Oscillators

Fundamentals

Oscillator vs Amplifier

An oscillator converts DC-supply power into a periodic AC output without any external periodic input. It contains an active gain element, a frequency-selective feedback network and an amplitude-limiting mechanism. Noise or the switch-on transient supplies the seed; the DC supply—not the feedback network—supplies the energy.

• Amplifier: needs a periodic input; output amplitude tracks the input; output frequency equals the input frequency.

• Oscillator: needs no input after start-up; amplitude is set by limiting/AGC; frequency is set by an $RC$, $LC$ or crystal network.

• An oscillator is essentially an amplifier with positive feedback that regenerates its own input.

Feature	Amplifier	Oscillator
Periodic input	Required for a periodic output	Not required after startup
Energy source	DC supply, controlled by input	DC supply, controlled by the loop
Feedback	Often negative for accuracy and linearity	Regenerative at one selected frequency
Frequency	Follows the applied signal within bandwidth	Set by RC, LC or crystal network
Amplitude	Proportional to input until limited	Set by AGC or nonlinearity
Principal design test	Gain, bandwidth, noise and distortion	Startup, loop phase, amplitude and stability

Oscillator and amplifier compared.

Positive Feedback

For forward gain $A(s)$ and feedback fraction $\beta(s)$ returned in the reinforcing sense, the loop obeys

$$x_o=A(s)\bigl[x_s+\beta(s)x_o\bigr], \qquad \frac{x_o}{x_s}=\frac{A(s)}{1-A(s)\beta(s)}.$$

A self-sustaining response (input removed) exists when the characteristic equation

$$\boxed{1-A(s)\beta(s)=0}$$

has a mode on the imaginary axis.

• Loop gain $A\beta$ is the fraction of output returned per round trip.

• Feedback must be regenerative (in phase) and selective, so only one frequency is reinforced.

Native oscillator loop and the distinction between startup growth and steady amplitude.

Barkhausen Criterion

For sustained oscillation at $\omega_0$ the loop gain must satisfy:

Caution

Key Formula — Barkhausen Conditions at $\omega_0$

$$\boxed{|A(j\omega_0)\beta(j\omega_0)|=1}, \qquad \boxed{\angle A(j\omega_0)\beta(j\omega_0)=2\pi k}, \quad k\in\mathbb Z.$$

• Magnitude $|A\beta|=1$: the returned signal exactly replaces per-cycle losses.

• Phase $=0^\circ$ (or $360^\circ$): the returned signal is in phase.

• Start-up: design $|A\beta|>1$ so noise grows; amplitude limiting then pulls the average loop gain down to unity:

$$\begin{array}{ccl} |A\beta|>1 & \Rightarrow & \text{amplitude grows}, \\ |A\beta|=1 & \Rightarrow & \text{steady sinusoid sustained}, \\ |A\beta|<1 & \Rightarrow & \text{amplitude decays}. \end{array}$$

Amplitude is settled by one of:

• a lamp or thermistor in the amplifier feedback path;

• a detector-controlled JFET/OTA or other automatic gain control (AGC);

• smooth transistor gain compression; or

• back-to-back diodes/Zeners or another explicit limiter.

Note

Key Point

Barkhausen is a necessary steady-state condition, not a sufficient one: it does not by itself guarantee start-up, a unique frequency, low distortion or a stable amplitude. Reliable design needs $|A\beta|>1$ at start-up plus amplitude control. (With a $1+A\beta$ summing convention the same physical condition reads $A\beta=-1$.)

Classification

• By waveform: sinusoidal ($RC$, $LC$, crystal) or non-sinusoidal / relaxation (multivibrator, UJT, 555 — square, triangular, sawtooth).

• By frequency-determining network:

  • $RC$ — Wien bridge, RC phase-shift (audio, low frequency);

  • $LC$ — Hartley, Colpitts, Clapp (radio frequency);

  • Crystal — Pierce, Miller (very high stability, fixed frequency);

  • Negative-resistance — tunnel diode, UJT (microwave/relaxation).

• By tuning: fixed, variable (VFO) or voltage-controlled (VCO).

