4.1 RF Oscillators, LC Tank Topologies, and Frequency Stability

4.1 RF Oscillators, LC Tank Topologies, and Frequency Stability

Every radio transmitter and superheterodyne receiver relies upon a stable source of alternating current known as an RF oscillator. An oscillator is an autonomous electronic circuit that converts direct current (DC) energy from a power supply into a periodic, continuous alternating current (AC) waveform at a predetermined radio frequency without requiring an external AC input signal. In modern amateur communications, the spectral purity, frequency accuracy, and thermal stability of the primary oscillator dictate receiver selectivity, adjacent-channel interference rejection, and transmitted signal bandwidth.

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The Physics of Oscillation and the Barkhausen Criterion

An electronic oscillator can be modelled as an amplifier with open-loop voltage gain $A(j\omega)$ coupled to a frequency-selective positive feedback network with transfer function $\beta(j\omega)$.

                     +-------------------+
  Summing Node       |                   |
     ( + ) --------->|  Amplifier A(jω)  |--------+-----> V_out
       ^             |                   |        |
       |             +-------------------+        |
       |                                          |
       |             +-------------------+        |
       |             |                   |        |
       +-------------| Feedback Net β(jω)|<-------+
                     |                   |
                     +-------------------+

The closed-loop transfer function of this feedback system is given by:

$$A_{\text{cl}}(j\omega) = \frac{V_{\text{out}}}{V_{\text{in}}} = \frac{A(j\omega)}{1 - A(j\omega)\beta(j\omega)}$$

For the circuit to generate an output voltage in the absence of an external driving signal ($V_{\text{in}} = 0$ with $V_{\text{out}} \ne 0$), the denominator must approach zero. This establishes the Barkhausen criterion for oscillation, which comprises two mandatory conditions:

1. Loop Gain Magnitude Condition: The magnitude of the loop gain at the operating frequency $\omega_0$ must equal unity:

$$|A(j\omega_0)\beta(j\omega_0)| = 1$$

In practical circuits, to obtain reliable start-up from thermal noise, the small-signal loop gain magnitude must initially exceed unity ($|A\beta| > 1$). As the amplitude of the oscillations increases, non-linearities in the active device (such as transistor saturation, transconductance reduction, or automatic gain control) compress the effective gain until the steady-state loop gain settles precisely to $|A\beta| = 1$. If $|A\beta| < 1$, oscillations die away exponentially; if $|A\beta| > 1$ persists, the output clips heavily against the supply rails, introducing severe harmonic distortion.

1. Loop Phase Shift Condition: The total phase shift around the closed loop at $\omega_0$ must be zero or an integer multiple of $360^\circ$ ($2\pi n$ radians, where $n \in \{0, 1, 2, \dots\}$):

$$\angle A(j\omega_0) + \angle \beta(j\omega_0) = 2\pi n = 0^\circ, 360^\circ, 720^\circ, \dots$$

Most common-emitter bipolar transistor and common-source FET amplifier configurations introduce an intrinsic $180^\circ$ phase inversion between their input (base/gate) and output (collector/drain). Consequently, the passive frequency-determining feedback network must provide an additional $180^\circ$ of phase shift at the desired oscillation frequency $f_0$, ensuring positive (regenerative) feedback.

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Fundamental LC Resonant Tank Topologies

At high frequencies (HF and VHF), resonant tank circuits consisting of inductors ($L$) and capacitors ($C$) serve as the frequency-determining feedback network. Three classic LC topologies dominate amateur radio equipment: the Hartley, Colpitts, and Clapp oscillators. The AC models below use an inverting amplifier whose input is I and output is O, both measured relative to the grounded tap. DC bias and coupling components are omitted; feedback ratios are magnitudes for light tank loading.

1. The Hartley Oscillator

The Hartley oscillator is characterized by a tapped inductor (or two discrete inductors, $L_1$ and $L_2$) connected in parallel with a single tuning capacitor $C$. The tap point is held at AC ground potential, forming an inductive autotransformer voltage divider.

