\pi \cdot 1000 \cdot 0.01 = 62.8 \text{ Ω}\]
\[I_R = \frac{U}{R} = \frac{12}{8} = 1.5 \text{ A}\]
\[I_L = \frac{U}{X_L} = \frac{12}{62.8} = 0.19 \text{ A}\]
\[I = \sqrt{I_R^2 + I_L^2} = \sqrt{1.5^2 + 0.19^2} = 1.51 \text{ A}\]
Partial currents dominated by R
Example 3: High Frequency

Given: L = 1 µH, R = 50 Ω, f = 100 MHz, U = 5 V

\[X_L = 2\pi \cdot 100 \times 10^6 \cdot 1 \times 10^{-6} = 628 \text{ Ω}\]
\[I_R = \frac{5}{50} = 0.1 \text{ A}\]
\[I_L = \frac{5}{628} = 0.008 \text{ A}\]
\[I = \sqrt{0.1^2 + 0.008^2} = 0.1 \text{ A}\]
Inductor dominates at HF
Important Conversions
Frequency units:
1 kHz = 1,000 Hz
1 MHz = 1,000,000 Hz
1 GHz = 1,000,000,000 Hz
Power units:
1 kW = 1,000 W
1 MW = 1,000,000 W
1 kVA = 1,000 VA
1 kvar = 1,000 var

RL Parallel Circuit - Theory and Formulas

What is an RL Parallel Circuit?

In an RL parallel circuit, an ohmic resistor R and an inductance L are connected in parallel. The same voltage is applied to both components, but the current is distributed according to the respective resistance values. The total current is the geometric sum of the partial currents.

Calculation Formulas

Current Triangle
Total Current
\[I = \sqrt{I_R^2 + I_L^2} = \frac{U}{Z}\]

Geometric addition of partial currents

Active Current
\[I_R = \frac{U}{R} = I \cdot \cos(φ)\]

Current through the ohmic resistor

Reactive Current
\[I_L = \frac{U}{X_L} = I \cdot \sin(φ)\]

Current through the inductance

Phase Angle
\[φ = \arctan\left(\frac{I_L}{I_R}\right)\]

Phase shift between U and I

Conductance Triangle

For parallel circuits, it's often easier to work with conductances:

Admittance
\[Y = \sqrt{G^2 + B_L^2} = \frac{1}{Z}\]

Total conductance of the circuit

Conductance
\[G = \frac{1}{R}\]

Active conductance (reciprocal of resistance)

Susceptance
\[B_L = \frac{1}{X_L} = \frac{1}{2\pi f L}\]

Reactive conductance of the inductance

Impedance (Total Resistance)
Total Impedance
\[Z = \frac{R \cdot X_L}{\sqrt{R^2 + X_L^2}}\]

Parallel connection of R and L

Reactance
\[X_L = 2\pi f L = \omega L\]

Frequency-dependent reactance

Power Triangle
Apparent Power
\[S = \sqrt{P^2 + Q^2} = U \cdot I\]

Total power of the circuit

Real Power
\[P = U \cdot I_R = \frac{U^2}{R}\]

Usable power in the resistor

Reactive Power
\[Q = U \cdot I_L = \frac{U^2}{X_L}\]

Oscillating power in the inductance

Power Factor
\[\cos(φ) = \frac{P}{S} = \frac{I_R}{I}\]

Ratio of real to apparent power

Practical Applications

Filters & Crossovers:
• Speaker crossovers
• EMC filters
• Audio filters
• Mains filters
Motor Circuits:
• Starting circuits
• Power factor correction
• Three-phase drives
• Transformer circuits
Measurement Technology:
• Impedance measurement
• Current dividers
• Load resistors
• Test circuits

Behavior at Different Frequencies

Frequency-Dependent Behavior
  • Low frequencies (f → 0): XL → 0, inductor acts like short circuit
  • Medium frequencies: XL ≈ R, both branches contribute to current
  • High frequencies (f → ∞): XL → ∞, resistor dominates
  • Resonance: Does not occur in RL parallel circuits (only in RLC)
  • Time constant: τ = L/R determines transient behavior

Design Guidelines

Important Design Aspects
  • Current distribution: At low frequencies, more current flows through the inductor
  • Losses: Power losses only occur in the resistor
  • Phase angle: Current leads voltage (inductive)
  • Reactive power: Inductor stores and returns energy
  • Power quality: Parallel connection improves power factor
  • Short-circuit behavior: Inductor limits current change


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