Sinusoidal steady state and phasors

Zero a hundred times a second

• Mains voltage crosses zero a hundred times a second, yet a heater on it gives out steady warmth. Once the switch-on transient dies away, every voltage and current repeats the same sinusoid.
• At one frequency, a sinusoid is fixed by two numbers: its size and its timing. This lesson turns AC circuits into arithmetic on those two numbers.

The phasor: amplitude and phase

• $v(t) = V_{m}\cos(\omega t + \varphi)$ becomes the phasor 相量 $\mathbf{V} = V_{m}\angle\varphi$. The frequency $\omega$ is the same everywhere in the circuit, so it is left out.
• A phasor is a complex number: $V_{m}\angle\varphi = V_{m}\cos\varphi + jV_{m}\sin\varphi$. Use polar form to multiply and divide, rectangular form to add.

Calculus becomes multiplication

• Differentiating $\cos(\omega t + \varphi)$ gives a sinusoid $\omega$ times larger and 90° ahead. With phasors, $\dfrac{d}{dt}$ becomes multiplication by $j\omega$.
• So the inductor's $v = L\dfrac{di}{dt}$ becomes $\mathbf{V} = j\omega L\,\mathbf{I}$, and the capacitor's $i = C\dfrac{dv}{dt}$ becomes $\mathbf{I} = j\omega C\,\mathbf{V}$. The differential equations are gone.

Impedance: Ohm's law with complex numbers

• Impedance 阻抗 is $Z = \dfrac{\mathbf{V}}{\mathbf{I}}$: $Z_{R} = R$, $Z_{L} = j\omega L$, $Z_{C} = \dfrac{1}{j\omega C}$.
• The resistor is flat at 0°. The inductor sits at +90° and grows with frequency; the capacitor sits at −90° and shrinks. Dividers, nodal analysis and Thévenin all carry over unchanged.

Worked example: R and L in series

• $R = 30\,\Omega$ in series with $\omega L = 40\,\Omega$, driven by $100\angle 0^{\circ}\,\text{V}$. $Z = 30 + j40 = 50\angle 53.13^{\circ}\,\Omega$.
• $\mathbf{I} = \dfrac{100\angle 0^{\circ}}{50\angle 53.13^{\circ}} = 2\angle -53.13^{\circ}\,\text{A}$: divide the sizes, subtract the angles. So $i(t) = 2\cos(\omega t - 53.13^{\circ})$ A; the current lags.

Worked example: R and L in parallel

• The same parts in parallel: $Z = \dfrac{30 \times j40}{30 + j40} = 19.2 + j14.4 = 24\angle 36.87^{\circ}\,\Omega$.
• The result is smaller than either branch (24 < 30 < 40), and its angle lies between the resistor's 0° and the inductor's 90°. Parallel always pulls both size and angle this way.

Three common slips

Recap