Feedback in Amplifiers & Waveform Oscillators

1. Fundamental Feedback Equation & Closed-Loop Desensitivity

In a feedback amplifier, a fraction $\beta$ of the output signal is sampled and combined with the external input source $x_s$. For negative (degenerative) feedback, the feedback signal is subtracted in phase from the source signal:

The open-loop amplifier provides gain $A = x_o / x_i$. Substituting into the transfer relation:

Thus, the closed-loop gain $A_f$ is:

The term $D = 1 + A\beta$ is called the desensitivity factor or amount of feedback ($F = 20 \log_{10}|1 + A\beta|\,\text{dB}$). When loop gain $A\beta \gg 1$:

The closed-loop gain becomes virtually independent of internal transistor parameters, device aging, temperature fluctuations, and supply voltages, governed solely by stable passive feedback resistors.

2. Advantages of Negative Feedback

• Gain Stabilization: Differentiating $A_f = A / (1 + A\beta)$:
  $$\frac{d A_f}{A_f} = \frac{1}{1 + A\beta} \left( \frac{dA}{A} \right)$$
  Internal open-loop variations are attenuated by the factor $(1 + A\beta)$.
• Bandwidth Extension: High-frequency cutoff extends while low-frequency cutoff drops:
  $$f_{H,f} = f_H (1 + A\beta), \quad f_{L,f} = \frac{f_L}{1 + A\beta} \implies BW_f \approx BW (1 + A\beta)$$
  The Gain-Bandwidth product remains constant ($A_f \cdot BW_f = A \cdot BW$).
• Harmonic Distortion Reduction: Nonlinear harmonic distortion generated within the amplifier is reduced to $D_f = D / (1 + A\beta)$.
• Noise Reduction: Internally generated spurious noise $N$ is reduced to $N_f = N / (1 + A\beta)$.

3. The Four Feedback Topologies

Feedback networks are classified by how the output is sampled (voltage or current) and how feedback is injected at the input (series voltage or shunt current):

1. Barkhausen Criterion for Sustained Oscillations

An electronic oscillator generates continuous sinusoidal AC waveforms without any external AC input signal, converting DC supply energy into AC oscillation. In a feedback loop where feedback is positive:

To sustain steady, unattenuated oscillations at a specific frequency $\omega_0$, the Barkhausen Criteria must be fulfilled:

1. Magnitude Condition: The loop gain magnitude must equal unity:
   $$|A \beta| = 1$$
   (To initiate oscillation from thermal noise, $|A\beta| > 1$ initially; nonlinear amplitude limiting stabilizes $|A\beta| = 1$ in steady state).
2. Phase Condition: The total net phase shift around the closed loop must be an integer multiple of $360^\circ$ (or $0^\circ$):
   $$\angle (A \beta) = 0^\circ \quad \text{or} \quad 2\pi n \quad (n = 0, 1, 2, \dots)$$
   If the active amplifier introduces a $180^\circ$ phase shift (such as a Common Emitter BJT), the feedback network must introduce an additional $180^\circ$ phase shift at $\omega_0$.

2. Hartley LC Oscillator

The Hartley oscillator employs a tapped inductive voltage divider ($L_1, L_2$) in parallel with a single tuning capacitor $C$. If mutual inductance between the coil segments is $M$, the effective inductance is $L_{eq} = L_1 + L_2 + 2M$.

The natural resonant frequency of oscillation is:

The feedback fraction is determined by the inductive tap ratio:

For sustained oscillations, the minimum required transistor current gain is:

3. Colpitts LC Oscillator

The Colpitts oscillator replaces the tapped inductor with a capacitive voltage divider ($C_1, C_2$) in parallel with a single continuous inductor $L$. The series combination of $C_1$ and $C_2$ forms the equivalent tank capacitance:

The oscillation frequency is derived from tank resonance:

The feedback voltage fed to the base is tapped across $C_1$, while output is taken across $C_2$. The feedback factor magnitude is:

To satisfy $|A\beta| \ge 1$, the minimum amplifier gain condition is:

Because capacitors can be fabricated with higher precision and less stray magnetic coupling than tapped inductors, the Colpitts oscillator offers superior frequency stability in high-frequency RF communications.

1. RC Phase-Shift Oscillator

At audio frequencies ($20\,\text{Hz} - 200\,\text{kHz}$), inductors become excessively bulky, heavy, and lossy. RC feedback networks are preferred.

A CE transistor amplifier introduces a $180^\circ$ phase shift. To achieve the requisite $360^\circ$ total loop phase shift, a cascade of three identical $RC$ high-pass filter sections is inserted in the feedback loop. Each section contributes a $60^\circ$ phase advance at the oscillation frequency $\omega_0$.

