Unit 4: Amplifiers and Oscillators

An amplifier raises signal power using a transistor biased in its active region; an oscillator is an amplifier with feedback engineered to sustain a signal with no external input. Both are governed by the same gain-and-feedback algebra, so this unit builds from the single-stage amplifier, adds feedback, and ends with oscillators as the limiting case of positive feedback.

I. Governing Principles

The behaviour of every circuit below follows from small-signal transistor gain and the closed-loop gain formula.

• Voltage gain (open loop): the ratio A = Vo / Vi, negative in a CE stage because of the 180° phase inversion.
• Closed-loop gain: for a fed-back stage A_f = A / (1 ± Aβ), where β is the fraction of output returned to the input.
• Loop gain: the product Aβ; its sign and magnitude decide whether feedback is negative (stabilising) or positive (regenerative).
• h-parameter model: small-signal analysis uses h_ie (input resistance), h_fe (current gain, β_ac), h_oe (output conductance).
• Barkhausen criterion: the boundary condition Aβ = 1 at 0° (or 360°) loop phase separates a stable amplifier from a self-starting oscillator.

II. Single-Stage RC Coupled CE Amplifier

Analysis and frequency response

The most common voltage amplifier: a common-emitter transistor with resistive load and capacitive (RC) coupling to source and next stage.

A. Circuit and mid-band analysis

The stage is a CE transistor stabilised by voltage-divider bias, with coupling capacitors blocking DC between stages and a bypass capacitor across the emitter resistor.

• Bias network: R1, R2 set base voltage; R_E gives DC stabilisation; C_E bypasses R_E for AC so gain is not degenerated.
• Coupling: C_C passes signal but blocks the DC operating point of one stage from disturbing the next.
• Small-signal gain: using the h-model,

A_v = -h_fe * (R_C || R_L) / h_ie

• Symbols: h_fe current gain, h_ie base input resistance, R_C collector resistor, R_L next-stage load. Minus sign = phase inversion.
  • Input impedance: Z_i = R1 || R2 || h_ie.
  • Output impedance: Z_o = R_C || (1/h_oe).

B. Frequency response

Gain is constant only over a middle band; capacitances roll it off at both ends.

• Low-frequency fall: coupling and bypass capacitors (C_C, C_E) have rising reactance as f falls, so more signal drops across them — gain declines below the lower cutoff f_L.
• High-frequency fall: transistor junction capacitances and the Miller-multiplied C_bc shunt the signal to ground, dropping gain above the upper cutoff f_H.
• Mid-band: all series capacitors act as shorts, all shunt capacitors as opens — gain is flat and maximum.
• Bandwidth: BW = f_H − f_L, measured between the two half-power (−3 dB) points where gain falls to 0.707 of mid-band.
• Bode plot: gain (dB) vs log frequency shows +20 dB/decade rise, a flat plateau, then −20 dB/decade fall.

III. Feedback Amplifiers

Sampling output and returning it to the input

Feedback mixes a scaled portion of the output signal back with the input; the sign of that mixing defines the amplifier's whole character.

A. Positive and negative feedback

The two feedback polarities are opposites in both algebra and purpose.

1. Negative feedback: the returned signal opposes the input, so A_f = A / (1 + Aβ).
   • Effect: gain reduced but stabilised; used in all practical amplifiers.
2. Positive feedback: the returned signal aids the input, so A_f = A / (1 − Aβ).
   • Effect: gain rises and, at Aβ = 1, becomes infinite — the basis of oscillation.
   • Feedback factor: β = V_f / V_o, the fraction fed back; V_f is the feedback voltage.

B. Effect of feedback on gain

Negative feedback trades gain for precision.

• Gain reduction: by the factor (1 + Aβ), called the desensitivity factor.
• Gain stability: the fractional change shrinks,

dA_f / A_f = (1 / (1 + Aβ)) * (dA / A)

• Meaning: a 10% drift in A with Aβ = 49 becomes a 0.2% drift in A_f — gain now set by the passive β network, not the transistor.

C. Effect of feedback on bandwidth

Negative feedback widens the useful frequency range.

• Bandwidth increase: BW_f = BW * (1 + Aβ) — cutoffs spread out as gain falls.
• Gain–bandwidth product: stays constant, so what is lost in gain is gained in bandwidth.
  • Example: A = 100, BW = 20 kHz, Aβ = 9 → A_f = 10, BW_f = 200 kHz; product 2 × 10^6 unchanged.

