RC oscillator calculator: Wien bridge and phase-shift oscillators

The oscillation frequency of an op-amp Wien-bridge, phase-shift or four-section bubba oscillator from its resistors and capacitors, the gain its RC network asks for and the loop gain your RF and RG give, the op-amp bandwidth and slew rate that frequency needs, and the parts for a target frequency rounded to an E-series. The equations are TI's SLOA060; the defaults are its Wien bridge, 1.59 kHz from 10 kΩ and 10 nF, and the page's tests hold the calculation to the values SLOA060, AN-31 and AN-20 print.

Fig 1 — Wien bridge: R1 = R2 = 10.0 kΩ, C1 = C2 = 10.0 nF, R_F 20.0 kΩ, R_G 10.0 kΩ. Below, the feedback network alone, SLOA060 Eq 12: its phase reaches 0° at f0 = 1.59 kHz, where |β| = 1/3, −9.54 dB, so the amplifier needs a gain of 3.00. It has 3.00, a loop gain of 1.000, exactly 1.

Calculate

Topology

Series resistor R1 (kΩ)

Series capacitor C1 (nF)

Shunt resistor R2 (kΩ)

Shunt capacitor C2 (nF)

Feedback resistor RF (kΩ)

Gain resistor RG (kΩ)

Op-amp gain-bandwidth GBW (MHz)

Peak output amplitude Vp (V)

Oscillation frequency f0: 1.592 kHz
Feedback network at f0: |β| · phase: 1/3 (−9.54 dB) · 0°
Gain needed, A = 1/|β| · RF/RG for it: 3 · 2
Gain of the stage, 1 + RF/RG: 3
Loop gain A·|β|: 1.000 (+0.00 %)
GBW needed for a Wien bridge, 43 × f0 (SLOA060 p. 7): 68.4 kHz
GBW needed for f0 a decade below GBW/A (SLOA060 Fig 3, derived): 47.7 kHz
Slew rate needed, 2π·Vp·f0 at 2.5 V peak: 25.0 mV/µs

The loop gain is within 2 % of 1. SLOA060 built its Wien bridge on that edge: with RF = 2RG it oscillated, clipping at both rails with 2.8 % distortion, and "oscillations ceased when RF was decreased by a mere 0.8%" (p. 11). A lamp, a JFET AGC or diodes in the feedback hold the gain at 3 with far less distortion (pp. 12–14).

How this is calculated

Standard: TI SLOA060, Sine-Wave Oscillator (Mancini, Palmer, 2001), §3–8 (Eq 6, 10–17; pp. 4–18); TI SNLA140D, AN-31 Amplifier Circuit Collection (Figures 2-1, 2-2, 2-7; pp. 15, 18); TI SNOA621C, AN-20 (Figure 26, p. 19); TI SBOA092, Handbook of Operational Amplifier Applications (p. 83)