Family	Examples	Typical range	Stability
$RC$	Wien bridge, phase-shift	$\sim 1\,\mathrm{Hz}$–$1\,\mathrm{MHz}$	Moderate
$LC$	Hartley, Colpitts, Clapp	$\sim 100\,\mathrm{kHz}$–hundreds of MHz	Good
Crystal	Pierce, Miller	fixed; kHz–tens of MHz	Excellent

Sinusoidal oscillator families at a glance.

Resonant Circuits

Resonance occurs when inductive and capacitive reactances cancel, leaving a purely resistive terminal impedance. It is the frequency-selecting basis of every $LC$ oscillator.

Resonant Frequency

For ideal $L,C$, reactance cancellation $\omega_0L=1/(\omega_0C)$ gives

$$\omega_0L=\frac1{\omega_0C}, \qquad \boxed{\omega_0=\frac1{\sqrt{LC}}}, \qquad \boxed{f_0=\frac1{2\pi\sqrt{LC}}}.$$

Series and parallel connections then behave oppositely at $\omega_0$.

Native series and ideal parallel RLC models. The location of the loss resistance determines the applicable $Q$ formula.

Series Resonance

With total series resistance $R_s$,

$$Z_s=R_s+j\left(\omega L-\frac1{\omega C}\right).$$

• At $\omega_0$: $Z=R_s$ is minimum, current is maximum, power factor $=1$.

• $V_L$ and $V_C$ are equal and opposite, each $Q_s$ times the source voltage — voltage magnification.

Caution

Key Formula — Series-RLC Quality and Magnification

$$\boxed{Q_s=\frac{\omega_0L}{R_s} =\frac1{\omega_0CR_s}=\frac1{R_s}\sqrt{\frac LC}}, \qquad \boxed{|V_L|=|V_C|=Q_sV}.$$

Half-power points ($I=I_{max}/\sqrt2$) give $|\omega L-1/(\omega C)|=R_s$, hence

$$\begin{aligned} \omega_{1,2} & =\sqrt{\omega_0^2+\left(\frac{R_s}{2L}\right)^2} \mp\frac{R_s}{2L}, \\ \omega_1\omega_2 & =\omega_0^2, & \Delta\omega & =\omega_2-\omega_1=\frac{R_s}{L}, \\ \boxed{BW} & =f_2-f_1=\frac{f_0}{Q_s}. \end{aligned}$$

Parallel Resonance

For $R_p,L,C$ in parallel,

$$Y_p=\frac1{R_p}+j\left(\omega C-\frac1{\omega L}\right).$$

• At $\omega_0$: $Z=R_p$ is maximum, line current is minimum.

• Equal, opposite reactive currents circulate in the tank — current magnification.

Caution

Key Formula — Ideal Parallel-RLC Quality and Magnification

$$\boxed{Q_p=\frac{R_p}{\omega_0L}=\omega_0CR_p =R_p\sqrt{\frac CL}}, \qquad \boxed{\frac{I_C}{I_R}=\frac{I_L}{I_R}=Q_p}.$$$$\boxed{\Delta\omega=\frac1{R_pC}}, \qquad \boxed{BW=\frac{f_0}{Q_p}}.$$

For a practical tank with coil series loss $r_s$ (branch $r_s+j\omega L$ in parallel with $C$), setting total susceptance to zero gives

$$\boxed{\omega_r=\sqrt{\frac1{LC}-\frac{r_s^2}{L^2}}}, \qquad \boxed{R_{p,eq}=\frac{L}{Cr_s}}.$$

For a high-$Q$ coil, $\omega_r\simeq\omega_0$ and

$$Q\simeq\frac{\omega_0L}{r_s}, \qquad R_{p,eq}\simeq\frac{(\omega_0L)^2}{r_s}.$$

Note

Key Point

Do not substitute the small series coil loss $r_s$ into the ideal parallel formula as if it were the parallel loss $R_p$; convert first with $R_{p,eq}\simeq(\omega_0L)^2/r_s$.

Native normalized response plots for $Q=5$: series impedance has a minimum, while ideal parallel impedance has a maximum.

Q-factor and Bandwidth

$Q=2\pi\,\dfrac{\text{energy stored}}{\text{energy lost per cycle}}$; it measures selectivity. Higher $Q$ means a sharper response and a narrower bandwidth.

• Half-power bandwidth: $\boxed{BW=f_0/Q}$ for both series and parallel resonance.

