4.4 Oscillators & Multivibrators

4.4 Oscillators & Multivibrators

In aircraft avionics, electronic signal generators are divided into two fundamental classes:

1. Sinusoidal Oscillators: Generate pure, single-frequency sine waves for radio frequency (RF) communications carrier waves, radar local oscillators, VHF omnidirectional range (VOR) navigation receivers, and synchro system 400 Hz excitation references.
2. Multivibrators (Relaxation Oscillators): Generate non-sinusoidal digital clock waveforms, rectangular pulses, and square waves used in flight management computer clocks, radar range gate timing, Distance Measuring Equipment (DME) pulse decoding, and cockpit indicator flashers.

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1. Principles of Sinusoidal Oscillation & The Barkhausen Criteria

An oscillator is essentially an amplifier that generates an AC output signal from a DC power supply without requiring any external AC input signal. It achieves this by taking a portion of its output voltage and feeding it back to the input in positive (regenerative) feedback.

The Two Barkhausen Criteria

For an amplifier-feedback loop to sustain continuous, stable sinusoidal oscillation at a designated frequency ($f_0$), the system must satisfy the Barkhausen Criteria:

1. Loop Gain Magnitude Criterion: $|A \cdot \beta| = 1$

   • At Startup: The initial loop gain must be slightly greater than unity ($|A\beta| > 1$). This allows tiny, broadband thermal noise voltages across the circuit components to be progressively amplified and filtered around $f_0$, rapidly building up the oscillation amplitude.
   • At Steady-State: As the oscillation reaches the desired amplitude, non-linearities in the transistor (such as saturation/cutoff limiting or automatic gain control) compress the amplifier gain until the effective loop gain stabilizes at exactly unity ($|A\beta| = 1$). If $|A\beta| < 1$, oscillations decay exponentially to zero; if $|A\beta| > 1$, the waveform grows until severe clipping occurs.

2. Phase Shift Criterion: $\angle (A \cdot \beta) = 0^\circ \quad \text{or} \quad 360^\circ$

   • A standard Common Emitter (CE) transistor amplifier stage introduces an intrinsic $180^\circ$ phase inversion between base and collector.
   • Therefore, the external frequency-determining feedback network must introduce an additional $180^\circ$ phase shift precisely at frequency $f_0$.
   • Total round-trip phase shift: $180^\circ + 180^\circ = 360^\circ \equiv 0^\circ$ (in-phase regenerative feedback).

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2. LC Resonant Tank Oscillators

At radio frequencies ($100\text{ kHz to }>500\text{ MHz}$), frequency determination relies on a parallel LC resonant tank circuit. In an ideal tank, energy continuously cycles between the electric field of capacitor $C$ ($E = \frac{1}{2} C V^2$) and the magnetic field of inductor $L$ ($E = \frac{1}{2} L I^2$) at the natural resonant frequency:

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

Because practical inductors possess internal winding resistance ($R_s$), energy is dissipated as heat, producing damped (decaying) oscillations. The active transistor injects a pulse of energy each cycle to replenish these losses.

A. The Hartley Oscillator

In a Hartley oscillator, the frequency-determining feedback network employs an inductive voltage divider formed by a center-tapped inductor ($L_1$ and $L_2$) tuned by a single parallel capacitor ($C$):

• Feedback Ratio: $\beta \approx \frac{L_1}{L_2}$ (ratio of feedback winding to collector winding).
• Total Inductance: $L_{eq} = L_1 + L_2 + 2M$ (accounting for mutual inductance $M$).
• Resonant Frequency: $f_0 = \frac{1}{2\pi \sqrt{L_{eq} C}}$
• Characteristics: Readily tuned over a wide frequency band using a single variable capacitor. However, parasitic inter-turn capacitance in the tapped coil can introduce harmonic distortion at VHF frequencies.

