RCL Series Circuit Calculator

Formulas

The formulas to calculate the total voltage, impedance, and phase angle in an RCL series circuit are:

Total Voltage:

\[ U = \sqrt{U_R^2 + (U_C - U_L)^2} \]

Impedance:

\[ Z = \sqrt{R^2 + (X_C - X_L)^2} \]

Phase Angle:

\[ \tan \phi = \frac{U_C - U_L}{U_R} \]

\[ \tan \phi = \frac{X_C - X_L}{R} \]

Where:

\( U \) is the total voltage in volts (V)
\( U_R \) is the voltage across the resistor in volts (V)
\( U_C \) is the voltage across the capacitor in volts (V)
\( U_L \) is the voltage across the inductor in volts (V)
\( Z \) is the impedance in ohms (Ω)
\( R \) is the resistance in ohms (Ω)
\( X_C \) is the capacitive reactance in ohms (Ω)
\( X_L \) is the inductive reactance in ohms (Ω)
\( \phi \) is the phase angle in degrees (°)

What is an RCL Series Circuit?

An RCL series circuit is a circuit containing a resistor, a capacitor, and an inductor connected in series. The total voltage in the circuit is the vector sum of the voltages across the resistor, capacitor, and inductor. The impedance is the square root of the sum of the squares of the resistance and the difference between the capacitive and inductive reactances. The phase angle indicates the phase difference between the total voltage and the current.

Example Calculation

Let's assume the following values:

Resistance (\( R \)): 500 Ω
Capacitance (\( C \)): 1 µF
Inductance (\( L \)): 10 H
Voltage (\( U \)): 220 V
Frequency (\( f \)): 50 Hz

Step 1: Calculate the inductive reactance:

\[ X_L = 2 \pi \times 50 \times 10 = 3141.59 \, \Omega \]

Step 2: Calculate the capacitive reactance:

\[ X_C = \frac{1}{2 \pi \times 50 \times 1 \times 10^{-6}} = 3183.10 \, \Omega \]

Step 3: Calculate the impedance:

\[ Z = \sqrt{500^2 + (3183.10 - 3141.59)^2} = \sqrt{500^2 + 41.51^2} = \sqrt{250000 + 1722.63} = \sqrt{251722.63} \approx 501.72 \, \Omega \]

Step 4: Calculate the current:

\[ I = \frac{220}{501.72} \approx 0.44 \, A \]

Step 5: Calculate the voltages:

\[ U_R = I \cdot R = 0.44 \times 500 = 220 \, V \]

\[ U_L = I \cdot X_L = 0.44 \times 3141.59 \approx 1382.30 \, V \]

\[ U_C = I \cdot X_C = 0.44 \times 3183.10 \approx 1400.56 \, V \]

Step 6: Calculate the phase angle:

\[ \tan \phi = \frac{U_C - U_L}{U_R} = \frac{1400.56 - 1382.30}{220} = \frac{18.26}{220} \approx 0.083 \]

\[ \phi = \arctan(0.083) \approx 4.76^\circ \]