• Higher $Q$ $\Rightarrow$ narrower $BW$, sharper selectivity, larger internal $V$/$I$ stress, slower settling.

• Whether a resonator acts band-pass or band-stop depends on where it is inserted and where the output is taken.

Property	Series resonance	Parallel resonance
Terminal quantity	$Z$ minimum, $Y$ maximum	$Z$ maximum, $Y$ minimum
Source current	Maximum	Minimum
Internal magnification	Voltage across $L,C$	Circulating $L,C$ current
Loss-resistance model	Series $R_s$	Parallel $R_p$
Typical role	Selective current path, matching	Oscillator/tuned-amplifier tank
Lossless limit	$Z(\omega_0)=0$	$Z(\omega_0)\to\infty$

Series and parallel resonance compared.

Wien Bridge Oscillator

Circuit

• A non-inverting op-amp with two feedback paths.

• Positive feedback: a series $RC$ and a parallel $RC$ (lead–lag network) to the $+$ input — frequency selective.

• Negative feedback: a resistive divider $R_f,R_g$ to the $-$ input — sets gain.

Native op-amp Wien-bridge oscillator with distinct frequency-selective positive feedback and gain-setting negative feedback.

Working

• The lead–lag network gives zero phase shift and maximum transfer at one frequency $f_0$.

• There, the amplifier’s $0^\circ$ plus the network’s $0^\circ$ satisfy the Barkhausen phase condition.

• Noise at $f_0$ is reinforced; other frequencies are phase-shifted/attenuated and die out.

Frequency Derivation

With $s=j\omega$ and equal components,

$$Z_s=R+\frac{1}{sC}=\frac{1+sRC}{sC}, \qquad Z_p=R\parallel\frac{1}{sC}=\frac{R}{1+sRC}.$$

The returned fraction is

$$\begin{aligned} \beta(s) & =\frac{Z_p}{Z_s+Z_p} =\frac{sRC}{s^2R^2C^2+3sRC+1}, \\ \beta(j\omega) & = \frac{1}{3+j\left(\omega RC-\dfrac{1}{\omega RC}\right)}. \end{aligned}$$

Zero bridge phase requires $\omega_0RC=1/(\omega_0RC)$.

Gain Condition

Caution

Key Formula — Equal-Component Wien Conditions

$$\boxed{f_0=\frac{1}{2\pi RC}},\qquad \boxed{\beta(\omega_0)=\frac13},\qquad \boxed{A_v=3}.$$

For a non-inverting op-amp,

$$A_v=1+\frac{R_f}{R_g}=3 \quad\Rightarrow\quad \boxed{R_f=2R_g}.$$

For unequal arms $R_1,C_1$ (series) and $R_2,C_2$ (shunt),

$$\beta(s)=\frac{sR_2C_1} {1+s(R_1C_1+R_2C_1+R_2C_2)+s^2R_1R_2C_1C_2}.$$

Caution

Key Formula — General Wien Balance

$$\boxed{f_0=\frac{1}{2\pi\sqrt{R_1C_1R_2C_2}}}, \qquad \boxed{A_v=1+\frac{R_1}{R_2}+\frac{C_2}{C_1}}.$$

Amplitude Stabilization

• Set small-signal gain slightly above 3 so oscillation starts.

• Lamp/thermistor in the $R_g$ leg lowers gain smoothly toward 3 as amplitude grows — lowest distortion.

• Back-to-back diodes clamp amplitude — compact but add harmonics.

• JFET/AGC gives electronic level control over a wide tuning range.

Frequency selectivity: below $f_0$ the series $C$ blocks feedback (bridge leads); at $f_0$ the phase is zero and $\beta=1/3$; above $f_0$ the shunt $C$ grounds the node (bridge lags). A convenient audio source with ganged tuning; high-frequency use is limited by op-amp gain–bandwidth and slew rate.

RC Phase-Shift Oscillator

• Circuit: one inverting stage (CE/CS/op-amp) plus a three-section $RC$ ladder in the feedback path.

• Working: the amplifier gives $180^\circ$ and the loaded three-section ladder adds the other $180^\circ$ at $f_0$, giving $360^\circ$ loop phase. ($60^\circ$ per section is only a mnemonic — sections load one another.)

Native FET and BJT forms of the three-section RC phase-shift loop.