           Hartley AC tank network
       O o---------+-----------+
                   |           |
                 [ L2 ]        |
                   |         [ C ]
      Ground ------o tap       |
                   |           |
                 [ L1 ]        |
                   |           |
       I o---------+-----------+
          L1 and L2 are series-aiding
          magnetically coupled sections.

• Resonant Frequency: The oscillation frequency is determined by the total tank inductance ($L_T$) and the tuning capacitance ($C$):

$$f_0 = \frac{1}{2\pi \sqrt{L_T C}}$$

where $L_T = L_1 + L_2 + 2M$, and $M$ represents the mutual inductance between the two coil sections.

• Feedback Factor: With output O across $L_2$ and input I across $L_1$, the lightly loaded feedback magnitude is:

$$|\beta|\approx\frac{L_1+M}{L_2+M}\approx\frac{N_1}{N_2}=\sqrt{\frac{L_1}{L_2}}\quad\text{for tight coupling}$$

• Operational Attributes: Because the center tap is AC grounded, the RF voltages at opposite ends of the inductor are $180^\circ$ out of phase with each other. This inversion, combined with the transistor's $180^\circ$ common-emitter phase shift, satisfies Barkhausen's $360^\circ$ phase criterion. The Hartley oscillator is simple to tune over a broad frequency range because only a single variable capacitor is required; however, fabricating a tapped inductor with precise mutual coupling is mechanically challenging, and harmonic content can be elevated compared to capacitive-divider circuits.

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2. The Colpitts Oscillator

The Colpitts oscillator replaces the tapped inductor with a tapped capacitive voltage divider comprising two series capacitors, $C_1$ and $C_2$, in parallel with a single continuous inductor $L$. The junction between $C_1$ and $C_2$ is referenced to AC ground.

          Colpitts AC tank network
       O o---------+-----------+
                   |           |
                 [ C1 ]        |
                   |         [ L ]
      Ground ------o tap       |
                   |           |
                 [ C2 ]        |
                   |           |
       I o---------+-----------+

• Equivalent Capacitance and Resonant Frequency: Because $C_1$ and $C_2$ are connected in series across inductor $L$, the equivalent tank capacitance $C_T$ is:

$$C_T = \frac{C_1 C_2}{C_1 + C_2}$$ $$f_0 = \frac{1}{2\pi \sqrt{L C_T}} = \frac{1}{2\pi \sqrt{L \left( \frac{C_1 C_2}{C_1 + C_2} \right)}}$$

• Feedback Factor: The feedback voltage developed across $C_2$ relative to the output voltage across $C_1$ is proportional to their reactive impedances:

$$\beta = \frac{X_{C2}}{X_{C1}} = \frac{\frac{1}{\omega C_2}}{\frac{1}{\omega C_1}} = \frac{C_1}{C_2}$$

• RF Stability and Swamping of Parasitics: The Colpitts topology is exceptionally popular in amateur HF/VHF designs because capacitor $C_2$ is connected directly across the transistor's base-emitter junction (in common-emitter) or gate-source junction (in FET configurations), while $C_1$ parallels the collector-emitter or drain-source output. Because $C_1$ and $C_2$ are intentionally selected to be large (typically hundreds of picofarads), they swamp the active device's small internal parasitic capacitances ($C_{be}, C_{ce} \approx 2\text{ to }10\text{ pF}$). As temperature or supply voltage fluctuates, changes in the internal semiconductor depletion capacitances represent an insignificant fraction of $C_T$, resulting in significantly superior frequency stability compared to the Hartley design.

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3. The Clapp (Gouriet-Clapp) Oscillator

The Clapp oscillator is an advanced refinement of the Colpitts topology designed to overcome the tuning and stability limitations of high-frequency LC tanks. In a standard Colpitts, making $C_1$ and $C_2$ exceedingly large to maximize parasitic swamping forces the equivalent capacitance $C_T$ to become large, requiring an impractically small inductor $L$ with a low loaded $Q$.

