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title: 'Lecture 39: Tank circuit'
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  ## DI02011011: Electronics Circuit and Application (ECA)
  Oscillators
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# DI02011011: Electronics Circuit and Application (ECA)
## Lecture 39: Tank circuit
### Oscillators

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Welcome to Lecture 39. Today we study the LC tank circuit, which forms the frequency-determining backbone of RF oscillators, tuners, and selective amplifiers. We will cover resonance physics, parallel impedance, quality factor Q, damped oscillations, tapped tanks, varactor tuning, and crystal resonators.
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# Agenda: Tank Circuit Fundamentals

- Operational Principles & Flywheel Energy Exchange in LC Tanks
- Derivation of Resonant Frequency $f_0 = \frac{1}{2\pi \sqrt{LC}}$
- Impedance & Admittance Analysis of Ideal vs Practical Tank Circuits
- Quality Factor ($Q$) Definition, Derivation, and Selectivity
- Differential Analysis of Damped Oscillations & Logarithmic Decrement
- Quantitative Numerical Example: Design & Parameter Calculation
- Tapped Tank Circuits: Hartley Inductive and Colpitts Capacitive Dividers
- Electronic Frequency Tuning via Varactor Diodes and Ferrite Cores
- Loading Effects and Loaded Quality Factor ($Q_L$)
- Quartz Crystal as an Ultra-High $Q$ Equivalent LC Tank Circuit
- Summary & Key Engineering Takeaways

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Here is our lecture outline. We will derive the resonance equations, examine practical coil loss, calculate Q and parallel impedance, study tapped tanks, and see how quartz crystals act as ultra-high Q tank equivalents.
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# Operational Principles & Flywheel Energy Exchange

- Energy Storage Elements: Capacitor stores electrostatic energy $E_C = \frac{1}{2} C V^2$; Inductor stores magnetic energy $E_L = \frac{1}{2} L I^2$.
- Flywheel Effect Cycle 1: Fully charged capacitor discharges current $i(t)$ into inductor, building up magnetic flux in inductor coil.
- Flywheel Effect Cycle 2: As capacitor voltage drops to zero, collapsing magnetic field induces electromotive force (EMF) $e = -L \frac{di}{dt}$, recharging capacitor with opposite polarity.
- Self-Sustained Sinusoidal Exchange: In ideal lossless tank ($R_L = 0$), continuous back-and-forth charge circulation maintains constant amplitude sinusoidal voltage waveform.

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An LC tank circuit operates like an electronic flywheel. Electrostatic energy stored in the capacitor's electric field transfers into magnetic energy in the inductor's magnetic field, and vice-versa, creating continuous sinusoidal oscillation.
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# Derivation of Resonant Frequency $f_0$

- Admittance Formulation: Inductive admittance $Y_L = \frac{1}{j\omega L} = -j \frac{1}{\omega L}$; Capacitive admittance $Y_C = j\omega C$.
- Total Parallel Admittance: $Y(j\omega) = Y_L + Y_C = j \left( \omega C - \frac{1}{\omega L} \right)$.
- Resonance Condition: Imaginary part of admittance vanishes: $\text{Im}(Y) = 0 \implies \omega_0 C - \frac{1}{\omega_0 L} = 0$.
- Resonant Frequency Formulas: $\omega_0^2 = \frac{1}{LC} \implies \omega_0 = \frac{1}{\sqrt{LC}} \implies f_0 = \frac{1}{2\pi \sqrt{LC}}$.

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Here we derive the resonant frequency formula. Setting the net imaginary admittance of the parallel combination to zero yields omega_0 = 1/sqrt(LC), giving the fundamental resonance frequency f_0 = 1/(2*pi*sqrt(LC)).
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# Impedance Analysis of Ideal vs Practical Tank Circuit

- Ideal Parallel Tank Impedance: At resonance, $Y(\omega_0) = 0 \implies Z_0 = \frac{1}{Y} \to \infty$ (infinite dynamic impedance, acts as open circuit).
- Practical Coil Resistance: Inductor winding has internal resistance $R_L$. Inductive branch impedance $Z_L = R_L + j\omega L$.
- Practical Parallel Admittance Derivation: $Y = j\omega C + \frac{1}{R_L + j\omega L} = \frac{R_L + j(\omega L - \omega C R_L^2 - \omega^3 L^2 C)}{R_L^2 + \omega^2 L^2}$.
- Equivalent Parallel Resistance ($R_p$): At resonance $\omega_0 \approx \frac{1}{\sqrt{LC}}$, real conductance $G_p = \frac{R_L}{\omega_0^2 L^2} \implies R_p = \frac{L}{R_L C} = Q^2 R_L$.