For the standard equal-$R$, equal-$C$, unbuffered lag ladder (low-resistance drive, high-resistance input, negligible device capacitance),

Caution

Key Formula — Standard RC Phase-Shift Conditions

$$\boxed{f_0=\frac{1}{2\pi RC\sqrt6}}, \qquad \boxed{|\beta(\omega_0)|=\frac1{29}}, \qquad \boxed{|A|\geq29\ \text{for startup}}.$$

• Set $|A|>29$ for start-up; limiting brings the average loop gain to unity.

• Simple and inductor-free, but high attenuation needs high gain, tuning is awkward, and stability/distortion are poorer than a Wien bridge.

Variants (do not mix with the standard formula):

• an equal-component lead (CR) ladder has $f_0=\sqrt6/(2\pi RC)$ under corresponding ideal loading;

• followers between equal lag sections remove mutual loading, so each section is a true $60^\circ$, $f_0=\sqrt3/(2\pi RC)$, $|\beta|=1/8$ and required gain is about 8;

• a directly loaded single-BJT form gives

$$f_0=\frac{1}{2\pi RC\sqrt{6+4R_C/R}},\qquad h_{fe}\geq23+29\frac{R}{R_C}+4\frac{R_C}{R}.$$

Property	Wien bridge	RC phase shift
Amplifier	Non-inverting	Inverting
Network phase at $f_0$	$0^\circ$	$180^\circ$ combined
Feedback magnitude	$1/3$ for equal parts	$1/29$ under standard assumptions
Required gain	steady, slightly above 3 to start	At least 29 to start
Tuning/distortion	Convenient ganged tuning; very low with lamp/AGC	Interacting parts; usually higher distortion

Wien bridge and standard unbuffered phase-shift oscillators.

LC Oscillators

LC Tank

• Energy oscillates between the capacitor ($W_C=\frac12Cv^2$) and inductor ($W_L=\frac12Li^2$) at $f_0=1/(2\pi\sqrt{LC})$.

• Loss would damp the swing; the active device tops up the lost energy from the DC supply each cycle.

Three-reactance criterion. For a lossless three-point tank at $\omega_0$:

Native general three-reactance oscillator network.

$$\boxed{X_1+X_2+X_3=0}.$$

• Divider arms $X_1,X_2$ have the same sign; the third arm $X_3$ the opposite sign.

• $C$–$C$–$L$ $\Rightarrow$ Colpitts; $L$–$L$–$C$ $\Rightarrow$ Hartley.

Native Colpitts and Hartley feedback tanks. The labelled voltage terminals define the ratios quoted in the text.

Hartley Oscillator

• Circuit: tapped inductor ($L_1,L_2$) with a single tuning capacitor $C$; feedback taken from the inductive divider.

• Working: the tank resonates at $f_0$; the coil tap returns in-phase voltage that replaces tank loss and sustains oscillation.

• Frequency:

Caution

Key Formula — Hartley Tank

$$\boxed{L_T=L_1+L_2\pm2M}, \qquad \boxed{f_0=\frac{1}{2\pi\sqrt{L_TC}}}.$$

• Feedback: $|\beta|\simeq L_2/L_1$ for negligible $M$; the mutual sign follows the coil dots ($+$ aiding, $-$ opposing).

Simple one-capacitor tuning, but sensitive to coil loss and mutual coupling.

Colpitts Oscillator

• Circuit: single inductor $L$ with a capacitive divider $C_1,C_2$; feedback taken from the capacitor tap.

• Working: the tank resonates at $f_0$; $C_2$ returns in-phase voltage to the device.

• Frequency:

Caution

Key Formula — Colpitts Tank

$$\boxed{C_T=\frac{C_1C_2}{C_1+C_2}}, \qquad \boxed{f_0=\frac{1}{2\pi\sqrt{LC_T}}}.$$

• Feedback: $|\beta|=C_1/C_2$ (the series capacitors carry equal charge); device/stray capacitance adds in parallel with the divider.

Convenient at higher RF; sensitive to device/stray capacitance and loading.

Clapp Oscillator

A Colpitts with a small $C_3$ in series with $L$; $C_1,C_2$ still set the feedback division.

Native Clapp tank: the added series capacitor controls frequency while $C_1,C_2$ retain the feedback ratio.