The Clapp oscillator solves this problem by inserting a small variable series capacitor ($C_3$) directly in series with the tank inductor $L$, such that $C_3 \ll C_1$ and $C_3 \ll C_2$.

            Clapp AC tank network
       O o---------+-----------+
                   |           |
                 [ C1 ]      [ C3 ]
                   |           |
      Ground ------o tap       |
                   |         [ L ]
                 [ C2 ]        |
                   |           |
       I o---------+-----------+

• Resonant Frequency: The total series equivalent capacitance $C_{\text{eq}}$ is:

$$\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}$$

Because $C_3$ is chosen to be an order of magnitude smaller than $C_1$ and $C_2$ (for example, $C_3 = 15\text{ pF}$ while $C_1 = C_2 = 470\text{ pF}$), the reciprocal summation is completely dominated by $1/C_3$, meaning $C_{\text{eq}} \approx C_3$. The resonant frequency is thus:

$$f_0 \approx \frac{1}{2\pi \sqrt{L C_3}}$$

• Stability Advantages:
  1. Reduced Parasitic Sensitivity: Large $C_1$ and $C_2$ have low reactance. When $C_3\ll C_1,C_2$, changes in capacitance across the divider arms have less effect on the total series capacitance.
  2. Separate Tuning Control: Varying $C_3$ tunes the tank while the ideal ratio $C_1/C_2$ remains fixed. Amplifier gain and loading still change with frequency, so start-up margin must be checked across the range.
  3. Remaining Loading: The active device and following stages still load the tank. A suitable output buffer reduces pulling; the topology alone does not eliminate it.

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Comparison of LC Oscillator Topologies

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Quartz Crystal Oscillators

While discrete LC tanks can achieve loaded $Q$ factors between $50$ and $250$, mechanical and thermal drift limit their long-term frequency stability to roughly $100\text{ to }1000\text{ parts per million}$ (ppm). For channelized amateur communications, repeaters, and digital modes, quartz crystal oscillators are utilized, providing loaded $Q$ values between $10^4$ and $10^6$ and stability figures better than $1\text{ to }10\text{ ppm}$.

The Piezoelectric Effect and Crystal Physics

Quartz (crystalline silicon dioxide, $\text{SiO}_2$) exhibits the piezoelectric effect: when mechanical pressure is applied across specific axes of the crystal lattice, an electric potential appears across its opposing faces. Conversely, when an alternating electric potential is applied across the crystal faces via metal electrodes, the crystal physically deforms, contracting and expanding in acoustic resonance with the applied electric field.

• Crystal Cuts: Raw quartz is sliced into wafer blanks at precise crystallographic angles relative to its optical, electrical, and mechanical axes. The most widely used cut in amateur radio is the AT-cut, which represents a shear-mode slice angled at $+35^\circ 15'$ relative to the optical Z-axis.
• Temperature Characteristic: An AT-cut crystal has an approximately cubic frequency-versus-temperature curve, with an inflection near room temperature. Fine changes in cut angle alter its slope and turning points. A small coefficient near the design temperature does not mean zero drift across the operating range.

The manufacturer's frequency-stability specification gives the allowed deviation over a stated temperature range. Include the oscillator circuit's temperature effects when assessing the complete frequency source; an AT-cut label alone does not establish a particular ppm limit.

Electrical Equivalent Circuit (Butterworth-Van Dyke Model)

Electrically, a quartz crystal resonator behaves as a complex Butterworth-Van Dyke (BVD) network consisting of a series motional arm shunted by a static electrostatic capacitance:

                +---[ L_m ]---[ C_m ]---[ R_m ]---+  (Motional Arm)
                |                                 |
       o--------+                                 +--------o
                |                                 |
                +--------------[ C_0 ]------------+  (Static Shunt Arm)

