2.2 Series and Parallel LC Resonance and Q Factor

2.2 Series and Parallel LC Resonance and Q Factor

Resonance is the fundamental phenomenon underlying almost all frequency-selective radio frequency (RF) equipment. From antenna tuning units and transmitter power amplifiers to intermediate frequency (IF) filters and receiver preselectors, resonant LC circuits isolate desired signals from the electromagnetic spectrum while rejecting unwanted frequencies. A thorough mastery of series and parallel resonance, quality factor ($Q$), and circuit bandwidth is vital for advanced amateur radio theory.

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The Physical Condition for Resonance

At resonance, energy alternates between the inductor's magnetic field and the capacitor's electric field. In the ideal resonator their maximum stored energies are equal. Reactive voltages cancel in a series circuit; reactive branch currents cancel in a parallel circuit. For ideal $L$ and $C$, both conditions give:

$$X_L = X_C$$

Mathematical Derivation of the Thomson Resonant Frequency Formula

Substituting the definitions of inductive and capacitive reactance into the equilibrium condition:

$$2\pi f_0 L = \frac{1}{2\pi f_0 C}$$

Multiplying both sides by $f_0$:

$$2\pi f_0^2 L = \frac{1}{2\pi C}$$

Dividing both sides by $2\pi L$:

$$f_0^2 = \frac{1}{4\pi^2 LC}$$

Taking the square root of both sides yields the classical Thomson resonant frequency formula:

$$f_0 = \frac{1}{2\pi \sqrt{LC}}$$

Expressed in terms of angular frequency ($\omega_0 = 2\pi f_0$):

$$\omega_0 = \frac{1}{\sqrt{LC}}$$

At $f_0$, series LC reactances sum to zero, whereas parallel LC susceptances sum to zero. A series resonator therefore has low input impedance; a parallel resonator has high input impedance. Resistance limits these ideal responses.

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Series Resonant Circuits: Low Impedance and Voltage Magnification

In a series resonant circuit, an alternating voltage source drives an inductor, capacitor, and equivalent series resistance ($R$) connected in a single loop.

       Series Resonant Circuit
         +---[ L ]---[ C ]---[ R ]---+
         |                           |
         +----------( ~ )------------+
                    V_in

Impedance and Current Characteristics

The total impedance of the series circuit is:

$$Z_s = R + j(X_L - X_C)$$

• At resonance ($f = f_0$): $X_L = X_C$, so $X_L - X_C = 0$. The impedance collapses to its theoretical minimum: $Z_s = R$. The circuit is purely resistive, the phase angle between generator voltage and current is $0^\circ$, and circuit current reaches its absolute maximum:

$$I_{\text{max}} = \frac{V_{\text{in}}}{R}$$

• Below resonance ($f < f_0$): Capacitive reactance exceeds inductive reactance ($X_C > X_L$). The circuit appears net capacitive; current leads applied voltage.
• Above resonance ($f > f_0$): Inductive reactance exceeds capacitive reactance ($X_L > X_C$). The circuit appears net inductive; current lags applied voltage.

Resonant Voltage Magnification

Because the series current at resonance is limited only by resistance $R$, very large alternating currents can circulate through the inductor and capacitor. The voltage drops across $L$ and $C$ are:

$$V_L = I_{\text{max}} X_L = \left(\frac{V_{\text{in}}}{R}\right) X_L = \left(\frac{X_L}{R}\right) V_{\text{in}} = Q \times V_{\text{in}}$$ $$V_C = I_{\text{max}} X_C = \left(\frac{V_{\text{in}}}{R}\right) X_C = \left(\frac{X_C}{R}\right) V_{\text{in}} = Q \times V_{\text{in}}$$

where $Q = X_L / R$ is the quality factor.

Although $V_L$ and $V_C$ are individually enormous, they are $180^\circ$ out of phase with each other ($+j V_L$ and $-j V_C$), cancelling at any instant across the combined LC pair: $V_L + V_C = 0$.

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Parallel Resonant Circuits (Tank Circuits): High Impedance and Circulating Current

A practical parallel resonant circuit, commonly termed a tank circuit, consists of an inductor with internal winding resistance $r_s$ placed in parallel with a low-loss capacitor $C$.

