In one line
In AC circuits a sinusoidal voltage or current is represented by a complex number (phasor), and each component by a complex impedance, so circuit calculations become simple complex-number arithmetic.
Explained simply
An AC voltage keeps rising and falling like a wave. Adding two waves that are out of step is messy with sines and cosines. Engineers instead picture each wave as a rotating arrow (a phasor) whose length is the amplitude and whose angle is the phase. Adding waves becomes adding arrows, which is exactly adding complex numbers. Electrical engineers write \(j\) instead of \(i\), because \(i\) is used for current.
Technically, \(V_m\cos(\omega t+\phi)=\mathrm{Re}\left[V_me^{j\phi}e^{j\omega t}\right]\), and the phasor is \(V_m\angle\phi=V_me^{j\phi}\). Resistance, inductance and capacitance have impedances \(R\), \(j\omega L\), \(\dfrac1{j\omega C}=-\dfrac{j}{\omega C}\). Ohm's law becomes \(V=IZ\) with complex numbers.
How it works
- Write each impedance: resistor \(R\), inductor \(jX_L\) with \(X_L=\omega L\), capacitor \(-jX_C\) with \(X_C=\dfrac1{\omega C}\).
- Series impedances add; parallel ones combine as \(\dfrac1Z=\dfrac1{Z_1}+\dfrac1{Z_2}\).
- Convert \(Z\) to polar form \(|Z|\angle\phi\).
- Current \(I=\dfrac VZ\): divide moduli, subtract angles.
Key formula
\(Z=R+j(X_L-X_C)\), \(|Z|=\sqrt{R^2+(X_L-X_C)^2}\), \(\phi=\tan^{-1}\dfrac{X_L-X_C}{R}\).
\(R,X_L,X_C,|Z|\) in ohms (Ω); \(\omega=2\pi f\) in rad/s; \(L\) in henry; \(C\) in farad; \(V\) in volts; \(I\) in amperes.
Quick example
A series circuit has \(R=3\,\Omega\) and \(X_L=4\,\Omega\), supplied with \(10\,\text{V}\) (phasor \(10\angle0^\circ\)).
\(Z=3+4j\), \(|Z|=5\,\Omega\), \(\phi=\tan^{-1}\frac43\approx53.13^\circ\).
\(I=\dfrac{10\angle0^\circ}{5\angle53.13^\circ}=2\angle-53.13^\circ\,\text{A}\). The current is 2 A and lags the voltage by about \(53^\circ\), as expected for an inductive circuit.
Common mistakes
- Adding magnitudes of impedances, e.g. \(3+4=7\,\Omega\), instead of adding complex numbers (\(|3+4j|=5\,\Omega\)).
- Giving the capacitor impedance as \(+jX_C\); it is \(-jX_C\).
- Mixing degrees and radians when converting between forms.
How MU asks it
This is a self-learning topic, so it is unlikely to be a long theory question in the end-semester paper; it may appear as an application-based short part or in tutorials. If asked, write the phasor idea, the impedance formulas, convert to polar form, and give the answer with units and the phase angle (leading or lagging).