1. Motional Inductance ($L_m$): Models the vibrating mechanical mass of the quartz wafer. Because quartz is dense, $L_m$ is extraordinarily large, typically ranging from $0.05\text{ H}$ to several henrys ($> 10\text{ H}$). This enormous inductance cannot be physically replicated with wire coils in comparable volumes.
2. Motional Capacitance ($C_m$): Models the mechanical elasticity (compliance) of the quartz plate. It is minuscule, typically measuring fractions of a picofarad ($0.005\text{ pF}$ to $0.05\text{ pF}$, or $5\text{ to }50\text{ fF}$).
3. Motional Resistance ($R_m$): Models the internal mechanical frictional losses (damping) of the acoustic wave. It is small, typically $10\ \Omega$ to $50\ \Omega$ for HF fundamental crystals.
4. Static Shunt Capacitance ($C_0$): Models the real electrostatic capacitance formed by the metal electrode platings deposited on the quartz blank, separated by the quartz dielectric, plus lead and holder packaging parasitics. It typically measures $2\text{ pF}$ to $7\text{ pF}$.

Series and Parallel Resonant Frequencies

Because of its BVD structure, every quartz crystal possesses two distinct resonant frequencies spaced closely together:

1. Series Resonant Frequency ($f_s$): Occurs when the inductive reactance of $L_m$ cancels the capacitive reactance of $C_m$ ($X_{Lm} = X_{Cm}$):

$$f_s = \frac{1}{2\pi \sqrt{L_m C_m}}$$

At $f_s$, the motional arm exhibits zero reactance, and the crystal's terminal impedance collapses to its minimal value—the purely resistive motional resistance $R_m$ ($10\text{ to }50\ \Omega$). Current through the crystal reaches a sharp maximum.

1. Parallel (Anti-Resonant) Frequency ($f_p$): Occurs when the net inductive reactance of the motional arm resonates with the static shunt capacitance $C_0$. The equivalent series capacitance of the parallel loop is $C_{\text{parallel}} = (C_m C_0) / (C_m + C_0)$:

$$f_p = \frac{1}{2\pi \sqrt{L_m \left( \frac{C_m C_0}{C_m + C_0} \right)}} = f_s \sqrt{1 + \frac{C_m}{C_0}} \approx f_s \left( 1 + \frac{C_m}{2 C_0} \right)$$

At $f_p$, the terminal impedance reaches an immense peak ($> 100\text{ k}\Omega$). Because $C_m \ll C_0$, the parallel resonance is typically only $0.1\%$ to $0.3\%$ ($10\text{ to }30\text{ kHz}$ at $10\text{ MHz}$) above $f_s$.

Fundamental vs. Overtone Modes

The fundamental mechanical resonance frequency of an AT-cut crystal is inversely proportional to the physical thickness ($t$) of the quartz wafer:

$$f_0 \approx \frac{1.67\text{ MHz}\cdot\text{mm}}{t}$$

For a $10\text{ MHz}$ fundamental crystal, $t \approx 0.167\text{ mm}$ ($167\ \mu\text{m}$). At frequencies above $30\text{ MHz}$, a fundamental wafer must be ground thinner than $50\ \mu\text{m}$, making it fragile, difficult to handle, and vulnerable to shattering under moderate RF drive levels.

To operate at VHF ($30\text{ to }200\text{ MHz}$), crystals are excited in overtone modes. Because the faces of the crystal must maintain opposite electrical charges, acoustic standing waves can only be sustained across odd mechanical multiples:

• 3rd Overtone: Operates from approximately $30\text{ MHz}$ to $90\text{ MHz}$.
• 5th Overtone: Operates from approximately $80\text{ MHz}$ to $150\text{ MHz}$.
• 7th Overtone: Operates up to approximately $200\text{ MHz}$.

The Pierce Crystal Oscillator Configuration

The Pierce oscillator is the most widely utilized crystal circuit in amateur radio, microcontrollers, and frequency synthesizers due to its minimal component count, exceptional stability, and low phase noise.