$A\beta = 1\angle{-180^\circ}\ \text{(negative feedback)}, \qquad A\beta = 1\angle{0^\circ}\ \text{(positive feedback)}$: The Barkhausen criterion, SLOA060 p. 4: the system "goes unstable when the denominator in equation 5 becomes zero, i.e., when 1 + Aβ = 0". A is the amplifier gain, β the fraction of the output the network feeds back.
$\frac{V_+}{V_{TEST}} = \frac{1}{1 + \frac{R_1}{R_2} + \frac{C_2}{C_1} + j\left(\frac{\omega_0}{\omega_1} - \frac{\omega_2}{\omega_0}\right)}, \qquad \omega_1 = \frac{1}{R_1 C_2},\ \omega_2 = \frac{1}{R_2 C_1}$: SLOA060 Eq 12 (p. 11): the Wien network, R1 and C1 in series from the output, R2 and C2 in parallel to ground. The calculator evaluates it at every frequency for the figure.
$f_0 = \frac{1}{2\pi\sqrt{R_1 R_2 C_1 C_2}}, \qquad A = 1 + \frac{R_1}{R_2} + \frac{C_2}{C_1} = 1 + \frac{R_F}{R_G}$: Eq 12 with its imaginary term zero (ω0² = ω1ω2) and its real part inverted: derived here for unequal arms. With equal parts it is SLOA060's f0 = 1/(2πRC) (p. 11) and Eq 13's β = 1/3, "requiring A = 3. RF must be set to twice the value of RG". SBOA092 p. 83 gives the same fO = 1/(2πRC).
$A\beta = A\left(\frac{1}{RCs + 1}\right)^3, \qquad \omega_0 = \frac{\tan 60^\circ}{RC} = \frac{1.732}{RC}, \qquad |\beta| = \left(\tfrac{1}{2}\right)^3, \quad A = \frac{R_F}{R_G} = 8$: SLOA060 Eq 14 (p. 15), three RC sections taken as independent, "−60°" each; AN-31 writes the same f = tan(60°)/(2πRC). With 10 kΩ and 10 nF it is 2.76 kHz, SLOA060's "2.76 kHz". Exact only when buffers stop the sections loading each other (p. 16); tan 60° = 1.732.
$A\beta = A\left(\frac{1}{RCs + 1}\right)^4, \qquad \omega_0 = \frac{1}{RC}, \qquad |\beta| = \frac{1}{\sqrt{2}^{\,4}} = \frac{1}{4}, \quad \phi = \tan^{-1}(1) = 45^\circ$: SLOA060 Eq 15–17 (p. 17), the four-section bubba oscillator: "The gain, A, must equal 4" (p. 18). Eq 16 prints the section as 1/(j + 4); the value it evaluates, 1/√2⁴, is that of 1/(1 + j).
$GBW > 43\,\omega_{OSC}\ \text{(Wien)}, \qquad f_0 \le 0.1\,\frac{GBW}{A}, \qquad SR > 2\pi V_P f_0$: SLOA060 p. 7: the op amp needs "a gain bandwidth at least one decade above the oscillation frequency", read from Figure 3 as f0 at most a tenth of the closed-loop corner GBW/A, and "The Wien bridge requires a gain bandwidth greater than 43 ωOSC" (both sides angular, so GBW in Hz above 43 f0). p. 8: "The slew rate must be greater than 2πVPf0".

Assumptions

The op amp is ideal at f0: SLOA060's equations "are valid when the op-amp open-loop gain is large and the oscillation frequency is less than 0.1 ω3dB" (p. 9). The bandwidth rows show how far that holds.
The phase-shift sections do not load each other. True for the buffered and bubba circuits; for the single-op-amp ladder SLOA060 measured a higher frequency and gain than Eq 14 gives (p. 15).
All sections of a ladder are equal, as in every circuit SLOA060 and AN-31 draw. The Wien arms may differ.
RG does not load the last section, and no input capacitance or large-resistor pole shifts the phase (SLOA060 pp. 8, 16).
Nominal part values: tolerance moves f0, as SLOA060's 1.57 kHz against 1.59 kHz shows.
The design mode uses equal R and equal C throughout, and rounds RF up so the loop gain is not below 1. Amplitude control (lamp, JFET, diodes) is not modelled.

What sets the frequency of an RC oscillator

An oscillator is an amplifier that has been made unstable on purpose, at one frequency. TI's application report SLOA060, Sine-Wave Oscillator, by Ron Mancini and Richard Palmer, starts from the feedback equation V_OUT/V_IN = A/(1 + Aβ), where A is the amplifier's gain and β the fraction of the output fed back. The system "goes unstable when the denominator in equation 5 becomes zero, i.e., when 1 + Aβ = 0, or Aβ = –1. The key to designing an oscillator is ensuring that Aβ = –1. This is called the Barkhausen criterion" (p. 4). Two conditions are packed into it: the loop gain has magnitude 1, and the loop's phase shift is exactly right, "Aβ = 1∠ –180° for a negative feedback system" and "Aβ = 1∠ 0°" for a positive one.

The phase condition sets the frequency, because only one frequency has the right phase. "Phase shift determines the oscillation frequency because the circuit oscillates at whatever frequency accumulates a 180° phase shift" (p. 5). The magnitude condition sets the gain: whatever the RC network loses at that frequency, the amplifier has to put back. That is the whole of the calculator. For each topology it finds the frequency where the network's phase is right, reads the network's loss there, and compares the gain your R_F and R_G give with the gain that loss asks for. The figure plots the network alone, gain and phase, so you can see both conditions being met at one point.

