Calculate RL Series Circuit

Calculator and formulas for calculating an RL series circuit

RL Series Circuit Calculator

RL Series Circuit

The calculator calculates voltages, powers, current, apparent and reactive resistance for a series circuit consisting of a resistor and an inductor. Enter the values for inductor, resistor, frequency and voltage.

Results
Reactance XL:
Total impedance Z:
Voltage UR:
Voltage UL:
Current I:
Real power P:
Reactive power Q:
Apparent power S:
Phase angle φ:

Circuit Diagram & Parameters

RL Series Circuit
Parameter Legend
UApplied voltage
URVoltage across resistor
ULVoltage across inductor
ICurrent
ROhmic resistance
XLInductive reactance
ZImpedance - total resistance
PReal power
QInductive reactive power
SApparent power
φPhase shift in °

Example Calculations

Practical Calculation Examples

Example 1: Low Frequency Motor

Given: L = 50 mH, R = 10 Ω, f = 50 Hz, U = 230 V

\[X_L = 2\pi f L = 2\pi \cdot 50 \cdot 0.05 = 15.7 \text{ Ω}\]
\[Z = \sqrt{R^2 + X_L^2} = \sqrt{10^2 + 15.7^2} = 18.7 \text{ Ω}\]
\[I = \frac{U}{Z} = \frac{230}{18.7} = 12.3 \text{ A}\]
\[U_R = I \cdot R = 12.3 \cdot 10 = 123 \text{ V}\]
\[U_L = I \cdot X_L = 12.3 \cdot 15.7 = 193 \text{ V}\]
Motor starting circuit
Example 2: Audio Crossover

Given: L = 1 mH, R = 4 Ω, f = 3 kHz, U = 12 V

\[X_L = 2\pi \cdot 3000 \cdot 0.001 = 18.8 \text{ Ω}\]
\[Z = \sqrt{4^2 + 18.8^2} = 19.2 \text{ Ω}\]
\[I = \frac{12}{19.2} = 0.625 \text{ A}\]
\[φ = \arctan\left(\frac{18.8}{4}\right) = 78.0°\]
\[P = I^2 \cdot R = 0.625^2 \cdot 4 = 1.56 \text{ W}\]
High-pass behavior
Example 3: RF Circuit

Given: L = 10 µH, R = 50 Ω, f = 10 MHz, U = 5 V

\[X_L = 2\pi \cdot 10 \times 10^6 \cdot 10 \times 10^{-6} = 628 \text{ Ω}\]
\[Z = \sqrt{50^2 + 628^2} = 630 \text{ Ω}\]
\[I = \frac{5}{630} = 7.94 \text{ mA}\]
\[U_R = 7.94 \times 10^{-3} \cdot 50 = 0.4 \text{ V}\]
\[U_L = 7.94 \times 10^{-3} \cdot 628 = 4.99 \text{ V}\]
Inductor dominates at HF
Important Conversions
Inductance units:
1 H = 1,000 mH
1 mH = 1,000 µH
1 µH = 1,000 nH
1 nH = 0.001 µH
Voltage units:
1 kV = 1,000 V
1 V = 1,000 mV
1 mV = 1,000 µV
1 µV = 0.001 mV

RL Series Circuit - Theory and Formulas

What is an RL Series Circuit?

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

Calculation Formulas

Voltage Triangle
Total Voltage
\[U = \sqrt{U_R^2 + U_L^2}\]

Geometric addition of partial voltages

Active Voltage
\[U_R = I \cdot R = U \cdot \cos(φ)\]

Voltage across ohmic resistor

Reactive Voltage
\[U_L = I \cdot X_L = U \cdot \sin(φ)\]

Voltage across inductance

Current
\[I = \frac{U}{Z} = \frac{U}{\sqrt{R^2 + X_L^2}}\]

Current is the same everywhere

Impedance Triangle
Total Impedance
\[Z = \sqrt{R^2 + X_L^2} = \frac{U}{I}\]

Total resistance of the circuit

Resistance
\[R = \sqrt{Z^2 - X_L^2} = Z \cdot \cos(φ)\]

Ohmic resistance

Reactance
\[X_L = 2\pi f L = Z \cdot \sin(φ)\]

Frequency-dependent reactance

Phase Angle
\[φ = \arctan\left(\frac{X_L}{R}\right)\]

Phase shift between U and I

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

Total power of the circuit

Real Power
\[P = U_R \cdot I = I^2 \cdot R\]

Usable power (only in resistor)

Reactive Power
\[Q = U_L \cdot I = I^2 \cdot X_L\]

Oscillating power in inductance

Power Factor
\[\cos(φ) = \frac{P}{S} = \frac{R}{Z}\]

Ratio of real to apparent power

Phase Relations
Voltage Phase
\[φ = \arctan\left(\frac{U_L}{U_R}\right)\]

Phase angle from voltages

Resistance Phase
\[φ = \arctan\left(\frac{X_L}{R}\right)\]

Phase angle from resistances

Phase Behavior
  • φ > 0°: Inductive behavior - current lags voltage
  • φ = 0°: Pure resistive behavior - current and voltage in phase
  • φ = 90°: Pure inductive behavior - current lags 90°
  • Typical: 0° < φ < 90° in RL circuits

Practical Applications

Motors & Drives:
• Starting circuits
• Current limiting
• Chokes
• Transformers
Filters & Crossovers:
• High-pass filters
• Frequency crossovers
• Interference filters
• Audio crossovers
RF Technology:
• Antenna tuners
• Impedance matching
• Oscillators
• Resonant circuits

Behavior at Different Frequencies

Frequency-Dependent Behavior
  • Low frequencies (f → 0): XL → 0, resistor dominates
  • Medium frequencies: XL ≈ R, both components important
  • High frequencies (f → ∞): XL → ∞, inductor dominates
  • Cutoff frequency: fc = R/(2πL) when XL = R
  • High-pass behavior: Low frequencies are attenuated

Design Guidelines

Important Design Aspects
  • Voltage distribution: UL can be larger than Utotal!
  • Losses: Power losses only occur in the resistor
  • Phase angle: Current lags voltage (inductive)
  • Resonance: No resonance in RL circuits
  • Time constant: τ = L/R determines transient behavior
  • Self-induction: Inductor generates voltage spikes when switching


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