How to Calculate the Phase Angle (Circuits, Phasors and Waveforms)

“Phase angle” means the angular offset between two sinusoids of the same frequency, or the angle of a phasor. Which formula you use depends on what you are given.

1. Series RLC circuit

The impedance of a series RLC circuit is

$$Z = R + j(X_L - X_C), \qquad X_L = \omega L,\quad X_C = \frac{1}{\omega C},\quad \omega = 2\pi f.$$

The phase angle of the voltage relative to the current is the angle of $Z$:

$$\varphi = \arctan\!\left(\frac{X_L - X_C}{R}\right), \qquad |Z| = \sqrt{R^2 + (X_L - X_C)^2}.$$

• $\varphi > 0$: inductive circuit, and the voltage leads the current.
• $\varphi < 0$: capacitive circuit, and the current leads the voltage.
• $\varphi = 0$: resonance ($X_L = X_C$), purely resistive.

The power factor is $\cos\varphi = R/|Z|$.

Example. $R = 30\ \Omega$, $X_L = 60\ \Omega$, $X_C = 20\ \Omega$:

$$\varphi = \arctan\!\left(\frac{60 - 20}{30}\right) = \arctan(1.333) \approx 53.13^\circ, \qquad |Z| = \sqrt{30^2 + 40^2} = 50\ \Omega.$$

The voltage leads the current by about $53.1^\circ$, and the power factor is $30/50 = 0.6$ (lagging).

For a parallel circuit, use the admittance instead. For parallel R and C, for example, $\varphi = \arctan(\omega R C)$ with the current leading the voltage.

2. A phasor or complex number

For $z = a + jb$, the phase angle is

$$\varphi = \operatorname{atan2}(b, a).$$

Use atan2, not $\arctan(b/a)$. The plain arctangent can’t tell $1 + j$ ($45^\circ$) from $-1 - j$ ($-135^\circ$). For $3 + 4j$: $\varphi = \operatorname{atan2}(4, 3) \approx 53.13^\circ$, with magnitude $5$.

3. Two waveforms on an oscilloscope

If $v_1$ and $v_2$ have the same frequency and the same feature (a zero crossing or a peak) of $v_2$ occurs $\Delta t$ after that of $v_1$, then

$$\varphi = 360^\circ \times \frac{\Delta t}{T} = 360^\circ \times f\,\Delta t.$$

Example. At $f = 50\ \text{Hz}$ ($T = 20\ \text{ms}$), a delay of $\Delta t = 2\ \text{ms}$ gives $\varphi = 360^\circ \times 2/20 = 36^\circ$, with $v_2$ lagging $v_1$.

Another method is the Lissajous figure: in X-Y mode the trace is an ellipse, and $\sin\varphi = y_0 / y_{\max}$, where $y_0$ is where the ellipse crosses the vertical axis.

4. From equations

If $v(t) = V_m \sin(\omega t + \alpha)$ and $i(t) = I_m \sin(\omega t + \beta)$, the phase difference is simply $\varphi = \alpha - \beta$. Write both in the same function first. Use $\cos\theta = \sin(\theta + 90^\circ)$ to convert, and remember that a negative amplitude adds $180^\circ$.

Common mistakes

• Mixing degrees and radians, especially when using $\omega = 2\pi f$ and then reporting the angle.
• Using $\arctan(b/a)$ for phasors in the second or third quadrant.
• Forgetting that $X_C$ enters with a minus sign in series circuits.
• Comparing waveforms of different frequencies. A phase difference is only constant between sinusoids of the same frequency.