4. Oscillators

Negative feedback stabilizes. Positive feedback at exactly the right gain produces sustained oscillation. Every oscillator is a feedback loop with $A\beta = 1$ at one specific frequency.

4.1 The Barkhausen criterion

For sustained oscillation in a linear feedback loop:

1. The loop gain magnitude $|A\beta| = 1$ exactly. Lower and the oscillation dies away exponentially. Higher and it grows until something saturates.
2. The loop phase is 0° (or any integer multiple of 360°). The signal that comes back has to be in phase with the original so it reinforces rather than cancels.

These two conditions together are the Barkhausen criterion, named after Heinrich Barkhausen who derived it in the 1920s. They are necessary conditions; sufficiency requires also that the loop be capable of starting up from noise, which means $|A\beta|$ should be slightly greater than 1 at startup so that an initial seed (thermal noise, power-up transient) grows.

In practice, designers make $|A\beta|$ slightly greater than 1 at startup. A nonlinear element (clipping in a transistor, an AGC circuit, a thermistor, an incandescent bulb's positive temperature coefficient of resistance, a JFET acting as a voltage-controlled resistor) eventually forces the effective loop gain back to exactly 1 once amplitude is large, and the system settles at sustained oscillation.

The clever trick of a Wien bridge oscillator (next subsection) is that Bill Hewlett's PhD thesis at Stanford in 1939 used a small incandescent bulb as the gain-stabilizing element. As the oscillation grows, the bulb's filament heats up, its resistance rises, and the loop gain is gently pulled back to 1. The bulb thermal time constant is much longer than the oscillation period, so the instantaneous loop gain stays linear, but the average loop gain self-stabilizes. The trick was so good that Hewlett-Packard's first product, the HP 200A audio oscillator (1939), used it, and it defined the company.

4.2 RC oscillators (audio range, kHz)

These use resistor-capacitor networks to set the oscillation frequency. Cheap, simple, no inductors. Frequencies from a fraction of Hz up to a few hundred kHz.

Phase-shift oscillator. Three RC sections each shifting roughly 60° give a total of 180°; combined with the inverting amplifier's 180°, the loop phase is 360° at the oscillation frequency:

              Rf
       ┌─────[ ]─────┐
       │             │
       │     ───|>───┤ ──+─[R]─[C]─[R]─[C]─[R]─[C]─┐
       │       Q1    │   │                        │
       │             ▼   │                        │
       └────output  GND  │                       output → input again

The frequency at which the phase is exactly 180° (so the loop sums to 360°) is

$f_{osc} = \frac{1}{2\pi RC \sqrt{6}}$

and the inverting amplifier gain must be at least 29 to compensate the RC network's attenuation. Decent stability, very simple. Used in low-cost signal generators and lab demonstrations.

Wien bridge oscillator. A series-RC and a parallel-RC bridge network feeds a non-inverting amplifier:

     Vout ─┬──[ R ]──[ C ]──┬── V+
           │                │
           │   ┌──[R]──[C]──┘
           │   │
           │   ▼ (parallel-RC to ground)
           │   GND
           │
          [Rf]
           │
          [Rg]
           │
          GND
                                    ┌──+ V+
                                    │
                                Op─Amp
                                    │
                              ──+──── V_out
                                    │
                                    └─ V−

The series-parallel RC network passes the input most readily at $f_0 = 1/(2\pi R C)$ with no phase shift (so the loop phase is 0° from the non-inverting amp). The amplifier needs gain exactly 3 to satisfy Barkhausen, achieved with a $R_f / R_g = 2$ ratio. The remaining piece is the amplitude-control trick: an incandescent bulb in the $R_g$ position, or a JFET, or a soft-clipping diode pair.

The Wien bridge produces astonishingly clean sine waves: THD below 0.001% in well-built ones. It is the classic audio test-equipment oscillator.

4.3 LC oscillators (RF range, MHz to GHz)

These use inductor-capacitor tank circuits. Higher frequency stability than RC because LC resonance Q can be much higher than RC Q.

Hartley oscillator. Tapped inductor as the feedback element, a single capacitor across the whole inductor as the resonator. Common in old AM superhet receivers as the local oscillator. Frequency $f = 1/(2\pi\sqrt{(L_1+L_2)C})$.