             R_F (bias feedback)
        +--------[R_F]---------+
        |                      |
        +----[ CMOS inverter ]-+----> Buffered output stage
        |       (inverting)    |
        |                   [R_S]
        |                      |
        +------[Crystal]-------+
        |                      |
     [C_L1]                 [C_L2]
        |                      |
       GND                    GND

• Circuit Architecture: In the CMOS Pierce implementation shown, an unbuffered inverter supplies gain. The high-value $R_F$ (commonly $1$–$10\text{ M}\Omega$) biases it near its linear operating point. Choose $R_S$ for adequate start-up gain and the crystal's permitted drive level; take the usable output through a separate buffer.
• The Phase-Shift Network: The crystal bridges inverter output and input and operates in its inductive region between $f_s$ and $f_p$. Together with the two load capacitors to ground, it forms the frequency-selective feedback network.
• Meeting Barkhausen: At the operating frequency, the inverter and feedback network together provide a full-cycle phase shift and unity steady-state loop gain. Device delay, resistance and loading affect the exact frequency and start-up margin.
• Load Capacitance ($C_L$): Crystal manufacturers calibrate parallel crystals for a specific circuit load capacitance ($C_L$, typically $16\text{ pF}, 20\text{ pF},\text{ or }30\text{ pF}$):

$$C_L = \frac{C_{L1} C_{L2}}{C_{L1} + C_{L2}} + C_{\text{stray}}$$

where $C_{\text{stray}}$ includes board trace and device pin capacitances (typically $3\text{ to }5\text{ pF}$). Trimming $C_{L1}$ or $C_{L2}$ pulls the oscillation frequency slightly (a process known as "rubbering"), permitting fine calibration.

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Frequency Stability, Phase Noise, and Drift Mitigations

An ideal oscillator produces a single, infinitely narrow spectral impulse at frequency $f_0$. Real-world oscillators suffer from short-term and long-term frequency instability, spreading RF power into adjacent sidebands and degrading transceiver performance.

Primary Causes of Frequency Instability

1. Thermal Variations: Temperature changes cause physical expansion of inductors, altering their cross-sectional area and winding pitch ($L \propto r^2$). Thermal drift also alters the dielectric constant of capacitors and shifts the built-in barrier potential and depletion layer widths of active semiconductor junctions.
2. Mechanical Vibration and Shock (Microphonics): Acoustic sound waves or physical bumping alter the physical spacing between capacitor plates and flex inductor turns, generating instantaneous frequency modulation.
3. Supply Voltage Fluctuations (Voltage Pushing): Changes in DC rail voltage alter the reverse-bias voltage across transistor collector-base and collector-emitter junctions, modulating the device's parasitic depletion capacitances ($C_{cb}, C_{ce}$) and shifting the tank frequency.
4. Load Impedance Changes (Load Pulling): Changes in the input impedance of subsequent stages (such as when a power amplifier transitions from receive to transmit, or when an antenna tuner adjusts) reflect back into the resonant tank through the active device's reverse transfer capacitance, pulling the oscillation frequency.
5. Phase Noise: Short-term, rapid random phase fluctuations caused by thermal noise ($4kTR$), semiconductor shot noise, and flicker ($1/f$) noise in the active device, producing continuous noise sidebands around the carrier.

Engineering Mitigations in Amateur Equipment

• Temperature Compensation with NP0/C0G Capacitors: High-stability LC tanks utilize ceramic capacitors with NP0 (Negative-Positive-Zero, EIA code C0G) dielectric formulations. NP0 capacitors maintain zero temperature coefficient (drift within $\pm 30\text{ ppm}/^\circ\text{C}$ from $-55^\circ\text{C}$ to $+125^\circ\text{C}$). Where copper inductors exhibit an unavoidable positive temperature coefficient, engineers deliberately pair them with controlled negative-temperature-coefficient capacitors (such as N750, $-750\text{ ppm}/^\circ\text{C}$) to establish complete thermal cancellation.
• Active Buffer Amplifiers: A buffer after the oscillator reduces load pulling. It should load the tank lightly and provide good reverse isolation at the operating frequency. Suitable arrangements include source followers, emitter followers and dual-gate MOSFET stages; actual isolation depends on the device, bias and layout.
• DC Supply Stabilization: Regulation and decoupling reduce frequency pushing from supply changes and PA current surges. Choose a suitable supply voltage and regulator noise performance. Regulators have finite ripple rejection, so local bypassing and sensible supply routing remain necessary.
• Rigid Mechanical Construction: Tank components must be housed in thick, milled aluminum die-cast enclosures. Inductors are wound on rigid ceramic or powdered-iron toroidal formers and potted with high-grade, low-loss dielectric varnish to prevent mechanical microphonics.