The frequency-selective part is made of resistors and capacitors rather than an inductor for a practical reason SLOA060 states directly: "LC and LR oscillators are not considered here because low frequency inductors are expensive, heavy, bulky, and highly nonideal" (p. 5). An RC section contributes at most 90°, so "at least two poles must be used". SLOA060 also says why op-amp RC oscillators stay at low frequencies: "Op-amp oscillators are restricted to the lower end of the frequency spectrum because op amps do not have the required bandwidth to achieve low phase shift at high frequencies" (p. 3). For higher frequencies see the LC resonance calculator and the crystal load capacitor calculator.

Wien bridge oscillator: f0 = 1/(2πRC) and a gain of 3

The Wien bridge feeds the output back to the non-inverting input through two RC arms: R1 in series with C1 from the output, and R2 in parallel with C2 from the input to ground. SLOA060 breaks the loop, applies a test voltage and derives the network's transfer function, its Eq 12 (p. 11): 1/(1 + R1/R2 + C2/C1 + j(ω/ω1 − ω2/ω)), with ω1 = 1/R1C2 and ω2 = 1/R2C1. With equal parts the imaginary term vanishes at ω = 1/RC, the real part is 3, and "This results in an overall feedback factor of β = 1/3". So: "The gain, A, of the negative feedback portion of the circuit must then be set such that |Aβ| = 1, requiring A = 3. R_F must be set to twice the value of R_G to satisfy this condition." TI's handbook SBOA092 gives the same f_O = 1/(2πRC) for its lamp-stabilised Wien bridge, over "100 to 6000 Hz" (p. 83).

Unequal arms are allowed. SLOA060 goes on to say that each capacitor "must each contribute 90° of phase shift toward the 180° required for oscillation at ω0. This requires that C1 = C2 and R1 = R2", but Eq 12 itself only needs its imaginary term to be zero, which happens at ω0 = √(ω1ω2), f₀ = 1/(2π√(R1R2C1C2)), whatever the ratios. The feedback factor there is 1/(1 + R1/R2 + C2/C1), so the gain needed rises as the arms move apart. The calculator takes all four parts and uses that general form; it is derived here from Eq 12, not printed in the report. The equal-part case is where the gain is lowest, which is the practical reason SLOA060 insists on it. Enter R2 and C2 equal to R1 and C1 for the textbook result.

Phase-shift oscillator: three RC sections, tan 60° and a gain of 8

The phase-shift oscillator puts an inverting amplifier, which supplies 180° of its own, in front of a ladder of RC sections that supplies the other 180°. SLOA060 writes the loop for three sections as its Eq 14, Aβ = A(1/(RCs + 1))³, and states the assumption it rests on: "The usual assumption is that the phase shift sections are independent of each other, allowing equation 14 to be written. The loop phase shift is –180° when the phase shift of each section is –60°. This occurs when ω = 2πf = 1.732/RC (tan 60° = 1.732…). The magnitude of β at this point is (1/2)³, so the gain, A, must be 8 for the system gain of unity" (p. 15). AN-31 prints the same frequency as tan(60°)/(2πRC) beside both of its versions (p. 15).

Whether that assumption holds is the difference between the calculator's two phase-shift topologies. On one op amp, each section drives the next directly and they load each other; SLOA060 built it and reported the gap (p. 15): "The oscillation frequency with the component values shown in Figure 14 is 3.76 kHz rather than the calculated oscillation frequency of 2.76 kHz. Also, the gain required to start oscillation is 27 rather than the calculated gain of 8. These discrepancies are partially due to component variations, however, the biggest factor is the incorrect assumption that the RC sections do not load each other." With op-amp buffers between sections, "the buffered phase-shift oscillator performs more nearly at the calculated frequency and gain" (p. 16). Neither SLOA060 nor AN-31 gives a formula for the loaded ladder, so this calculator does not offer one: on the one-op-amp setting it shows Eq 14's figure and says that the bench will differ, by SLOA060's measurement +36 % in frequency.

AN-31 handles the difference in its gain rather than its frequency. Its single-amplifier circuit asks for R2/R1 "≫ 8" and uses 150; its buffered one asks for "8 ≤ R2/R1 ≤ 10" and uses 9.09 (p. 15).

The bubba oscillator: four sections of 45°

With four buffered sections each supplies 45°, the loop phase condition is met at ω = 1/RC, and the loss is smaller. SLOA060 Eq 15–17 (p. 17): Aβ = A(1/(RCs + 1))⁴, |β| = 1/√2⁴ = 1/4 and φ = tan⁻¹(1) = 45°, so "The gain, A, must equal 4 for oscillation to occur" (p. 18). The report calls it "the most stable RC oscillator configuration", because four sections give the steepest phase slope dφ/dω at the oscillation frequency (p. 6), and a quad op amp gives quadrature outputs from alternate sections for free. Eq 16 is printed with the section as 1/(j + 4); the value it evaluates, 1/√2⁴, is that of 1/(1 + j), which is what the calculator uses.