Caution

Key Formula — Clapp Frequency

$$\boxed{\frac1{C_T}=\frac1{C_1}+\frac1{C_2}+\frac1{C_3}}, \qquad \boxed{f_0=\frac1{2\pi\sqrt{LC_T}}}.$$

If $C_3\ll C_1,C_2$, then $C_T\simeq C_3$.

• A stable $C_3$ dominates the tank, so junction/stray capacitances shift frequency less — better stability than a basic Colpitts.

• Tuning $C_3$ leaves the feedback ratio nearly fixed, so the start-up margin varies little.

Crystal Oscillator

A crystal oscillator uses a piezoelectric quartz resonator as the frequency-selective element of a feedback loop. Quartz is passive with a very high $Q$, so it fixes frequency sharply while the amplifier replaces loss.

Piezoelectric Effect

• Direct effect: mechanical stress produces charge/voltage.

• Inverse effect: an applied voltage produces mechanical strain.

• An AC voltage excites a natural elastic vibration; the cut (e.g. AT cut) sets the frequency and its temperature behaviour.

Equivalent Circuit

• Motional arm $R_m,L_m,C_m$ models loss, vibrating mass and compliance.

• Electrode/holder capacitance $C_0$ sits in parallel with the whole motional arm.

Native quartz equivalent circuit and reactance regions. Impedance is minimum at $f_s$ and maximum near $f_p$.

Series Resonance

Neglecting $R_m$, the motional arm is series-resonant when $\omega_sL_m=1/(\omega_sC_m)$:

$$\boxed{f_s=\frac1{2\pi\sqrt{L_mC_m}}}.$$

At $f_s$ the crystal impedance is minimum ($\approx R_m$); series-mode oscillators operate here.

Parallel Resonance

Above $f_s$ the arm is inductive and antiresonates with $C_0$. With $C_{eq}=C_mC_0/(C_m+C_0)$,

Caution

Key Formula — Crystal Resonances and Motional $Q$

$$\boxed{f_p=\frac1{2\pi\sqrt{L_mC_{eq}}} =f_s\sqrt{1+\frac{C_m}{C_0}}}, \qquad f_p>f_s,$$$$\boxed{Q_m\simeq\frac{\omega_sL_m}{R_m} =\frac1{\omega_sC_mR_m}}.$$

• Reactance: capacitive below $f_s$, inductive only for $f_s<f<f_p$, capacitive again above $f_p$.

• Because $C_m\ll C_0$, $f_s$ and $f_p$ are very close ($\sim0.1$–$0.5\%$ apart).

• Parallel-mode frequency is set within this window by the external load capacitance.

Native series-feedback and parallel-resonant crystal oscillator forms.

In a Pierce network the load capacitance is

$$\boxed{C_L\simeq\frac{C_1C_2}{C_1+C_2}+C_{stray}}.$$

The crystal must be specified for series or load-capacitance operation.

Working

Native crystal oscillator family: Pierce, Miller and op-amp forms.

• Pierce: inverting FET/CMOS stage ($180^\circ$) with the crystal and $C_1,C_2$; a large $R_f$ self-biases the inverter. Standard clock oscillator (fewest parts).

• Miller: a drain $LC$ network near antiresonance; gate–drain (Miller) capacitance closes the loop.

• Op-amp / modified Colpitts: the crystal is the high-$Q$ element; limiting or a back-to-back Zener clamp can give a square-wave clock.

Stability

• Very high $Q$ ($10^4$–$10^6$, versus tens–hundreds for an $LC$ tank) gives a steep phase slope and excellent stability.

• High stability is not the same as absolute accuracy: frequency still shifts with cut tolerance, temperature, aging, load/stray $C$, supply pulling and drive level.

• Excess drive heats and ages the crystal (nonlinearity/fracture). Fundamental cut below $\sim30$ MHz; odd overtone modes above.

Applications

Microprocessor/clock references, watches, radio carriers, test equipment and frequency synthesisers.

Phase-Locked Loop

A phase-locked loop (PLL) is a negative-feedback control system that adjusts an oscillator until its divided-output phase tracks a reference phase. In lock, the compared frequencies are equal and the phase error is constant or bounded; the output need not equal the input frequency when a divider is used.

Native PLL with optional divide-by-$N$ feedback and a linearised phase detector characteristic.