          I_line -->
    +---------------+------------+
    |               |            |
  ( ~ )           [ r_s ]       [ C ]
 V_line             |            |
    |             [ L ]          |
    |               |            |
    +---------------+------------+

Dynamic Resistance ($R_p$)

The total admittance of the practical parallel circuit is the sum of the inductive branch admittance and capacitive branch admittance:

$$Y = \frac{1}{r_s + j\omega L} + j\omega C = \frac{r_s - j\omega L}{r_s^2 + \omega^2 L^2} + j\omega C$$

Grouping real and imaginary parts:

$$Y = \frac{r_s}{r_s^2 + \omega^2 L^2} + j\left(\omega C - \frac{\omega L}{r_s^2 + \omega^2 L^2}\right)$$

Resonance occurs when the imaginary susceptance term equals zero:

$$\omega_0 C = \frac{\omega_0 L}{r_s^2 + \omega_0^2 L^2} \implies r_s^2 + \omega_0^2 L^2 = \frac{L}{C}$$

At this resonant frequency, the admittance is purely real: $Y_0 = r_s / (r_s^2 + \omega_0^2 L^2) = r_s / (L/C) = C r_s / L$. The equivalent parallel impedance, called the dynamic resistance ($R_p$ or $Z_{\text{dynamic}}$), is the reciprocal of $Y_0$:

$$R_p = \frac{L}{C r_s}$$

Recalling that $Q \approx \omega_0 L / r_s$ and $\omega_0^2 = 1/(LC)$, dynamic resistance can be expressed directly in terms of quality factor:

$$R_p = Q^2 r_s = Q X_L = Q X_C$$

Line Current vs. Circulating Tank Current

• Impedance: At resonance, the parallel tank exhibits maximum impedance ($Z_p = R_p$). Below resonance it is inductive; above resonance it is capacitive (the exact inverse of the series resonant circuit).
• Line Current ($I_{\text{line}}$): The current drawn from the external generator reaches its minimum:

$$I_{\text{line}} = \frac{V_{\text{line}}}{R_p}$$

• Circulating Current ($I_{\text{circ}}$): Energy bounces continuously between the magnetic field of $L$ and the electric field of $C$. The internal circulating current is magnified by $Q$:

$$I_{\text{circ}} = \frac{V_{\text{line}}}{X_L} = \frac{I_{\text{line}} R_p}{X_L} = I_{\text{line}} \left(\frac{Q X_L}{X_L}\right) = Q \times I_{\text{line}}$$

In a high-power transmitter delivering $1000\text{ W}$ into an antenna matching tank where $Q = 15$, circulating tank currents often reach $20\text{ to }30\text{ amperes RMS}$. Tank coils must therefore be wound with heavy copper tubing or silver-plated wire to prevent destructive $I^2 r_s$ heating.

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Circuit Quality Factor ($Q$)

The Quality Factor ($Q$) is a dimensionless parameter quantifying the efficiency of an energy-storage element or resonant system:

$$Q = 2\pi \times \frac{\text{Maximum Energy Stored per Cycle}}{\text{Energy Dissipated per Cycle}}$$

Formulas for Q in Idealized and Practical Configurations

Unloaded Q ($Q_0$) vs. Loaded Q ($Q_L$)

In real RF hardware, we must distinguish between two operating conditions:

1. Unloaded Q ($Q_0$): The intrinsic quality factor of the isolated resonant components, governed almost exclusively by the AC resistance of the inductor (skin effect, core eddy currents, dielectric loss in coil formers):

$$Q_0 = \frac{\omega_0 L}{r_{\text{internal}}}$$

Typical unloaded $Q_0$ values for well-constructed air-wound or toroidal RF inductors range from $150$ to $400$. 2. Loaded Q ($Q_L$): When the resonant circuit is connected to an external source resistance ($R_s$) and load resistance ($R_L$), these external resistances dissipate stored energy, shunting the tank and reducing the effective parallel resistance:

$$R_{\text{total}} = R_p \parallel R_s \parallel R_L$$ $$Q_L = \frac{R_{\text{total}}}{X_L}$$

Finite source and load resistances reduce $R_{\text{total}}$ below $R_p$, so $Q_L<Q_0$. Choose loaded Q for the required bandwidth and matching conditions; there is no general $10$–$20$ range for both transmitters and receivers. For example, a $455\text{ kHz}$ resonator with $10\text{ kHz}$ bandwidth needs $Q_L=45.5$.