RC oscillator frequency table

f₀ for round values of R and C: the Wien bridge (and the bubba oscillator, which shares 1/(2πRC)) and the three-section phase-shift oscillator, which is tan 60° = 1.732 times higher for the same parts. The one-op-amp ladder is the Eq 14 figure, which SLOA060 found low.

R	C	Wien, bubba: 1/(2πRC)	Phase shift: tan 60°/(2πRC)
1.0 kΩ	1.0 nF	159 kHz	276 kHz
1.0 kΩ	10 nF	15.9 kHz	27.6 kHz
1.0 kΩ	100 nF	1.59 kHz	2.76 kHz
1.0 kΩ	1.0 µF	159 Hz	276 Hz
10 kΩ	1.0 nF	15.9 kHz	27.6 kHz
10 kΩ	10 nF	1.59 kHz	2.76 kHz
10 kΩ	100 nF	159 Hz	276 Hz
10 kΩ	1.0 µF	15.9 Hz	27.6 Hz
100 kΩ	1.0 nF	1.59 kHz	2.76 kHz
100 kΩ	10 nF	159 Hz	276 Hz
100 kΩ	100 nF	15.9 Hz	27.6 Hz
100 kΩ	1.0 µF	1.59 Hz	2.76 Hz

Frequency scales as 1/RC, so a decade in either part is a decade in f₀. Very large resistors and very small capacitors are where the model gives out first: see below.

Worked example: SLOA060's Wien bridge oscillator

SLOA060's Figure 8 is the calculator's default: R = 10 kΩ, C = 10 nF, R_F = 20 kΩ, R_G = 10 kΩ, a TLV2471 on +5 V, "with component values selected to provide an oscillation frequency of ω0 = 2πf0, where f0 = 1/(2πRC) = 1.59 kHz" (p. 11).

f0        1/(2π × 10 kΩ × 10 nF)                 = 1.592 kHz
network   β at f0, Eq 13                         = 1/3 (−9.54 dB)
gain      1 + RF/RG = 1 + 20 kΩ/10 kΩ            = 3.000
loop      Aβ = 3 × 1/3                           = 1.000
RF +1 %   1 + 20.2/10, × 1/3                     = 1.0067
RF −0.8 % 1 + 19.84/10, × 1/3                    = 0.9947
GBW       43 × f0 (p. 7)                         = 68.4 kHz
slew      2π × 2.5 V × f0 (p. 8)                 = 25.0 mV/µs

The report measured "1.57 kHz, caused by varying component values", −1.4 % from the calculation. The loop gain of exactly 1 is the interesting part. With R_F = 2R_G the circuit did oscillate, but with its output clipped at both rails and 2.8 % distortion. Then "The feedback resistor was then adjusted ±1%. … The distortion grew as the saturation increased with increasing R_F, and oscillations ceased when R_F was decreased by a mere 0.8%." The calculator reproduces the boundary: R_F at +1 % gives a loop gain of 1.0067, at −0.8 % 0.9947, below 1. A bare Wien bridge has no gain margin to spare, which is why SLOA060's next three circuits are all ways of holding the gain at 3: a lamp for R_G (less than 0.1 % distortion, p. 12), and a JFET automatic gain control, "less than 0.2%" (p. 13). In the AGC circuit, Figure 12, the report gives the JFET-off minimum as "2.87 (1+R_F/R_G1)"; with its R_F of 18.2 kΩ and R_G1 of 10 kΩ that expression is 2.82.

The bandwidth rows are comfortable here: 68.4 kHz is the 43 × f₀ SLOA060 asks of a Wien bridge's op amp, and the TLV247x has 2.8 MHz. The slew rate needed for a 2.5 V peak is 25.0 mV/µs, which is nothing; the slew rate calculator does the same sum for any amplitude.