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While an ideal tank has infinite impedance at resonance, practical inductors have winding resistance R_L. The equivalent parallel resistance R_p at resonance is R_p = L / (R_L * C) = Q^2 * R_L. This finite impedance determines the voltage gain in tuned amplifiers.
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# Quality Factor ($Q$) & Selectivity

- Quality Factor Definition: $Q = 2\pi \frac{\text{Maximum Energy Stored}}{\text{Energy Dissipated per Cycle}} = \frac{\omega_0 L}{R_L} = \frac{1}{\omega_0 C R_L}$.
- Parallel Resistance Form: Expressed in terms of equivalent parallel impedance: $Q = \frac{R_p}{\omega_0 L} = R_p \omega_0 C = R_p \sqrt{\frac{C}{L}}$.
- 3dB Bandwidth Relation: $BW = f_2 - f_1 = \frac{f_0}{Q}$.
- Selectivity & Phase Slope: Higher $Q$ produces narrower 3dB bandwidth, sharper frequency selection, and steeper phase transition slope $\left. \frac{d\phi}{d\omega} \right|_{\omega_0} = \frac{2Q}{\omega_0}$.

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The Quality Factor Q measures energy storage relative to resistive energy loss per cycle. Higher Q values mean lower damping, higher parallel resistance R_p, sharper selectivity, and narrower 3dB bandwidth.
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# Differential Analysis of Damped Oscillations

- Differential Equation: Applying KVL around practical RLC tank: $L \frac{d^2 q}{dt^2} + R_L \frac{dq}{dt} + \frac{q}{C} = 0$.
- Damping Factor & Damped Natural Frequency: Damping coefficient $\alpha = \frac{R_L}{2L}$; damped resonant frequency $\omega_d = \sqrt{\omega_0^2 - \alpha^2} = \omega_0 \sqrt{1 - \frac{1}{4Q^2}}$.
- Transient Voltage Response: Under-damped response ($Q > 0.5$): $v(t) = V_0 e^{-\alpha t} \cos(\omega_d t)$.
- Logarithmic Decrement ($\delta$): Natural logarithm of ratio of successive peak amplitudes: \delta = \ln\left(\frac{V_n}{V_{n+1}}\right) = \alpha T_d \approx \frac{\pi}{Q}.

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Without continuous energy injection from active feedback, internal coil resistance R_L causes oscillations to decay exponentially. The decay rate depends on alpha = R_L / (2L), giving an exponentially damped sine wave envelope.
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# Quantitative Numerical Example: Design & Parameter Calculation

- Problem Statement: An LC tank circuit uses inductor $L = 100\ \mu\text{H}$ with winding resistance $R_L = 5\ \Omega$, and tuning capacitor $C = 200\ \text{pF}$. Calculate $f_0$, $Q$, $R_p$, and $BW$.
- 1. Resonant Frequency: $f_0 = \frac{1}{2\pi \sqrt{100 \times 10^{-6} \times 200 \times 10^{-12}}} = \frac{1}{2\pi \sqrt{2 \times 10^{-14}}} = 1.1254\ \text{MHz}$.
- 2. Reactance & Quality Factor: $X_L = 2\pi f_0 L = 2\pi (1.1254 \times 10^6)(10^{-4}) = 707.1\ \Omega \implies Q = \frac{707.1}{5} = 141.42$.
- 3. Equivalent Parallel Impedance: $R_p = Q^2 R_L = (141.42)^2 \times 5 = 100\ \text{k}\Omega$ (or $R_p = \frac{L}{R_L C} = \frac{10^{-4}}{5 \times 2 \times 10^{-10}} = 100\ \text{k}\Omega$).
- 4. 3dB Bandwidth: $BW = \frac{f_0}{Q} = \frac{1.1254 \times 10^6\ \text{Hz}}{141.42} = 7.958\ \text{kHz}$.

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Let's walk through this complete design calculation. With L = 100 uH, C = 200 pF, and R_L = 5 ohms, we find resonant frequency f_0 = 1.125 MHz, quality factor Q = 141.4, parallel resistance R_p = 100 k-ohms, and bandwidth BW = 7.96 kHz.
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# Tapped Tank Circuits: Inductive & Capacitive Dividers

- Hartley Inductive Tap: Tank inductor split into two series coils $L_1$ and $L_2$ with mutual inductance $M$. Voltage tap fraction $\beta = \frac{L_1 + M}{L_1 + L_2 + 2M}$.
- Colpitts Capacitive Tap: Tank capacitor split into series pair $C_1$ and $C_2$. Voltage tap fraction $\beta = \frac{V_f}{V_{out}} = \frac{C_1}{C_2}$.
- Impedance Transformation Ratio: Tapping acts as step-down transformer ($n = \frac{N_1}{N_2}$ or $\frac{C_2}{C_1}$), matching high tank impedance $R_p$ ($100\ \text{k}\Omega$) to low transistor input impedance ($1\ \text{k}\Omega$).
- Preservation of Tank $Q$: Tapped impedance transformation prevents heavy loading of main tank resonator by low transistor input impedance.

