RC Phase-Shift Oscillator Calculator
Calculate and audit the classic loaded three-section RC phase-shift oscillator. The calculator solves frequency, feedback attenuation, required inverting amplifier gain, gain dB, R/C component values, loop-gain startup margin, and matched equal-component tolerance range.
The V1 model uses a documented high-pass RC ladder and complex nodal analysis. It does not model three independent unloaded RC filters, BJT bias, op-amp compensation, amplitude stabilization, phase noise, or full transient startup behavior.
Engineering tool
RC Phase-Shift Oscillator Calculator
Analyze and design the classic three-section loaded RC phase-shift oscillator with an inverting amplifier.
Mode
Parameter panel
Result console
- Solved R
- 6.49747 kΩ
- Solved C
- 10 nF
- Oscillation Frequency
- 1 kHz
- Period
- 1000 µs
- Network β
- 0.0344828V/V
- Required Gain Magnitude
- 29V/V
- Required Gain
- 29.248dB
- Amplifier Polarity
- Inverting
- Loop Gain
- 1|Aβ|
- Startup Margin
- 3.6953e-10%
- Phase at Selected f
- 180deg
- Phase Error
- -1.0368e-10deg
Equal-RC design uses the classic loaded three-section formula f0 = 1/(2πRC√6) and |β| = 1/29.
RC Phase-Shift Formula Audit
- Adopted Topology
- Three identical or specified loaded RC high-pass sections feeding an ideal high-input-impedance inverting amplifier.
- RC Section Orientation
- Series capacitors between nodes with each node shunted by a resistor to ground.
- Amplifier Polarity
- Inverting amplifier; the amplifier contributes 180° phase inversion.
- Loaded Network Model
- Three-node complex nodal analysis; the sections are not treated as independent unloaded first-order filters.
- Classic Frequency
- For equal sections, f0 = 1 / (2πRC√6).
- Classic Beta
- For the adopted equal-RC loaded network at f0, |β| = 1/29.
- Required Gain
- |Arequired| = 1 / |β|, therefore approximately 29 V/V for equal RC.
- General Method
- Unequal R/C values use deterministic bounded phase-crossing search on the nodal transfer function.
- Phase Crossing
- The RC network crossing is β phase = -180° / 180° so the inverting amplifier completes the loop phase.
- Loop Gain
- L = |Aβ|.
- Startup Margin
- Margin = L - 1, displayed as percent above or below threshold.
- Tolerance Model
- Matched equal-RC approximation: fmin uses Rmax and Cmax; fmax uses Rmin and Cmin.
- GBW Boundary
- Op-amp GBW, phase margin, loading, slew rate, amplitude limiting, and distortion are not full SPICE-modeled.
- Formula Used
- 1 kHz, |β| 0.0344828, |Areq| 29.
Formula reference
RC Phase-Shift Oscillator Formulas
The adopted oscillator uses three loaded high-pass RC sections and an inverting amplifier. Equal-RC mode reduces to the classic f0 = 1/(2πRC√6) and required gain magnitude of about 29.