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Worked Engineering Calculations

Example 1: Colpitts Oscillator Tank and Feedback Calculation

Problem: A nominal $10\text{ MHz}$ Colpitts oscillator uses an inductor $L = 2.50\ \mu\text{H}$ and a capacitive divider consisting of $C_1 = 120\text{ pF}$ and $C_2 = 680\text{ pF}$. Calculate:

1. The equivalent tank capacitance ($C_T$).
2. The nominal resonant oscillation frequency ($f_0$).
3. The feedback factor ($\beta$) and the minimum active device voltage gain ($|A_{\text{min}}|$) required to initiate oscillation.

Solution:

1. Equivalent Tank Capacitance:

$$C_T = \frac{C_1 C_2}{C_1 + C_2} = \frac{120\text{ pF} \times 680\text{ pF}}{120\text{ pF} + 680\text{ pF}} = \frac{81600}{800}\text{ pF} = 102.0\text{ pF}$$

1. Resonant Frequency:

$$f_0 = \frac{1}{2\pi \sqrt{L C_T}} = \frac{1}{2\pi \sqrt{(2.50 \times 10^{-6}\text{ H}) \times (102.0 \times 10^{-12}\text{ F})}}$$ $$\sqrt{L C_T} = \sqrt{2.550 \times 10^{-16}} \approx 1.59687 \times 10^{-8}\text{ s}$$ $$f_0 = \frac{1}{2\pi \times 1.59687 \times 10^{-8}} \approx \frac{1}{1.00334 \times 10^{-7}} \approx 9.9667 \times 10^6\text{ Hz} = 9.967\text{ MHz}$$

1. Feedback Factor and Required Gain:

$$\beta = \frac{C_1}{C_2} = \frac{120\text{ pF}}{680\text{ pF}} \approx 0.1765 \quad (-15.06\text{ dB})$$

The ideal unity-loop-gain threshold is $|A\beta|=1$:

$$|A_{\text{min}}| = \frac{1}{\beta} = \frac{1}{0.1765} \approx 5.67 \quad (+15.06\text{ dB})$$

In this lightly loaded model, $5.67$ is the unity-loop-gain threshold. The small-signal gain must exceed it for reliable start-up; amplitude limiting then reduces the effective loop gain to unity.

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Example 2: Clapp vs. Colpitts Sensitivity to Transistor Capacitance Drift

Problem: Compare the frequency drift of the Colpitts oscillator from Example 1 ($C_1 = 120\text{ pF}, C_2 = 680\text{ pF}, C_T = 102.0\text{ pF}$) against a Clapp oscillator having identical $C_1$ and $C_2$, but with a small series capacitor $C_3 = 15.0\text{ pF}$ and an inductor $L_{\text{Clapp}} = 17.5\ \mu\text{H}$ chosen to resonate near $10.5\text{ MHz}$. Assume thermal heating of the transistor causes its base-emitter capacitance across $C_2$ to increase by $\Delta C_{\text{in}} = +3.0\text{ pF}$.