Colpitts oscillator. Tapped capacitor (two caps in series, with the tap as the feedback point), a single inductor across the whole tank. More frequency-stable than Hartley because a capacitive divider is less affected by stray inductance than an inductive divider is by stray capacitance:

       V_CC
        │
       [L]      Tank inductor
        │
        ├───────────── output
        │
       [C1]
        │
        ├──── feedback to base of transistor
        │
       [C2]
        │
       GND

Frequency: $f_0 = 1/(2\pi\sqrt{L \cdot (C_1 C_2)/(C_1 + C_2)})$. Used in many crystal oscillators (with the crystal in place of the inductor, exploiting the crystal's series-resonant behavior).

Clapp oscillator. A modified Colpitts with an extra series cap in the tank, even more stable.

Armstrong oscillator. Tickler-coil feedback, the original Edwin Armstrong design from 1912. Largely historical now, but it was the first practical regenerative oscillator, the foundation of all amateur radio in the 1920s.

4.4 Crystal oscillators

Replace the LC tank with a quartz crystal, a piezoelectric resonator with $Q$ factors of 10,000 to over 1,000,000. The crystal has both series and parallel resonances very close together; the oscillator is built so the crystal operates near series resonance, where its impedance is purely resistive and very low.

Frequency stability of 10 to 100 ppm typical; temperature-compensated (TCXO) and oven-controlled (OCXO) crystals reach 1 ppb. Used in:

• CPU and microcontroller clocks. Every chip with timing has a crystal somewhere; the laptop you are reading this on has a 25 MHz crystal feeding a PLL that synthesizes the GHz CPU clock from it.
• Watches and timekeeping (the 32.768 kHz tuning-fork crystal is the canonical wristwatch oscillator).
• Frequency standards in test equipment (10 MHz reference).
• GPS receivers, where the local oscillator must be very stable so that the GPS code-correlation can resolve sub-microsecond timing.
• Radio transceivers, where TX and RX frequencies must align tightly across thousands of channels.

Hardware-security tie-in. The internal jitter of a free-running ring oscillator (a chain of inverters wrapped in a feedback loop, just enough loop gain and phase to oscillate) is dominated by random thermal noise on each inverter's threshold. Two ring oscillators on the same chip have slightly different frequencies because of fabrication-level random variations. This is exploited in two security primitives. (1) True random number generators (TRNGs) use a slow ring oscillator sampled by a fast one; the unpredictable sample is a true random bit, harvesting thermal noise as entropy. (2) Physical Unclonable Functions (PUFs) measure the relative frequencies of pairs of ring oscillators, which depend on chip-level manufacturing variations and are essentially impossible to clone or model. A "ring oscillator PUF" can give a chip a unique fingerprint usable as a hardware identity. Both primitives are oscillators with $A\beta = 1$, just put to surprising use.

4.5 Voltage-controlled oscillators (VCOs)

An oscillator whose frequency varies with a control voltage. Built around varactor diodes (whose capacitance varies with reverse bias), or around current-controlled relaxation oscillators (a comparator and an integrator with switchable polarity).

VCOs are at the heart of phase-locked loops (PLLs), FM modulators, function generators, frequency synthesizers, and software-defined radio front-ends. The PLL chip in your CPU package multiplies the 25 MHz crystal up to several GHz by using a VCO that locks to a divided version of the crystal frequency.

4.6 Frequency stability comparison

Pick based on the requirement. A blinking LED needs nothing better than RC. A USB device needs about 500 ppm. WiFi needs 25 ppm. Cellular base stations need OCXO, sometimes locked to GPS. Atomic clocks anchor financial trading, telecom, and the GPS constellation itself. Tests of general relativity demand the best.

4.7 SPICE example: a simple Colpitts oscillator

Here is a netlist for a classic 10 MHz Colpitts. Each line is one component, parsed by ngspice or LTspice.

* Colpitts oscillator at ~10 MHz
V1   vcc 0      DC 12
R1   vcc b      DC 100k         ; bias
R2   b   0      DC 22k          ; bias
RE   e   0      DC 1k
CE   e   0      1u              ; emitter bypass
C1   c   e      100p            ; tank cap C1
C2   e   0      330p            ; tank cap C2
L1   c   vcc    2.2u            ; tank inductor
Q1   c   b   e  Q2N3904
.MODEL Q2N3904 NPN(BF=300)
.IC V(c)=11
.TRAN 1n 5u
.END

Run this and watch V(c) ring up from the initial condition into a stable sinusoid at $f_0 = 1/(2\pi\sqrt{L_1 \cdot C_1 C_2/(C_1+C_2)}) \approx 10$ MHz. This is the same topology used in the 10 MHz reference oscillators of every benchtop frequency counter built before the 1990s.