The phase detector/PFD compares reference and feedback phase/frequency. The low-pass loop filter removes detector ripple and sets bandwidth, damping and acquisition dynamics. The VCO converts control voltage into angular frequency:

$$v_d=K_d(\phi_r-\phi_f), \qquad \omega_o=\omega_{free}+K_vv_c.$$

Since phase is the integral of frequency, the VCO contributes $K_v/s$ in the small-signal phase model. Define

$$G(s)=\frac{K_dF(s)K_v}{s}, \qquad L(s)=\frac{G(s)}{N}.$$

Then the linearised reference-to-output phase transfer is

$$\boxed{\frac{\Theta_o(s)}{\Theta_r(s)} =\frac{G(s)}{1+G(s)/N}=\frac{NL(s)}{1+L(s)}}.$$

Loop-filter poles and zeros set crossover, damping and phase margin. Wider bandwidth usually gives faster acquisition/tracking but passes more reference and detector noise; narrower bandwidth filters high-frequency input jitter but captures more slowly and tolerates less rapid drift.

Acquisition, lock and ranges

1. Free run: without correction, the VCO runs near $f_{free}$.

2. Acquisition: a frequency/phase error creates a varying detector output whose low-frequency component steers the VCO toward the reference-related frequency.

3. Lock: the compared frequencies become equal and a constant phase error supplies the required tuning voltage.

4. Tracking: slow reference or VCO changes create phase error; negative feedback corrects them within loop bandwidth.

For unity feedback,

$$\boxed{N=1:\quad f_o=f_i}.$$

With divide-by-$N$ feedback,

$$\boxed{\frac{f_o}{N}=f_i}, \qquad \boxed{f_o=Nf_i}.$$

The capture or pull-in range is the set of initial input frequencies from which an unlocked loop can acquire lock under specified conditions. The lock, hold-in or tracking range is the set over which an already locked loop remains locked.

Capture is normally narrower: a large beat frequency is attenuated by the loop filter and must produce a net tuning correction before the phases slip again. Once locked, the detector supplies a DC/low-frequency correction and the loop can often track farther. Exact ranges depend on detector, filter, VCO tuning limits, signal level and acquisition method, so no single range formula applies to every PLL.

Note

Example — PLL Frequency Synthesiser

A $10\,\mathrm{MHz}$ crystal reference is divided by 100 to give $f_i=100\,\mathrm{kHz}$. For an FM local oscillator at $91.5\,\mathrm{MHz}$, choose

$$N=\frac{f_o}{f_i} =\frac{91.5\times10^6}{100\times10^3}=915.$$

In lock the feedback divider presents $91.5\,\mathrm{MHz}/915=100\,\mathrm{kHz}$ to the PFD. Incrementing $N$ by one changes the output by the $100\,\mathrm{kHz}$ reference spacing.

PLL applications include programmable frequency synthesis, FM demodulation (the control voltage follows instantaneous frequency deviation), carrier recovery for coherent detection, clock/data recovery, narrowband tracking and jitter/noise filtering. Practical limits include acquisition time, cycle slips or false lock, finite capture/hold ranges, reference spurs, divider/VCO phase noise, tuning nonlinearity and inadequate phase margin.

Comparison

Wien Bridge vs LC

Feature	Wien bridge ($RC$)	LC (Hartley/Colpitts)
Frequency element	$R$ and $C$	$L$–$C$ tank
Typical range	Audio, $\sim 1\,\mathrm{Hz}$–$1\,\mathrm{MHz}$	RF, $\sim 100\,\mathrm{kHz}$–hundreds of MHz
Amplifier	Non-inverting, $A_v=3$	Inverting/tuned, gain $>1/\beta$
Waveform purity	Very low distortion (with AGC)	Moderate
Inductor	Not needed	Required
Tuning	Ganged $R$ or $C$	Variable $C$ (or $L$)

Wien bridge versus LC oscillators.

Hartley vs Colpitts

Feature	Hartley	Colpitts
Divider	Tapped inductor $L_1,L_2$	Capacitor pair $C_1,C_2$
Single element	One capacitor $C$	One inductor $L$
Frequency	$f_0=1/(2\pi\sqrt{L_TC})$	$f_0=1/(2\pi\sqrt{LC_T})$
Feedback ratio	$\beta\simeq L_2/L_1$	$\beta=C_1/C_2$
Tuning	Simple (one $C$)	Vary $C_1,C_2$ (or add Clapp $C_3$)
Main weakness	Coil loss, mutual coupling	Stray/device capacitance
Typical use	Lower RF	Higher RF, better stability

Hartley versus Colpitts oscillator.