AN-31's Wien bridge with AGC (Figure 2-7, p. 18) uses 2 kΩ and 10 nF, f = 1/(2πC1R5) = 7.96 kHz, and a "Set Gain = 3.1". The gain its resistors give is (R3 + R4)/R3 = 3.13 with R3 = 4.7 kΩ to the JFET and R4 = 10 kΩ from the output; the note prints the expression as "(R3 + R4)/R4", which would be 1.47. The 3.1 is the R3 form.

Worked example: SLOA060's phase-shift and bubba oscillators

SLOA060 builds all three ladder circuits with the same 10 kΩ and 10 nF and the same 1.5 MΩ R_F, changing only R_G. Select each topology in the calculator and the defaults are that figure's parts.

f0        tan 60° / (2π × 10 kΩ × 10 nF)         = 2.757 kHz
network   |β| = (1/2)³                           = 1/8 (−18.06 dB)
one amp   RF/RG = 1.5 MΩ / 55.2 kΩ               = 27.17
buffered  RF/RG = 1.5 MΩ / 180 kΩ                = 8.33
bubba     f0 = 1/(2π × 10 kΩ × 10 nF)            = 1.592 kHz
          RF/RG = 1.5 MΩ / 360 kΩ                = 4.17

One op amp, Figure 14: Eq 14 gives 2.76 kHz and a gain of 8; SLOA060 measured 3.76 kHz (+36 %) and needed 27, the 27.17 of its 1.5 MΩ/55.2 kΩ. The distortion was "a low 0.46%" (p. 15).
Buffered, Figure 16: "The circuit oscillated at 2.9 kHz compared to an ideal frequency of 2.76 kHz, and it oscillated with a gain of 8.33 compared to an ideal gain of 8" (p. 16), +5.2 % and the 8.33 of 1.5 MΩ/180 kΩ. The report puts the remainder down to R_F and R_G: the large R_F "created a pole with C_INof approximately 5 kHz", and R_G still loaded the last section. "Adding a buffer between the last RC section and V_OUTlowered the gain and the oscillation frequency to the calculated values."
Bubba, Figure 18: "The test circuit oscillated at 1.76 kHz rather than the ideal frequency of 1.72 kHz when the gain was 4.17 rather than the ideal gain of 4" (p. 18). The 4.17 is 1.5 MΩ/360 kΩ = 4.17. The 1.72 kHz does not follow from the report's own ω = 1/RC with the parts drawn, which gives 1.59 kHz; against that the measurement is +11 %. The report does not say which figure is meant. The quadrature oscillator on the same page, with the same parts and the same ω = 1/RC, is given as "the calculated 1.59 kHz" and measured at 1.65 kHz, +3.7 %.
AN-31's versions use 2.71 kΩ and 10 nF, which Eq 14 puts at 10.2 kHz.

Worked example: AN-20's Wien bridge with unequal arms

AN-20, An Applications Guide for Op Amps, draws an amplitude-stabilised Wien bridge (Figure 26, p. 19) whose arms are nothing like equal: 300 kΩ and 0.068 µF in series, 10 kΩ and 2.2 µF in parallel, with the output labelled "16.5 V_PP, 10 Hz". The note gives no equation. Entered as R1 = 300 kΩ, C1 = 68 nF, R2 = 10 kΩ, C2 = 2200 nF, Eq 12's general form gives 7.51 Hz, the frequency of an equal-arm bridge of54.8 kΩ and 387 nF, and a feedback factor of 1/63.4: the amplifier needs a gain of 63.4 rather than 3. AN-20's FET, biased by a peak detector on the output, sets that gain. The calculated frequency+33 % short of the labelled 10 Hz, a gap AN-20 does not discuss. The example is worth having for the gain alone: lopsided arms cost a factor of 21 in the gain the amplifier has to supply.

Designing an RC oscillator for a frequency

Switch the calculator to "Parts for a frequency" and it solves f₀= k/(2πRC) for R or C, with k = 1 for the Wien bridge and bubba and tan 60° for three sections, rounds the result to an E-series, and gives the frequency the rounded part produces. It then sets R_F from your R_G for the required gain and rounds it up, never down, so the loop gain cannot fall below 1. For a 1 kHz tone on SLOA060's 10 nF:

Wien bridge: R = 15.92 kΩ. E24 gives 16.0 kΩ and 995 Hz(−0.53 %); E96 gives 15.8 kΩ and 1.01 kHz(+0.73 %). R_F is 20.0 kΩ for R_G = 10 kΩ.
Buffered phase shift: R = 27.57 kΩ. E24 gives 27.0 kΩ and 1.02 kHz(+2.10 %); E96 gives 27.4 kΩ and 1.01 kHz(+0.61 %). With R_G = 180 kΩ the exact R_Fis 1.44 MΩ, rounded up to 1.50 MΩ.
Solving for C instead, with R = 10 kΩ: C = 15.92 nF, and E12 gives 15.0 nF,1.06 kHz (+6.1 %). Capacitors come in coarser steps than resistors, which is why choosing C and solving for R is the usual order.