B. The Colpitts Oscillator

In a Colpitts oscillator, the feedback network uses a capacitive voltage divider consisting of two series capacitors ($C_1$ and $C_2$) in parallel with a single untapped inductor ($L$):

• Series Equivalent Capacitance: $C_{eq} = \frac{C_1 \cdot C_2}{C_1 + C_2}$
• Resonant Frequency: $f_0 = \frac{1}{2\pi \sqrt{L \cdot C_{eq}}}$
• Feedback Fraction: $\beta \approx \frac{C_1}{C_2}$ (voltage tapped across $C_2$ for base feedback).
• Superiority in Avionics: The capacitors in the divider provide a low-reactance shunt to ground for high-order harmonics, yielding a substantially cleaner sinusoidal waveform than the Hartley. It exhibits superior frequency stability at high frequencies, making it the premier LC architecture in aircraft VHF communications transceivers (118.000–136.975 MHz).

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3. Quartz Crystal Oscillators

While discrete LC oscillators achieve frequency stabilities of roughly $1%$ to $0.1%$, modern aviation communications, navigation, and radar demand tolerances better than $0.0001%$ (1 to 10 parts per million, ppm). This precision is achieved using quartz crystal oscillators.

The Piezoelectric Effect & Equivalent Circuit

Quartz ($\text{SiO}_2$) exhibits the piezoelectric effect: applying a mechanical stress across the crystal wafer generates a proportional voltage across its faces; conversely, applying an alternating voltage causes the crystal to vibrate mechanically at its natural acoustic resonant frequency.

Electrically, a mounted quartz crystal behaves as a complex RLC network:

• Motional Arm ($L_m, C_m, R_s$): $L_m$ represents the vibrating mechanical mass of the quartz (often several Henries); $C_m$ represents mechanical elasticity/compliance (fractions of a picofarad); $R_s$ represents internal acoustic frictional losses (a few ohms).
• Static Capacitance ($C_0$): Represents the electrostatic capacitance formed by the mounting metal electrodes sandwiching the quartz dielectric.

Crystal Equivalent Electrical Model:
          ---[ Lm ]---[ Cm ]---[ Rs ]---   (Motional Arm)
     |---|                              |---|
     |    ------------[ C0 ]------------    |  (Shunt Arm)

Dual Resonance and Extreme Q-Factor

The crystal exhibits two resonant frequencies separated by only a few kilohertz:

1. Series Resonant Frequency ($f_s$): The motional arm impedance drops to its minimum ($R_s$): $f_s = \frac{1}{2\pi \sqrt{L_m C_m}}$
2. Parallel Antiresonant Frequency ($f_p$): The inductance $L_m$ resonates with the series combination of $C_m$ and $C_0$, creating an extremely high parallel impedance.

Why Crystals Dominate Aviation

• Quality Factor ($Q$): The ratio of stored reactive energy to dissipated energy is $Q = \frac{\omega L_m}{R_s}$. Discrete LC tanks achieve $Q \approx 50\text{ to }200$. A quartz crystal achieves $Q > 10,000\text{ to }1,000,000$.
• Stability: Extreme $Q$ yields an extraordinarily steep phase-versus-frequency curve: any circuit drift attempts to shift phase, but a minute frequency shift restores equilibrium, locking the frequency.
• Aviation Standards: Temperature-Compensated Crystal Oscillators (TCXO) and Oven-Controlled Crystal Oscillators (OCXO) serve as master clocks in GPS navigation receivers, Mode-S transponders, and Traffic Collision Avoidance Systems (TCAS).

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4. Multivibrators: Non-Sinusoidal Relaxation Oscillators

Unlike sinusoidal oscillators that cycle energy between reactive components, multivibrators are relaxation circuits that operate by alternately switching cross-coupled active transistors between full saturation (ON) and cutoff (OFF).

A. Astable Multivibrator (Free-Running Clock)

• State Characteristics: Zero stable states; two quasi-stable states.
• Operation: Two cross-coupled BJT stages have their collectors connected to opposing bases via timing capacitors ($C_1, C_2$) and pull-up resistors ($R_1, R_2$). The circuit continuously toggles between states as the capacitors alternately charge and discharge.
• Output: Continuous square-wave pulse train.
• Frequency and Period: $T_1 = 0.693 R_1 C_1 \qquad T_2 = 0.693 R_2 C_2$ $T_{total} = T_1 + T_2 \approx 1.386 R C \quad (\text{for symmetric } R_1=R_2, C_1=C_2)$ $f = \frac{1}{T_{total}} = \frac{1}{1.386 R C} \approx \frac{0.72}{R C}$
• Avionics Application: Master clock pulse generation for digital logic circuits, flasher units for warning annunciator lights.