D. Effect of feedback on noise

Feedback improves the signal-to-noise handling of the stage.

• Noise reduction: internally generated noise is divided by (1 + Aβ), the same factor as the signal.
• Net benefit: SNR improves only if lost gain is restored by a low-noise pre-stage; otherwise signal and noise scale together.

E. Effect of feedback on input and output impedances

The impedance change depends on how the signal is sampled and mixed.

1. Series (voltage) mixing at input: raises input impedance, Z_if = Z_i (1 + Aβ).
2. Shunt (current) mixing at input: lowers input impedance, Z_if = Z_i / (1 + Aβ).
   • Voltage sampling at output: lowers output impedance, Z_of = Z_o / (1 + Aβ) — desirable for a stiff voltage source.
   • Current sampling at output: raises output impedance, Z_of = Z_o (1 + Aβ).

IV. Oscillators

Amplifiers with self-sustaining positive feedback

An oscillator converts DC to a periodic AC output using an amplifier, a frequency-selective feedback network, and no external input signal.

A. Condition for sustained oscillation

Oscillation is the exact balance point of positive feedback, stated by the Barkhausen criterion.

• Magnitude condition: |Aβ| = 1 — the loop replaces exactly the energy it loses each cycle.
• Phase condition: total loop phase shift = 0° or 360°, so feedback is truly regenerative.
• Starting condition: at switch-on |Aβ| > 1 so noise builds; amplitude then drives the transistor toward saturation, reducing gain until |Aβ| = 1 settles a steady amplitude.

B. R-C phase shift Oscillator

Uses resistor–capacitor sections to build the phase shift a CE stage needs to oscillate.

• Principle: the CE amplifier gives 180°; three cascaded RC sections each contribute ~60° to add another 180°, totalling 360°.
• Frequency:

f = 1 / (2πRC√6)

• Symbols: R, C per identical section; √6 from the three-section network.
  • Gain requirement: amplifier must supply |A| ≥ 29 to overcome network attenuation.
  • Use: low and audio frequencies where inductors would be bulky.

C. Hartley Oscillator

An LC oscillator identified by a tapped (split-inductor) tank.

• Tank: two inductors L1, L2 and one capacitor C; the tap sets the feedback fraction.
• Frequency:

f = 1 / (2π√(L_T · C)),  L_T = L1 + L2 + 2M

• Symbols: L_T total inductance, M mutual inductance between coils.
  • Feedback factor: β ≈ L1 / L2.
  • Use: RF signal generators and receiver local oscillators.

D. Colpitts Oscillator

The dual of Hartley — a split-capacitor tank instead of split-inductor.

• Tank: one inductor L, two capacitors C1, C2 in series.
• Frequency:

f = 1 / (2π√(L · C_eq)),  C_eq = C1·C2 / (C1 + C2)

• Symbols: C_eq series capacitance of the pair.
  • Feedback factor: β ≈ C1 / C2.
  • Advantage: better frequency stability at high RF than Hartley because capacitors are less affected by stray inductance.

E. Wien Bridge Oscillators

An RC oscillator using a balanced bridge for a stable, low-distortion audio sine.

• Network: series RC in one arm, parallel RC in another, forming a lead–lag filter that gives 0° phase at one frequency.
• Frequency:

f = 1 / (2πRC)

• Symbols: matched R and C in both arms.
  • Gain requirement: amplifier gain must be exactly 3 (the lead–lag network attenuates by 1/3 at f).
  • Amplitude stabilisation: a lamp or diode network in the negative-feedback arm holds gain at 3, keeping distortion low.
  • Use: precision audio-frequency sine sources.

F. Negative Resistance oscillator

Sustains oscillation by cancelling tank losses with a device whose current falls as voltage rises.

• Principle: a device (tunnel diode, UJT) exhibiting a region where dV/dI < 0 presents negative resistance that offsets the positive resistance of an LC tank.
• Loss cancellation: when |−R_device| = R_tank_loss, net tank resistance is zero and oscillation persists.
• Frequency: set by the tank, f = 1 / (2π√(LC)).
• Feature: needs no feedback loop or phase network — the device itself supplies the energy each cycle, allowing very high frequency operation.