The phasor method

In DC circuits voltages “add up” like ordinary numbers; in AC this rule breaks down, because two sinusoidal voltages of equal frequency but different phase do not add like scalars. The graphical trick that solves the problem is called the phasor method.

To every sinusoidal quantity $x(t) = X_0\cos(\omega t + \phi)$ we associate a vector of length $X_0$ that rotates in the plane with angular velocity $\omega$, forming an angle $\phi$ with the horizontal axis at instant $t = 0$. The projection of the vector onto the horizontal axis reproduces the function $x(t)$.

Phasor diagram for a series RLC circuit (case $X_L > X_C$). The phasor $V_R$ is in phase with the current; $V_L$ leads by $90^\circ$, $V_C$ lags by $90^\circ$. The vector sum of the three is the phasor of the total voltage $V_0$, rotated by $\varphi$ with respect to $I_0$.

Phasor rules

• For a resistor, the voltage phasor is parallel to the current phasor (zero phase shift).
• For an inductor, the voltage phasor is rotated by $+90^\circ$ relative to the current (voltage leads).
• For a capacitor, the voltage phasor is rotated by $-90^\circ$ relative to the current (voltage lags).

In a series RLC the current is the same in every element: we use its phasor $I_0$ as reference. The three voltage phasors $V_R = R\,I_0$, $V_L = X_L\,I_0$ and $V_C = X_C\,I_0$ add vectorially, because $\Delta V_\text{tot}(t) = V_R(t) + V_L(t) + V_C(t)$. The modulus of the sum is precisely $V_0 = I_0\sqrt{R^2 + (X_L - X_C)^2} = Z\,I_0$, and the angle with respect to $I_0$ is $\varphi$: we recover the formulas obtained analytically, this time with a drawing.

Phasors as complex numbers

The representation becomes compact if we identify the phasor plane with the complex plane. To every sinusoidal quantity $x(t) = X_0\cos(\omega t + \phi)$ we associate the complex number $\tilde{X} = X_0\,e^{i\phi}$; differentiating in time corresponds to multiplying by $i\omega$. Then $\tilde{V}_R = R\,\tilde{I}$, $\tilde{V}_L = i\omega L\,\tilde{I}$, $\tilde{V}_C = -\dfrac{i}{\omega C}\,\tilde{I}$. The complex impedances are $Z_R = R$, $Z_L = i\omega L$, $Z_C = 1/(i\omega C)$; in series they add, and Ohm’s law becomes $\tilde{V} = Z\tilde{I}$ with $Z = R + i(X_L - X_C)$. The modulus $|Z| = \sqrt{R^2 + (X_L - X_C)^2}$ recovers the real formula; the argument $\arg Z = \varphi$ is the phase shift.

Having discovered that a varying $\vv{B}$ generates an $\vv{E}$, in the next chapter we shall see that, by symmetry, a varying $\vv{E}$ generates a $\vv{B}$: the result that James Clerk Maxwell placed at the foundation of his equations, explaining electricity, magnetism and light in a single stroke.

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Topics: Electromagnetic induction Concepts: Alternating current Skills: Phasor method

Related exercises: Problem — Reactances of an RLC as omega varies · Problem — Impedance and phase shift of a series RLC · Problem — Phasor diagram (XL greater than XC)