The Miller Effect
A tiny capacitance between grid and plate becomes enormous at the input — multiplied by voltage gain. This is the dominant bandwidth limitation in tube amplifiers, and understanding it is essential to designing fast, wide-band circuits.
The Miller Effect — Bandwidth Killer
How grid-to-plate capacitance is multiplied by voltage gain
In any inverting amplifier, the capacitance between input and output (C_gp) appears at the input multiplied by the voltage gain plus one. This is the Miller effect, and it is the dominant bandwidth limitation in tube amplifiers.
For a 12AX7 with C_gp = 1.7pF and a gain of 60: C_Miller = 1.7 × 61 = 103.7pF. That tiny 1.7pF physical capacitance becomes over 100pF at the input — a 60× multiplication! Combined with the source impedance, this forms an RC low-pass filter that rolls off the high frequencies.
With a 47kΩ source (typical phono cartridge or preceding stage output impedance) and 103.7pF Miller capacitance: f_-3dB = 1/(2π × 47k × 103.7p) = 32.7kHz. Barely above the audio band! With a 100kΩ volume pot, it drops to 15.3kHz — clearly audible rolloff.
Ref: Horowitz & Hill, "The Art of Electronics" 3rd Ed. §2.4.4 — Morgan Jones, "Valve Amplifiers" 4th Ed. Ch.3
Miller Capacitance Calculator
See how gain and capacitance combine to limit bandwidth
Adjust the parameters below. Notice how even a small C_gp becomes enormous at the input when multiplied by gain. The source impedance R_s then sets the bandwidth: higher R_s means lower bandwidth.
Bandwidth vs Gain — Bode Plot
Higher gain = more Miller C = lower bandwidth. Drag the gain slider.
Miller Effect by Tube Type
Why your tube choice determines bandwidth
The table below shows the devastating impact of Miller effect across common tubes. Notice how the EF86 pentode has virtually zero Miller capacitance — its screen grid shields the plate from the control grid, reducing C_gp by a factor of 300×.
| Tube | C_gp | C_gk | μ | Av (typ) | C_Miller | BW @ 47kΩ |
|---|---|---|---|---|---|---|
| 12AX7 | 1.7pF | 1.6pF | 100 | 60 | 103.7pF | 32158360.7kHz |
| 12AT7 | 1.5pF | 2.2pF | 60 | 40 | 61.5pF | 53159739.2kHz |
| 12AU7 | 1.5pF | 1.5pF | 17 | 14 | 22.5pF | 141094807.7kHz |
| 6SN7 | 3.9pF | 4pF | 20 | 16 | 66.3pF | 48168924.4kHz |
| 6SL7 | 3.2pF | 3.2pF | 70 | 43 | 140.8pF | 23515801.3kHz |
| EF86 (pentode) | 0.005pF | 4.3pF | 37 | 200 | 1.0pF | 638317697.4kHz |
BW calculated for 47kΩ source impedance. Red = below 30kHz (audible rolloff).
Fighting the Miller Effect
Four proven techniques from circuit design practice
1. Cascode — The Best Solution
In a cascode, the lower tube operates as a common-cathode stage but its plate swings are tiny (Av ≈ 1) because the upper tube’s low input impedance (~1/gm) clamps the plate voltage. So C_Miller = C_gp × (1 + 1) = 2 × C_gp — just 3.4pF instead of 103pF for a 12AX7! The full voltage gain comes from the upper tube, which operates as common-grid (no Miller effect). Bandwidth improvement: 30× or more.
2. Pentode Mode — Screen Grid Shielding
A pentode’s screen grid electrostatically shields the plate from the control grid. The EF86 has C_gp = 0.005pF compared to 1.7pF for a 12AX7. Even with a gain of 200, C_Miller is only 1pF — essentially zero. The tradeoff: pentodes have higher noise (partition noise from screen current) and more complex power supply requirements.
3. Lower Source Impedance
Since bandwidth = 1/(2π·Rs·Cin), lowering R_s raises the bandwidth proportionally. Use a cathode follower buffer before a high-gain stage: typical output impedance of ~500Ω instead of 47kΩ gives 94× more bandwidth. This is why cathode followers are essential between stages in high-fidelity designs.
4. Neutralization
Feed an inverted copy of the output signal back to the input through a small capacitor equal to C_gp. This cancels the Miller current. Used in RF amplifiers and some hi-fi preamps, but tricky to adjust — overcompensation causes peaking and potential oscillation. Rarely used in audio compared to cascode.
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What is the Miller capacitance formula?
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.