Formulas and description for RL in parallel
The total resistance of the RL parallel circuit in AC is called
impedance Z. Ohm's law applies to the entire circuit.
Current and voltage are in phase at the ohmic resistance.
The inductive reactance of the capacitor lags
the current the voltage by −90 °.
The total current I is the sum of the geometrically added partial currents.
For this purpose, both partial flows form the legs of a right triangle.
Its hypotenuse corresponds to the total current I.
The resulting triangle is called the current triangle or vector diagram of the currents.
Current triangle
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\(\displaystyle I=\sqrt{ {I_R}^2+{I_L}^2} \)
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\(\displaystyle =\frac{U}{Z} \)
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\(\displaystyle I_R=\sqrt{I^2-{I_L}^2} \)
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\(\displaystyle =\frac{U}{R} \)
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\(\displaystyle =I·cos(φ) \)
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\(\displaystyle I_L=\sqrt{I^2-{I_R}^2} \)
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\(\displaystyle =\frac{U}{X_L} \)
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\(\displaystyle =I·sin(φ) \)
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Conductance triangle
When connected in parallel, the partial currents behave like the conductance values
of resistances.
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\(\displaystyle Y=\sqrt{G^2+{B_L}^2} \)
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\(\displaystyle =\frac{1}{Z} \)
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\(\displaystyle G=\sqrt{Y^2-{B_L}^2} \)
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\(\displaystyle =\frac{1}{R} \)
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\(\displaystyle B_L=\sqrt{Y^2-G^2} \)
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\(\displaystyle =\frac{1}{X_L} \)
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\(\displaystyle φ=\frac{B_L}{G} \)
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Resistance triangle
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\(\displaystyle Z=\frac{R· X_L}{\sqrt{R^2+{X_L}^2}} \)
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\(\displaystyle =\frac{1}{Y}\)
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\(\displaystyle R=\frac{Z· X_L}{\sqrt{Z^2-{X_L}^2}} \)
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\(\displaystyle =\frac{1}{G}\)
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\(\displaystyle X_L=\frac{Z· R}{\sqrt{Z^2-R^2}} \)
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\(\displaystyle =\frac{1}{B_L}\)
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Power triangle
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\(\displaystyle S=\sqrt{P^2+{Q_L}^2} \)
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\(\displaystyle =U·I \)
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\(\displaystyle P=\sqrt{S^2-{Q_L}^2} \)
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\(\displaystyle =U_R·I_R \)
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\(\displaystyle Q_L=\sqrt{S^2-P^2} \)
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\(\displaystyle =U_L·I_L \)
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Power factor
\(\displaystyle cos(φ)=\frac{P}{S}=\frac{I_R}{I}\)