LC vs Crystal

Feature	LC	Crystal
Resonator	$L$–$C$ tank	Piezoelectric quartz
$Q$	Tens–hundreds	$10^4$–$10^6$
Stability	Good	Excellent
Tunability	Wide (variable $L$/$C$)	Essentially fixed
Cost/size	Low	Higher
Typical use	Tunable RF (VFO)	Precise fixed clocks/carriers

LC versus crystal oscillator.

Formula Validity Map

Topic	Formula or condition	Assumption / validity condition
Barkhausen	$\lvert A\beta\rvert=1,\ \angle A\beta=2\pi k$	Necessary steady-state loop condition; startup requires $\lvert A\beta\rvert>1$ and amplitude limiting.
Wien bridge	$f_0=1/(2\pi RC)$, $\beta=1/3$, $A_v=3$	Equal lead–lag parts and non-inverting amplifier; set gain slightly above 3 to start. General $f_0=1/[2\pi\sqrt{R_1C_1R_2C_2}]$.
RC phase shift	$f_0=1/(2\pi RC\sqrt6)$, $\lvert\beta\rvert=1/29$, $\lvert A\rvert\ge29$	Equal unbuffered RC sections with loading included.
Colpitts	$C_T=C_1C_2/(C_1+C_2)$, $f_0=1/(2\pi\sqrt{LC_T})$	If output is across $C_1$ and feedback across $C_2$, $\beta=C_1/C_2$; swapped labels invert the ratio.
Hartley	$L_T=L_1+L_2\pm2M$, $f_0=1/(2\pi\sqrt{L_TC})$	Mutual sign follows dots. Only for negligible $M$ does the labelled ratio reduce to $\beta\simeq L_2/L_1$.
Clapp	$C_T^{-1}=C_1^{-1}+C_2^{-1}+C_3^{-1}$	$C_3\ll C_1,C_2$ gives $C_T\simeq C_3$; parasitics are reduced in influence, not eliminated.

Oscillator formulas with the condition that makes each one valid.

Topic	Formula or condition	Assumption / validity condition
Series RLC	$Q_s=\omega_0L/R_s$, $BW=f_0/Q_s$	$R_s$ is total series loss; minimum $Z$, maximum line current and voltage magnification at $f_0$.
Ideal parallel RLC	$Q_p=R_p/(\omega_0L)=\omega_0CR_p$	$R_p$ is a parallel loss; maximum finite $Z=R_p$, minimum line current and current magnification. Lossless $Z$ is unbounded.
Coil-loss tank	$Q\simeq\omega_0L/r_s$, $R_{p,eq}\simeq(\omega_0L)^2/r_s$	High-$Q$ approximation for series coil loss $r_s$; do not confuse $r_s$ with parallel $R_p$.
Crystal series	$f_s=1/(2\pi\sqrt{L_mC_m})$	Minimum impedance of the motional arm; series-feedback oscillators operate near this frequency.
Crystal parallel	$f_p=f_s\sqrt{1+C_m/C_0}$	Low-loss equivalent circuit. Crystal is inductive only for $f_s<f<f_p$; external load capacitance sets the practical parallel-mode frequency.
PLL lock	$N=1:\ f_o=f_i$; divider: $f_o=Nf_i$	Only compared frequencies are equal. Capture (pull-in) is normally narrower than hold/lock range; a PLL is a negative-feedback phase-control loop.

Resonance, crystal and PLL formula map.

Problems

Resonance

Note

Example — Matched Series and Parallel Resonators

Let $L=10\,\mathrm{mH}$ and $C=1\,\mathrm{\mu F}$. Then

$$\omega_0=\frac1{\sqrt{LC}}=10^4\,\mathrm{rad/s}, \qquad f_0=1.5915\,\mathrm{kHz}.$$

For a series circuit with $R_s=10\,\Omega$ and $V=10\,\mathrm{V}$,

$$Q_s=\frac{(10^4)(0.01)}{10}=10, \quad BW=159.15\,\mathrm{Hz}, \quad I_{max}=1\,\mathrm A, \quad |V_L|=|V_C|=100\,\mathrm V.$$

For an ideal parallel model with $R_p=1.0\,\mathrm{k\Omega}$ at the same $10\,\mathrm V$, $Q_p=1000/[(10^4)(0.01)]=10$ and the bandwidth is again $159.15\,\mathrm{Hz}$. The line-loss current is $10\,\mathrm{mA}$ while each reactive branch carries $Q_pI_R=100\,\mathrm{mA}$ at resonance.