Applying mesh analysis to the three-stage ladder network:

Setting $s = j\omega$ and requiring the imaginary part of the denominator to vanish for zero net phase difference ($\angle \beta = -180^\circ$ relative to inverting input):

Taking loading into account across the ladder yields:

Evaluating the attenuation of the network at this frequency:

To satisfy Barkhausen's criterion $|A\beta| \ge 1$:

The amplifier must maintain a voltage gain of at least $29$ to overcome the passive $RC$ ladder attenuation and sustain audio oscillations.

2. Wien Bridge Oscillator

The Wien Bridge oscillator is the premier laboratory standard for pure low-distortion audio sinusoidal generation. It employs a four-arm AC bridge consisting of:

• A series $R-C$ branch in series with a parallel $R-C$ branch (the frequency-selective lead-lag arm).
• A resistive divider branch ($R_1, R_2$) that sets the closed-loop non-inverting gain.

The transfer function of the lead-lag network is:

Substituting $s = j\omega$:

For the phase shift to be strictly $0^\circ$ (real $\beta$), the imaginary term in the denominator must equal zero:

At resonance $\omega = \omega_0$:

Because the network introduces zero phase shift, a non-inverting operational amplifier or two-stage CE amplifier is used. To meet $|A\beta| = 1$:

Automatic gain control using a tungsten filament bulb, thermistor, or JFET channel in the negative feedback arm stabilizes the amplitude and suppresses harmonic distortion below $0.01\%$.

1. Piezoelectric Effect in Natural and Synthetic Quartz

Quartz ($\text{SiO}_2$) exhibits the direct piezoelectric effect: applying mechanical stress or compressive force across specific crystallographic axes produces electric polarization and charges on opposite faces. Conversely, by the inverse piezoelectric effect, applying an alternating voltage creates mechanical vibrations and acoustic standing waves within the quartz plate.

When the frequency of the applied AC voltage matches the mechanical resonant frequency of the wafer (determined by thickness $t$, cut angle like AT-cut, and acoustic velocity $v_a$):

The crystal vibrates with extreme mechanical amplitude and exhibits electrical resonance with unmatched frequency stability.

2. Electrical Equivalent Circuit of a Crystal

Electrically, a mounted quartz crystal is represented by the Butterworth-Van Dyke (BVD) model:

• $L$ (Motional Inductance): Represents the vibrating mechanical mass/inertia of the quartz wafer (very large: hundreds of millihenries to tens of henries).
• $C_s$ (Motional Capacitance): Represents the mechanical compliance/elasticity of the crystal (very small: femtofarads to picofarads, $\sim 0.01\text{ to }0.1\,\text{pF}$).
• $R$ (Motional Resistance): Represents internal mechanical friction and acoustic losses (small: $10\,\Omega \text{ to } 100\,\Omega$).
• $C_p$ (Parallel Electrostatic Capacitance): Represents the physical capacitance formed by the metal mounting electrodes sandwiching the dielectric quartz slab (typically $3\text{ to }8\,\text{pF}$, where $C_p \gg C_s$).

3. Series and Parallel Resonant Frequencies

Because the crystal contains both series and parallel reactive branches, it possesses two closely spaced resonant frequencies:

1. Series Resonant Frequency ($f_s$): Occurs when the motional branch impedance drops to minimum (pure resistance $R$):
   $$f_s = \frac{1}{2\pi \sqrt{L C_s}}$$
   At $f_s$, the crystal exhibits very low series impedance, operating as a series bandpass filter.
2. Parallel (Anti-resonant) Frequency ($f_p$): Occurs when the net inductive reactance of the motional branch resonates with parallel electrostatic capacitance $C_p$:
   $$C_{total} = \frac{C_s C_p}{C_s + C_p}$$
   $$f_p = \frac{1}{2\pi \sqrt{L C_{total}}} = f_s \sqrt{1 + \frac{C_s}{C_p}} \approx f_s \left( 1 + \frac{C_s}{2 C_p} \right)$$
   Between $f_s$ and $f_p$, the crystal reactance is strictly inductive ($X_L > X_C$), enabling it to replace inductors in Pierce or Colpitts oscillator circuits.

4. Quality Factor $Q$ and Unsurpassed Frequency Stability

The figure of merit of a resonator is its Quality Factor $Q$:

While traditional LC resonant tanks achieve $Q \approx 50 \text{ to } 200$, quartz crystals achieve $Q = 10,000 \text{ to } 1,000,000$ because motional inductance $L$ is colossal while resistance $R$ is tiny. This gigantic $Q$-factor makes the phase-frequency slope $d\phi/df$ extraordinarily steep, ensuring clock frequency drift is kept below $1\text{ part per million (ppm)}$ per year in digital microprocessors and telecommunications transmitters.