7.2 Phasors, the Series LCR Circuit and Resonance

Turn the tuning knob of an old radio and you are turning a variable capacitor. Hundreds of stations broadcast at once, and the aerial picks up all of them; yet the speaker plays just one. The tuner is a coil and a capacitor in series with the small resistance of the wires, and it responds strongly to one frequency only — the frequency at which the reactance of the coil equals that of the capacitor. Move the capacitor plates and that frequency moves, and with it the station you hear. The same resonance selects the channel in a phone, a Wi-Fi card and a wireless charger, and must be avoided in power networks, where it can build up dangerously large voltages.

In Section 7.1 each component was alone. Now we join a resistor, an inductor and a capacitor in series to one source. The same current flows through all three, but their voltages are out of step with one another, so they cannot simply be added as numbers. Phasors solve the problem: we add the voltages as arrows, find the impedance $Z$ and the phase angle, and then study resonance and how sharp it is — measured by the bandwidth and the quality factor $Q$.

The Series LCR Circuit and Phasor Addition

A source $v=v_m\sin\omega t$ drives a resistor $R$, an inductor $L$ and a capacitor $C$ in series. There is only one path, so the same current $i=i_m\sin(\omega t+\phi)$ flows through every element; $\phi$ is the phase of the current relative to the source voltage, still to be found. From Section 7.1, the peak voltages are $v_{Rm}=i_mR$, $v_{Lm}=i_mX_L$, $v_{Cm}=i_mX_C$, and relative to the current:

• $v_R$ is in phase with $i$;
• $v_L$ leads $i$ by $\pi/2$;
• $v_C$ lags $i$ by $\pi/2$.

Kirchhoff's loop rule holds at every instant: $v=v_R+v_L+v_C$. Instantaneous values are vertical projections of phasors, and the projection of a sum of arrows is the sum of their projections, so the phasor of $v$ is the arrow sum of the three voltage phasors. Draw the current phasor $\mathbf I$ as reference. $\mathbf V_R$ lies along it; $\mathbf V_L$ points $90^\circ$ ahead and $\mathbf V_C$ $90^\circ$ behind, so these two are exactly opposite and combine into one arrow of length $|v_{Lm}-v_{Cm}|$ perpendicular to $\mathbf V_R$. By Pythagoras, $$v_m^2=v_{Rm}^2+(v_{Cm}-v_{Lm})^2$$ $$=i_m^2\left[R^2+(X_C-X_L)^2\right].$$

The same results hold for any two of the three elements: put $X_C=0$ for an RL circuit (no capacitor) and $X_L=0$ for an RC circuit (no inductor). Note that the rms voltages across the parts do not add up to the supply voltage — they add as arrows.

Resonance

As the frequency rises, $X_L=\omega L$ grows and $X_C=1/\omega C$ falls. At low frequency the capacitor dominates (capacitive circuit, current leading); at high frequency the coil dominates (inductive, current lagging). In between there is one frequency at which the two reactances are equal and cancel: the impedance is then as small as it can be, $Z=R$, and the current is at its largest.

Resonance is useful because the current is large only near $\omega_0$. A tuning circuit with an adjustable capacitor responds strongly to the station whose frequency equals its $\omega_0$ and weakly to the others. It can also be dangerous: in a power circuit near resonance the voltage across a capacitor or a coil can be many times the supply voltage and break down the insulation.

Sharpness of Resonance: Bandwidth and Quality Factor

How selective is a resonant circuit? Away from $\omega_0$ the current falls, and it falls faster when $R$ is small. A convenient measure uses the half-power frequencies $\omega_1\lt\omega_0\lt\omega_2$ at which the current amplitude has fallen to $1/\sqrt2$ of its peak value. Since the power in $R$ is proportional to $I^2$, the power there is half its maximum. At these frequencies $Z=\sqrt2R$, so $|X_L-X_C|=R$: $$\omega L-\frac{1}{\omega C}=\pm R\quad\Longrightarrow\quad L\omega^2\mp R\omega-\frac1C=0 .$$ Taking the positive root of each quadratic, $$\omega_{2,1}=\frac{\pm R+\sqrt{R^2+4L/C}}{2L},$$ and the difference is exactly $\omega_2-\omega_1=R/L$.

Exercises