Equal RC: f0 = 1 / (2πRC√6)Equal RC: |β| = 1/29|Arequired| = 1 / |β|Equal RC: |Arequired| ≈ 29Inverting gain magnitude: |A| = Rf / RinRf = |A| RinL = |Aβ|Startup margin = L - 1Voltage gain dB = 20log10(|A|)Variable definitions
- R1, R2, R3
- shunt resistors in the loaded RC ladder
- C1, C2, C3
- series capacitors between network nodes
- β
- RC network voltage feedback magnitude
- A
- inverting amplifier gain magnitude
- Rin
- input resistor of the inverting amplifier
- Rf
- feedback resistor of the inverting amplifier
- L
- loop gain magnitude |Aβ|
RC Phase-Shift Formula Audit
| Adopted Topology | Three cascaded loaded high-pass RC sections feeding an ideal high-input-impedance inverting amplifier. |
|---|---|
| RC Section Orientation | Series capacitors between stages with each node shunted to ground by a resistor. |
| Amplifier Polarity | Inverting amplifier; use gain magnitude for loop calculations, but the voltage gain has negative polarity. |
| Loaded Network Model | Complex nodal analysis of the three-node ladder; not three independent first-order RC filters. |
| Classic Frequency | For equal R and C, f0 = 1 / (2πRC√6). |
| Classic Beta | For equal R and C at f0, |β| = 1/29. |
| Required Gain | |Arequired| = 1/|β|, approximately 29 V/V or 29.24796 dB. |
| General Network Method | Unequal components use bounded deterministic phase-crossing search on the nodal transfer function. |
| Phase Crossing | The network crossing is β phase = -180° / 180°, allowing the inverting amplifier to complete the loop phase. |
| Loop Gain | L = |Aβ|. |
| Startup Margin | Margin = L - 1, displayed as (L - 1) x 100%. |
| Tolerance Model | Matched equal-component approximation: fmin uses Rmax/Cmax and fmax uses Rmin/Cmin. |
| GBW Boundary | Op-amp GBW, input loading, phase lag, amplitude limiting, slew rate, and distortion require datasheet and simulation checks. |
Worked Examples
Classic frequency
Known: R = 10 kΩ, C = 10 nF
f0 = 1 / (2π x 10 kΩ x 10 nF x √6) ≈ 649.747 Hz.
Classic beta
Known: Three equal loaded RC sections at f0
|β| ≈ 1/29 ≈ 0.0344828.
Required gain
Known: |β| = 1/29
|Arequired| = 29 V/V.
Amplifier polarity
Known: RC network phase ≈ 180°
The amplifier must be inverting to complete the loop phase.
Inverting Rf
Known: Rin = 10 kΩ, gain magnitude = 29
Rf = |A|Rin ≈ 290 kΩ.
Startup gain
Known: |A| = 30, β = 1/29
Loop gain ≈ 1.03448 and startup margin ≈ 3.448%.
Insufficient gain
Known: |A| = 28, β = 1/29
Loop gain ≈ 0.96552, below the ideal startup threshold.
Solve R
Known: Target f = 1 kHz, C = 10 nF
R = 1/(2πfC√6), then analyzer returns approximately 1 kHz.
Solve C
Known: Target f = 1 kHz, R = 10 kΩ
C = 1/(2πfR√6), then analyzer returns approximately 1 kHz.
Classic phase
Known: Equal R/C at f0
Network phase is approximately -180° / 180°, with near-zero phase error.
Below f0
Known: Evaluate at f0 / 2
The phase error moves to one side of the 180° condition.
Above f0
Known: Evaluate at 2f0
The phase error moves to the opposite side of the 180° condition.
Tolerance
Known: R ±1%, C ±5%
fmin < fnominal < fmax in the matched equal-component approximation.
Unit conversion
Known: 10 nF = 0.01 µF
Both capacitance entries produce the same oscillation frequency.
Frequency units
Known: 1000 Hz = 1 kHz
Both target frequency entries solve the same R or C value.
Unequal components
Known: R1/R2/R3 and C1/C2/C3 differ
The calculator uses nodal phase crossing and does not apply the equal-RC formula.
Design round trip
Known: Target frequency -> solved R/C -> analyzer
The analyzer recovers the target frequency within numerical tolerance.
Gain dB
Known: Gain magnitude = 29
20log10(29) ≈ 29.24796 dB, using 20log10 for voltage gain.
Engineering Notes
RC phase-shift oscillator
The circuit combines an inverting amplifier with an RC network that contributes the remaining phase shift.
Three-section network
The V1 calculator adopts the classic three-section loaded high-pass RC ladder.
Barkhausen criterion
At steady state, the ideal loop has approximately unity magnitude and zero phase modulo 360 degrees.
Feedback attenuation
The classic loaded network attenuates strongly, so the amplifier needs high gain magnitude.