Rounding R_F up is a choice, not a rule from the sources. A Wien bridge with R_F exactly 2R_G sits on the edge SLOA060 measured, and a few per cent over is the price of starting reliably without amplitude control. For a closer pair the E-series calculator finds series and parallel combinations.

Where the RC oscillator model stops being valid

The op amp is not ideal. SLOA060's circuit equations "are valid when the op-amp open-loop gain is large and the oscillation frequency is less than 0.1 ω3dB" (p. 9). Its section 6 explains why: the op amp's own phase shift "affects the performance of the oscillator circuit by lowering the oscillation frequency, and the reduction in A_CLcan make Aβ < 1 and the oscillator then ceases to oscillate" (p. 7). Hence "the op amp should be chosen with a gain bandwidth at least one decade above the oscillation frequency", and for the Wien bridge, "a gain bandwidth greater than 43 ω_OSC to maintain the gain and frequency within 10% of the ideal values". The report's Figure 4 shows the effect on distortion with three op amps of 0.4, 2.8 and 10 MHz in the same Wien bridge (p. 8). The calculator reads Figure 3 as f₀ at most a tenth of the closed-loop corner GBW/A, which matters most for the one-op-amp phase shifter: at a gain of 27.2 and 2.76 kHzthat is 749 kHz of GBW.

Slew rate. "The slew rate must be greater than 2πV_Pf₀, where V_P is the peak output voltage and f₀ is the oscillation frequency; otherwise, distortion of the output signal results" (p. 8).

Large resistors. "Care must be taken when using large feedback resistors because they interact with the input capacitance of the op amp to create poles with negative feedback, and both poles and zeros with positive feedback. Large resistor values can move these poles and zeros into the neighborhood of the oscillation frequency and affect the phase shift" (p. 8). The buffered phase shifter's 1.5 MΩ R_F is SLOA060's own example of it.

Loading. Eq 14 and Eq 15 assume the sections do not load each other and that R_G does not load the last one. The one-op-amp ladder breaks the first; any ladder whose R_G is not much larger than R bends the second, and the calculator flags an R_G under ten times R.

Amplitude. The linear model says nothing about the output level. With loop gain above 1 the amplitude grows until something limits it, and in a bare circuit that is the op amp's output hitting the rails. SLOA060 p. 6: "When the gain is too low, oscillations cease under worst case conditions, and when the gain is too high, the output wave form looks more like a square wave than a sine wave." Low distortion needs a gain that is held, not merely set: a lamp, a JFET AGC, or diodes, as in SLOA060 §8.1, AN-20 Figure 26 and AN-31 Figure 2-7.

Common RC oscillator mistakes

Using 1/(2πRC) for a three-section phase shifter. That is the Wien bridge; three buffered sections oscillate tan 60° = 1.732 times higher for the same R and C.
Trusting the independent-section formula on a one-op-amp ladder. SLOA060 measured +36 % in frequency and a gain of 27 against 8.
Setting the Wien gain to exactly 3 and expecting a clean sine. It sits on the edge: 0.8 % low in R_F stopped SLOA060's circuit, and at 3 it clipped. Add amplitude control.
Mixing up the gain forms. The Wien stage is non-inverting, 1 + R_F/R_G = 3, so R_F = 2R_G; the phase-shift stages are inverting, R_F/R_G = 8 or 4. The op-amp gain calculator has both.
Choosing an op amp by its GBW alone for a high-gain ladder. The closed-loop bandwidth is GBW divided by the gain, and a 27× stage uses up bandwidth 27 times as fast.
Loading the output. SLOA060 p. 16: the purest sine is "taken from the output of the last RC section. This is a high-impedance node, so a high impedance input is mandated to prevent loading and frequency shifting with load variations."
Confusing an RC sine oscillator with a relaxation oscillator. A 555 astable or a comparator with RC feedback produces a square wave with a different timing equation; see the 555 timer calculator.