B. Monostable Multivibrator (One-Shot Pulse Generator)

• State Characteristics: One permanently stable state; one quasi-stable state.
• Operation: Normally rests in its stable state with $Q_1$ OFF and $Q_2$ ON. An external trigger pulse flips the circuit into its quasi-stable state ($Q_1$ ON, $Q_2$ OFF) for a calibrated time duration ($t_w$) dictated by an RC timing network, after which it automatically snaps back to its resting state.
• Pulse Width Equation: $t_w = \ln(2) \cdot R C \approx 0.693 R C$
• Avionics Application: Generating calibrated timing delays, pulse-shaping radar trigger pulses, stretching narrow sensor spikes, and Distance Measuring Equipment (DME) pulse-pair decoders.

C. Bistable Multivibrator (Flip-Flop Data Latch)

• State Characteristics: Two permanently stable states ($Q = 1, \bar{Q} = 0$ or $Q = 0, \bar{Q} = 1$).
• Operation: The circuit does not oscillate spontaneously. It remains indefinitely in one stable state until an external trigger pulse forces it to toggle into the opposing state.
• Significance: The fundamental 1-bit digital storage element in aviation flight management computers, binary counters, frequency dividers, and register files.

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Comparison: Oscillators and Multivibrators

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Worked Engineering Calculation: Colpitts Oscillator & Astable Clock

Problem 1: Colpitts Resonant Frequency

An aircraft VHF receiver local oscillator utilizes a Colpitts configuration. The circuit parameters are:

• Inductor: $L = 1.0\ \mu\text{H}$
• Feedback Capacitors: $C_1 = 200\text{ pF}$ and $C_2 = 800\text{ pF}$

Calculate the series equivalent capacitance ($C_{eq}$) and the fundamental oscillation frequency ($f_0$).

Solution 1:

1. Equivalent Series Capacitance ($C_{eq}$): $C_{eq} = \frac{C_1 \cdot C_2}{C_1 + C_2} = \frac{200\text{ pF} \times 800\text{ pF}}{200\text{ pF} + 800\text{ pF}} = \frac{160,000}{1,000} = 160\text{ pF} = 160 \times 10^{-12}\text{ F}$

2. Oscillation Frequency ($f_0$): $f_0 = \frac{1}{2\pi \sqrt{L \cdot C_{eq}}} = \frac{1}{2\pi \sqrt{(1.0 \times 10^{-6}\text{ H}) \times (160 \times 10^{-12}\text{ F})}}$ $\sqrt{1.6 \times 10^{-16}} = 1.2649 \times 10^{-8}\text{ s}$ $f_0 = \frac{1}{2\pi \times 1.2649 \times 10^{-8}} = \frac{1}{7.9477 \times 10^{-8}} \approx 12,582,200\text{ Hz} \approx 12.58\text{ MHz}$

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Problem 2: Astable Multivibrator Clock Frequency

A symmetric astable multivibrator used to clock an avionics data logger employs timing resistors $R_1 = R_2 = 10\text{ k}\Omega$ and timing capacitors $C_1 = C_2 = 4.7\text{ nF}$.

Calculate the total period ($T$) and the clock frequency ($f$).

Solution 2:

1. Total Period ($T$): $T = 1.386 \cdot R \cdot C = 1.386 \times (10 \times 10^3\ \Omega) \times (4.7 \times 10^{-9}\text{ F}) = 1.386 \times 4.7 \times 10^{-5}\text{ s} \approx 65.14\ \mu\text{s}$

2. Clock Frequency ($f$): $f = \frac{1}{T} = \frac{1}{65.14 \times 10^{-6}\text{ s}} \approx 15,350\text{ Hz} \approx 15.35\text{ kHz}$