Bootstrap Circuits
By feeding the output back to the top of a load resistor, you make it appear much larger — a 100kΩ resistor can look like 2MΩ. This is positive feedback used constructively, and it transforms cathode followers, input impedances, and more.
What Is a Bootstrap Circuit?
Using positive feedback to multiply effective impedance
By AC-coupling the output of a gain stage back to the top of a load resistor, the voltage across that resistor stays nearly constant. If both ends of a resistor move together, very little current flows through it — so the resistor appears much larger than its DC value.
Where A_v is the voltage gain of the feedback path (close to 1 for a cathode follower). For example, a cathode follower with A_v = 0.95 turns a 100kΩ resistor into an effective 2MΩ — a 20× multiplication.
This is positive feedback used constructively. Unlike oscillation-prone positive feedback, bootstrap feedback is inherently stable because A_v is always less than unity — the loop gain never reaches 1.
Ref: Horowitz & Hill, "The Art of Electronics" 3rd Ed. §2.4.3 — Morgan Jones, "Valve Amplifiers" 4th Ed.
Bootstrap Designer
Compute effective impedance and low-frequency limit
The bootstrap multiplies the effective resistance by 1/(1−Av). A cathode follower with Av = 0.95 turns 100kΩ into 2MΩ effective.
Cathode Follower Bootstrap
The classic application
In a cathode follower, the plate resistor connects to B+. By AC-coupling the cathode output back to the top of this resistor (via a large capacitor), the voltage across the resistor stays nearly constant for AC signals. The resistor “disappears” from an AC perspective.
This creates a virtuous cycle: higher effective plate load → higher gain (closer to unity) → better bootstrap → even higher effective load. The gain converges to a value very close to 1.
The coupling capacitor must be large enough to maintain the bootstrap effect at the lowest frequency of interest. For 20Hz with a 100kΩ plate resistor, C ≥ 1/(2π × 20 × 100k) ≈ 80nF — typically 1–10µF film or electrolytic is used for margin.
Input Impedance Multiplication
Bootstrap the grid resistor for ultra-high Z_in
By bootstrapping the grid resistor from the cathode, input impedance is multiplied: Z_in = R_grid / (1 − Av). With Av = 0.95 and R_grid = 1MΩ, the input impedance reaches 20MΩ.
This is crucial for driving high-impedance sources: condenser microphones (50–200MΩ), piezo pickups (1–10MΩ), and oscilloscope probe inputs. Without bootstrap, the grid resistor would load the source and distort the signal.
Where Bootstrap Excels
Oscilloscope Probes
10× probes use bootstrap to achieve 10MΩ input impedance while keeping capacitance low.
Mic Preamps
Tube mic preamps bootstrap the grid resistor for condenser mics that need >10MΩ load.
Phono Preamps
Moving-coil cartridge preamps use bootstrap for high input Z without excessive Johnson noise.
Push-Pull Stages
Bootstrap increases the drive voltage swing in push-pull output stages, improving power and linearity.
Practical Tips
1. Capacitor Sizing
The bootstrap cap must be large enough for the lowest frequency: C ≥ 1/(2π × f_low × R). For 20Hz and 100kΩ, that’s 80nF minimum. Use 1–10µF for margin.
2. Oscillation Risk
Very high bootstrap ratios (Av > 0.98) can cause instability. The positive feedback loop may oscillate if phase shift accumulates at high frequencies. A small series resistor in the bootstrap path can add damping.
3. DC Stability
Bootstrap only works for AC signals — the coupling capacitor blocks DC. The DC operating point is set entirely by the physical resistor value. This is actually an advantage: AC and DC are independently controlled.
Testez vos connaissances
What does bootstrapping a resistor achieve?
References
- Paul Horowitz & Winfield Hill, The Art of Electronics, 3rd ed., Cambridge University Press, 2015. ISBN 978-0521809269Canonical reference for analog design — covers tubes in Ch. 2.4 & Ch. 3.
- Morgan Jones, Valve Amplifiers, 4th ed., Newnes, 2012. ISBN 978-0080966403Modern engineering treatment of tube audio design.