Inverting amplifier
The gain magnitude is positive in calculations, but the physical amplifier must invert the signal.
Component loading
The RC sections load one another; this is why independent 60-degree section reasoning is not exact.
Component matching
Matching affects frequency, phase balance, attenuation, required gain, and distortion.
Op-amp GBW
High gain magnitude makes gain-bandwidth product and op-amp phase shift important.
Startup margin
Startup usually needs loop gain slightly above unity; excessive margin can increase clipping.
Amplitude limiting
Real amplitude is controlled by amplifier nonlinearity or explicit stabilization.
Slew rate
Large sine-wave output at high frequency can be slew-rate limited even if small-signal gain seems sufficient.
Verification
Critical designs should be checked with SPICE, datasheets, tolerance review, and bench measurement.
Common Mistakes
Support reference
FAQ
What is an RC phase-shift oscillator?
An RC phase-shift oscillator uses an inverting amplifier and a frequency-selective RC feedback network. The amplifier supplies 180 degrees of phase inversion and the RC network supplies the remaining 180 degrees at the oscillation frequency.
How do I calculate its oscillation frequency?
For the adopted classic three-section loaded equal-RC high-pass network, f0 = 1 / (2πRC√6). This formula applies only to the documented topology and assumptions.
Why are three RC sections used?
A practical RC network needs multiple sections to approach the required 180 degrees of phase shift while also providing a usable feedback signal to the inverting amplifier.
Why is the frequency formula divided by √6?
The √6 factor comes from the loaded three-section network transfer function. It is not obtained by treating the sections as three independent unloaded 60-degree filters.
Why is the required amplifier gain about 29?
For the adopted equal-RC loaded network, the feedback magnitude at the phase condition is approximately 1/29. The inverting amplifier therefore needs gain magnitude of about 29 for unity loop gain.
Why does the amplifier need to be inverting?
The RC network contributes approximately 180 degrees of phase shift at f0. The inverting amplifier contributes another 180 degrees so the total loop phase is 0 degrees modulo 360 degrees.
Do the three RC sections each contribute exactly 60 degrees?
Not in the classic unbuffered loaded network. The sections load each other, so the exact result must come from the loaded network model rather than three independent single-pole stages.
Why does loading between RC stages matter?
Loading changes both the phase response and attenuation of the network. That is why the classic formula and gain condition differ from a simple multiplication of three independent RC sections.
Will a gain of exactly 29 always start oscillation?
No. A gain magnitude of 29 is the ideal steady-state threshold for the equal-RC model. Real startup usually needs slightly more small-signal loop gain, followed by amplitude limiting or stabilization.
How do I choose R and C for a target frequency?
For equal sections, choose a practical capacitor or resistor value, then use R = 1/(2πf0C√6) or C = 1/(2πf0R√6). Verify the solved values with the analyzer.
Can the RC values be unequal?
Yes, but unequal values change the phase crossing, feedback attenuation, and required gain. This calculator uses nodal analysis and a bounded phase-crossing search for unequal networks.
How do component tolerances affect the oscillator?
For matched equal-section tolerance estimates, fmin uses maximum R and C and fmax uses minimum R and C. Independent mismatch also affects phase condition and required gain.
How does op-amp bandwidth affect oscillation?
The op-amp must still provide enough gain and acceptable phase shift at the oscillation frequency. High required gain makes gain-bandwidth product and phase margin important.
What is the difference between an RC phase-shift oscillator and a Wien Bridge oscillator?
An RC phase-shift oscillator uses an inverting amplifier and a three-section RC network with required gain about 29. A Wien Bridge oscillator uses a lead-lag bridge with a non-inverting amplifier and ideal equal-component gain of 3.
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Engineering Disclaimer
This calculator uses an ideal loaded RC ladder model for first-pass oscillator design. It does not guarantee startup, amplitude stability, low distortion, op-amp suitability, or production tolerance performance. Verify critical designs with simulation, component datasheets, and measured prototypes.