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Tapped tank circuits serve two functions: providing positive feedback voltage division and transforming impedance. By tapping the inductor (Hartley) or capacitor (Colpitts), we match high tank impedance to low transistor input impedance without degrading Q.
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# Electronic Frequency Tuning & Variable Tank Circuits

- Varactor Diode Tuning: Reverse-biased PN junction capacitance $C_v(V_R) = \frac{C_0}{(1 + V_R / V_{bi})^n}$. Varying DC tuning voltage $V_R$ sweeps resonant frequency $f_0(V_R) = \frac{1}{2\pi \sqrt{L C_v(V_R)}}$.
- Permeability Tuning: Mechanically shifting high-permeability ferrite core inside inductor coil alters effective inductance $L = \mu_r L_0$, adjusting $f_0$.
- Switched Capacitor Banks: Digitally switched PIN diode / MOSFET capacitor arrays for wideband multi-band RF synthesizers.
- Air Trimmer Capacitors: Precision mechanical tuning capacitors ($5-30\ \text{pF}$) for high-stability frequency calibration.

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To tune an LC tank electronically, we use varactor diodes. A reverse-biased varactor acts as a voltage-controlled capacitor C(V_R), allowing continuous electronic frequency tuning in Voltage-Controlled Oscillators (VCOs).
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# Loading Effects and Loaded Quality Factor ($Q_L$)

- Unloaded vs Loaded Tank: Unloaded tank impedance is $R_p = Q_{unloaded} \omega_0 L$. Connecting external load $R_{load}$ places $R_{load}$ in parallel with $R_p$.
- Total Parallel Load Resistance: $R_{total} = R_p || R_{load} = \frac{R_p R_{load}}{R_p + R_{load}}$.
- Loaded Quality Factor Formula: $Q_L = \frac{R_{total}}{\omega_0 L} = Q_{unloaded} \cdot \left( \frac{R_{load}}{R_p + R_{load}} \right) = \frac{Q_{unloaded}}{1 + R_p / R_{load}}$.
- Impact on Performance: Heavy loading ($R_{load} < R_p$) drastically reduces $Q_L$, widening bandwidth $BW_L = \frac{f_0}{Q_L}$ and degrading phase noise.

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When an LC tank is connected to a load resistor R_load, the effective parallel resistance drops to R_total = R_p || R_load. This reduces the loaded quality factor Q_L, broadening the bandwidth and degrading phase noise.
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# Quartz Crystal as an Equivalent Ultra-High $Q$ LC Tank

- Piezoelectric Quartz Wafer: Electromechanical coupling converts mechanical vibrations into high-precision electrical resonance.
- Butterworth-Van Dyke (BVD) Model: Motional branch with series $L_m$ (Henries), $C_m$ (femtoFarads), $R_m$ (Ohms), in parallel with static electrode capacitance $C_0$ (picoFarads).
- Dual Resonance Characteristics: Series resonant frequency $f_s = \frac{1}{2\pi \sqrt{L_m C_m}}$; Parallel resonant frequency $f_p = \frac{1}{2\pi \sqrt{L_m \frac{C_m C_0}{C_m + C_0}}}$.
- Ultra-High Quality Factor: Motional inductance $L_m$ is large ($mH$ to $H$) while motional resistance $R_m$ is small ($10-50\ \Omega$), achieving $Q = \frac{\omega_0 L_m}{R_m} \sim 10,000$ to $500,000$.

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Quartz crystals act as ultra-high Q LC tank circuits. Their BVD equivalent circuit shows enormous motional inductance L_m and tiny motional resistance R_m, yielding quality factors Q up to 500,000—orders of magnitude higher than discrete LC tanks.
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# Summary: Key Takeaways on LC Tank Circuits

- Resonance Mechanism: Parallel LC tank provides energy exchange between $C$ and $L$ with resonant frequency $f_0 = \frac{1}{2\pi \sqrt{LC}}$.
- Impedance & Loss: Practical tank has finite equivalent parallel resistance $R_p = \frac{L}{R_L C} = Q^2 R_L$ at resonance.
- Quality Factor Impact: Quality factor $Q = \frac{\omega_0 L}{R_L}$ controls selectivity, loaded bandwidth $BW = \frac{f_0}{Q_L}$, and phase slope.
- Advanced Configurations: Tapped tanks (Hartley/Colpitts) provide impedance transformation; varactor diodes enable electronic tuning; quartz crystals provide ultra-high $Q$ resonance.

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To summarize: LC tank circuits are fundamental resonance building blocks. Understanding R_p, Q factor, tapped impedance matching, varactor tuning, and crystal equivalents allows us to design high-performance RF oscillators and tuned circuits.
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