Wien Bridge

Note

Example — Design of a $1\,\mathrm{kHz}$ Wien Oscillator

Choose equal $C_1=C_2=10\,\mathrm{nF}$. Then

$$R_1=R_2=\frac{1}{2\pi f_0C} =\frac{1}{2\pi(10^3)(10\times10^{-9})} =15.9\,\mathrm{k\Omega}.$$

Take $R_g=10\,\mathrm{k\Omega}$ and an amplitude-controlled feedback resistance whose settled value is $R_f=20\,\mathrm{k\Omega}$. Its settled gain is $1+20/10=3$. A cold-lamp/JFET setting near $R_f=20.5\,\mathrm{k\Omega}$ gives $A_v=3.05$ for startup, after which level control returns it toward $20\,\mathrm{k\Omega}$.

Hartley

Note

Example — Hartley Design at $1\,\mathrm{MHz}$

Let $C=1.00\,\mathrm{nF}$ and neglect mutual coupling. The required inductance is

$$L_T=\frac{1}{(2\pi f_0)^2C} =\frac{1}{[2\pi(10^6)]^2(10^{-9})}=25.33\,\mathrm{\mu H}.$$

Choose $L_1=20.0\,\mathrm{\mu H}$ and $L_2=5.33\,\mathrm{\mu H}$. With output across $L_1$ and feedback across $L_2$, $\beta\simeq0.266$, so the active-stage magnitude must exceed $1/\beta=3.75$ at startup. If the coils are coupled, recompute both $L_T$ and the tap ratio using the measured $M$ and dot sign.

Colpitts

Note

Example — Colpitts Design at $1\,\mathrm{MHz}$

Choose $L=100\,\mathrm{\mu H}$. The required series capacitance is

$$C_T=\frac{1}{(2\pi f_0)^2L}=253.3\,\mathrm{pF}.$$

For a nominal ratio $C_2=4C_1$, $C_T=0.8C_1$; hence $C_1=316.6\,\mathrm{pF}$ and $C_2=1.266\,\mathrm{nF}$. Standard values $330\,\mathrm{pF}$ and $1.3\,\mathrm{nF}$ give $C_T=263.2\,\mathrm{pF}$ and $f_0\simeq981\,\mathrm{kHz}$. With the labelled ratio $\beta=C_1/C_2\simeq0.254$, startup needs $|A|>3.94$ before loading and parasitic corrections.

[!NOTE]

Example — Clapp Conversion of the $1\,\mathrm{MHz}$ Tank

Keep $L=100\,\mathrm{\mu H}$, so $C_T=253.3\,\mathrm{pF}$, and choose $C_1=C_2=10\,\mathrm{nF}$. Then

$$\frac1{C_3}=\frac1{253.3\,\mathrm{pF}}-\frac{2}{10\,\mathrm{nF}}, \qquad C_3=266.9\,\mathrm{pF}.$$

Here $C_3/C_1\simeq0.0267$, so the small series capacitor controls frequency while the equal $10\,\mathrm{nF}$ pair fixes the nominal divider ratio.

Crystal

Note

Example — How Close Are $f_s$ and $f_p$?

Let $L_m=0.50\,\mathrm H$, $C_m=0.020\,\mathrm{pF}$, $C_0=4.0\,\mathrm{pF}$ and $R_m=50\,\Omega$. Then

$$f_s=\frac1{2\pi\sqrt{(0.50)(0.020\times10^{-12})}} =1.59155\,\mathrm{MHz},$$$$f_p=f_s\sqrt{1+0.020/4.0}=1.59552\,\mathrm{MHz}.$$

The separation is only about $3.97\,\mathrm{kHz}$ or $0.25\%$, while $Q_m\simeq\omega_sL_m/R_m=100{,}000$.