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Bandwidth and Selectivity: Half-Power (-3 dB) Points

The bandwidth ($B$ or $\Delta f$) of a tuned circuit describes the frequency span across which the circuit delivers at least half its maximum power (or $1/\sqrt{2} \approx 70.7\%$ of its peak voltage):

$$B = f_2 - f_1 = \frac{f_0}{Q_L}$$

where $f_1$ and $f_2$ are the lower and upper half-power ($-3\text{ dB}$) cutoff frequencies:

$$f_1 \approx f_0 - \frac{B}{2}, \quad f_2 \approx f_0 + \frac{B}{2}$$

Relative voltage
1.000 |                 * (f0)
      |              /     \
0.707 |-----------*-----------*---------  -3 dB
      |          /|           |\
      |         / |           | \
0     +-----------+-----+-----+--------> Frequency
                  f1    f0    f2
                  |<--- B --->|

Shape Factor

Selectivity describes a circuit's ability to discriminate between closely spaced frequencies. While $-3\text{ dB}$ bandwidth defines the passband width, the shape factor describes the steepness ("skirts") of the response curve:

$$\text{Shape Factor} = \frac{\text{Bandwidth at } -60\text{ dB}}{\text{Bandwidth at } -6\text{ dB}}$$

An ideal "brick-wall" filter exhibits a shape factor of $1.0$. A single tuned LC tank has a poor shape factor (frequently exceeding $15$ to $20$), whereas multi-pole crystal or mechanical ladder filters achieve shape factors of $1.5$ to $2.0$.

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Step-by-Step Worked Numerical Example: High-Frequency Tank Circuit

An amateur radio transmitter power amplifier uses an output tank circuit resonant on the 80-meter band at $f_0 = 3.65\text{ MHz}$. The tank uses an inductance of $L = 10.0\ \mu\text{H}$ with a winding RF resistance of $r_s = 2.50\ \Omega$.

Step 1: Calculate the Required Resonating Capacitance ($C$):

$$C = \frac{1}{4\pi^2 f_0^2 L} = \frac{1}{4\pi^2 \times (3.65 \times 10^6)^2 \times (10.0 \times 10^{-6})} \approx 1.901 \times 10^{-10}\text{ F} \approx 190.1\text{ pF}$$

Step 2: Calculate Inductive Reactance ($X_L$):

$$X_L = 2\pi f_0 L = 2\pi \times (3.65 \times 10^6\text{ Hz}) \times (10.0 \times 10^{-6}\text{ H}) \approx 229.3\ \Omega$$

Step 3: Determine the Unloaded Quality Factor ($Q_0$) and Dynamic Resistance ($R_p$):

$$Q_0 = \frac{X_L}{r_s} = \frac{229.3\ \Omega}{2.50\ \Omega} \approx 91.72$$ $$R_p = Q_0^2 r_s = (91.72)^2 \times 2.50\ \Omega \approx 21,031\ \Omega \approx 21.03\text{ k}\Omega$$

Alternatively, verifying with $L / (C r_s)$:

$$R_p = \frac{10.0 \times 10^{-6}}{(1.901 \times 10^{-10}) \times 2.50} \approx 21,041\ \Omega \approx 21.04\text{ k}\Omega$$

Step 4: Calculate the Unloaded Bandwidth ($B_0$):

$$B_0 = \frac{f_0}{Q_0} = \frac{3650\text{ kHz}}{91.72} \approx 39.80\text{ kHz}$$

Step 5: Determine Loaded Q ($Q_L$) and Loaded Bandwidth ($B_L$) when driving a $2500\ \Omega$ load:

Connecting a load of $R_L = 2500\ \Omega$ across the tank places $R_L$ in parallel with $R_p$:

$$R_{\text{total}} = R_p \parallel R_L = \frac{21,031 \times 2500}{21,031 + 2500} = \frac{52,577,500}{23,531} \approx 2234.4\ \Omega$$ $$Q_L = \frac{R_{\text{total}}}{X_L} = \frac{2234.4\ \Omega}{229.3\ \Omega} \approx 9.74$$ $$B_L = \frac{f_0}{Q_L} = \frac{3650\text{ kHz}}{9.74} \approx 374.7\text{ kHz}$$

External resistive loading broadens the bandwidth from $39.8\text{ kHz}$ to $374.7\text{ kHz}$, illustrating how circuit loading directly governs operational selectivity.

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Comparison Table: Series Resonance vs. Parallel Resonance

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Practical Amateur Radio Applications

1. Receiver Front-End Preselectors: Tuned parallel LC circuits installed between the antenna and the RF amplifier attenuate strong out-of-band signals (such as local AM broadcast stations), preventing front-end overload and intermodulation distortion.
2. Transmitter Output Pi-Networks: Resonant tank circuits match the high output impedance of power amplifier devices (tubes or transistors) to standard $50\ \Omega$ coaxial feedlines while attenuating second and higher harmonic emissions.
3. Multiband Antenna Traps: Parallel LC traps embedded along a wire dipole or vertical element present a high impedance at higher band frequencies, effectively isolating the outer antenna sections, while acting as an inductive loading coil on lower frequencies.
4. Variable Frequency Oscillators (VFO): High-$Q$ inductors combined with temperature-stable capacitors establish precise, low-phase-noise reference signals essential for stable radio reception and transmission.