Solution:

1. Colpitts Shift:
   $C_2$ increases from $680\text{ pF}$ to $683\text{ pF}$. The new equivalent capacitance is:

$$C_T' = \frac{120 \times 683}{120 + 683} = \frac{81960}{803} \approx 102.0672\text{ pF}$$ $$\Delta C_T = 102.0672\text{ pF} - 102.0000\text{ pF} = +0.0672\text{ pF}$$

The fractional frequency change is:

$$\frac{\Delta f}{f_0} \approx -\frac{1}{2} \frac{\Delta C_T}{C_T} = -\frac{1}{2} \left( \frac{0.0672}{102.0} \right) \approx -3.294 \times 10^{-4} = -329.4\text{ ppm}$$

At $f_0 \approx 9.967\text{ MHz}$, the frequency shift is:

$$\Delta f = -329.4\text{ ppm} \times 9.967\text{ MHz} \approx -3,283\text{ Hz} \quad (-3.28\text{ kHz})$$

1. Clapp Shift:
   The original equivalent capacitance of the Clapp circuit was:

$$\frac{1}{C_{\text{eq}}} = \frac{1}{102.0\text{ pF}} + \frac{1}{15.0\text{ pF}} = 0.0098039 + 0.0666667 = 0.0764706\text{ pF}^{-1} \implies C_{\text{eq}} \approx 13.0769\text{ pF}$$

With $C_2$ shifting to $683\text{ pF}$, the reciprocal series capacitance becomes:

$$\frac{1}{C_{\text{eq}}'} = \frac{1}{102.0672\text{ pF}} + \frac{1}{15.0\text{ pF}} = 0.0097975 + 0.0666667 = 0.0764642\text{ pF}^{-1} \implies C_{\text{eq}}' \approx 13.0780\text{ pF}$$ $$\Delta C_{\text{eq}} = 13.0780\text{ pF} - 13.0769\text{ pF} = +0.0011\text{ pF}$$

The fractional frequency change is:

$$\frac{\Delta f}{f_0} \approx -\frac{1}{2} \left( \frac{0.0011}{13.0769} \right) \approx -4.206 \times 10^{-5} = -42.1\text{ ppm}$$

At $f_0 \approx 10.5\text{ MHz}$, the frequency shift is:

$$\Delta f = -42.1\text{ ppm} \times 10.5\text{ MHz} \approx -442\text{ Hz}$$

Conclusion: For these component values and the assumed $+3\text{ pF}$ change across $C_2$, the ideal Clapp model shifts about $442\text{ Hz}$ rather than $3283\text{ Hz}$. This illustrates reduced divider-capacitance sensitivity; it does not compare coil drift, loading or complete oscillator stability.

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Example 3: Quartz Crystal BVD Equivalent Circuit and Q Factor

Problem: An AT-cut quartz resonator for a local oscillator has these BVD parameters:

• Motional inductance: $L_m=0.850\text{ H}$.
• Motional capacitance: $C_m=0.0152\text{ pF}=1.52\times10^{-14}\text{ F}$.
• Motional resistance: $R_m=28.0\ \Omega$.
• Static shunt capacitance: $C_0=4.20\text{ pF}$.

Calculate its series resonance, parallel anti-resonance and unloaded Q.

1. Series resonance:

$$\sqrt{L_mC_m}=\sqrt{0.850\times1.52\times10^{-14}}=\sqrt{1.292\times10^{-14}}\approx1.13666\times10^{-7}\text{ s}$$ $$f_s=\frac{1}{2\pi\sqrt{L_mC_m}}\approx1.40020\times10^6\text{ Hz}=1.40020\text{ MHz}$$

2. Parallel anti-resonance:

$$f_p=f_s\sqrt{1+\frac{C_m}{C_0}}\approx1.400196\text{ MHz}\sqrt{1+\frac{0.0152}{4.20}}\approx1.40273\text{ MHz}$$

The separation is approximately $2.53\text{ kHz}$. Between these resonances the crystal presents inductive reactance, enabling operation with an external load capacitance.

3. Unloaded Q:

$$Q=\frac{2\pi f_sL_m}{R_m}=\frac{2\pi(1.400196\times10^6)(0.850)}{28.0}\approx267073\approx2.67\times10^5$$

This is roughly $1335$ times the Q of an LC tank with $Q=200$, or about three orders of magnitude higher. Such high Q improves frequency discrimination, while temperature, loading and drive level still affect the complete oscillator.