Multiplication in the Wien-bridge Oscillator The Wien-bridge oscillator earns its name from the typical bridge arrangement of the feedbac loops (fig.). This configuration is capable of delivering a clean output sine wave using a low- frequency-determining R-C networ and some negative feedbac. We are interested in computing a figure of merit or for the oscillator that will account for harmonic reection at the output, finding its relationship with the R-C networ s. We shall start considering a signal V fed bac from the output to the amplifier s inputs and resulting in a differential input signal (V - V -). We may write: V V V RC R C RC 3 RNL R R NL ( ) K () Here, it is assumed that the differential amplifier s input-impedance is very high. We can recognize the R-C frequency-sensitive networ as being a nd order bandpass filter. This type of filter has a transfer function in the Laplace domain given by: () s bs as cs with a, b and c being circuit constants. For steady-state sinusoidal operation the above expression may be written as: with s. a ( ) () b c
The resonant frequency o is given by: b ain at resonance is: c a The 3dB bandwidth can be shown to be: b c The networ s is: c b (3) Then, eq.() may be written as: (4) The amplitude-frequency response is described by: The phase-angle response may be obtained from eq.(4): tan π Φ We need now calculate the derivative of Φ() with respect to. From tables for derivatives we find that:
d dx tan ( y) y dy dx Then: dφ d Evaluating Φ () at o : or: Φ ' Φ ' ' Φ ( ) (5) At this point we can verify, using eq.(3), that the of the frequency-sensitive networ is /3. In the next section we will see how a multiplication taes place due to bridge operation in the oscillator. Multiplication Multiplying eq.() by A d yields the condition that must be satisfied for oscillations to tae place: [ ( ) K ] A (6) () is the transfer function of the frequency-sensitive networ. K is the transfer function of the non-linear networ. A d is the amplifier s open-loop gain. At the oscillation s frequency, () and K must be real if A d is a real quantity. For ideal OP-AMPS, A d is considered a real number, actually very large. For real-world devices with internal frequency compensation, A d is a complex quantity having a lowfrequency pole, and its magnitude rolls-off at db per decade above the corner frequency. It may be shown that A d can be considered to be a real quantity in eq.(6) if: d BW/f osc >>9 where BW is the gain-bandwidth product of the OP-AMP and f osc is the oscillation s frequency in hertz.
Selectivity of the frequency-dependent feedbac loop is given by its [eq.(5)]: dφ d Total selectivity resulting from the action of the two feedbac loops may be described by: dφ d For small variations of frequency and phase angles: Φ Φ From fig..b we may write: and for small phase shifts: ( ) sin Φ ( ) K sin Φ ( ) Φ ( ) K Φ We may deduce that: ( ) ( ) K (7) At the oscillation s frequency: ( ) K A d
and: ( ) 3 Then: A d (8) 3 Thus, the bridge is very nearly at balance and is many times. Typical open-loop voltage gain variation with frequency is indicated in fig.3 for an OP- AMP with internal frequency compensation. Here, o is the DC voltage gain expressed in decibels and f o is the low-frequency pole. is the voltage gain in decibels at frequency f. f u is the unity-gain frequency. The following holds due to the db per decade roll-off: BW A f A f f (9) do o d u At a frequency f, the open-loop voltage gain is: Substituting into eq. (8): BW A d () f BW 3 f
Then: BW 9 f The effective then varies inversely with frequency. A typical multiplication factor at Hz, with a 4MHz gain-bandwidth product OP- AMP is: 4 3 6 3 333.33 This would give a value of 444.44 for. For the case of the modified Wien-bridge oscillator using a single variable resistor for frequency control: ( ) RC R C ( ) [ ( ) ] RC () ( ) RC Eq.(7) yields the multiplication factor: () ( ) A d (3) ( ) is then given by: A d ( ) ( ) BW f [ ( ) ]
RC BW (4) π [ ( ) ] If ( )>>: π RC BW (5) is then approximately constant over one decade. ( ) Using eq.(4) we may calculate the ratio when varies between min and MAX. Then: Ratio ( MAX ) ( ) min MAX min min MAX ( ) ( ) ( ) ( ) 4 (6) Table I summarizes Ratio and values as given by eqs. (6) and (5), with as a parameter, for a Wien-bridge oscillator designed for operation over the Hz to Hz decade. TABLE I Ratio (aprox.).5.645 4.47.5.36 39.74.4 3.6.67 9.87.8 3.3 Calculations for have been made with BW 4MHz, R 47 ohms, and C 7.57nF. For other values of, C has been changed accordingly, so the same Hz to Hz decade may be tuned. From the total selectivity point of view, low values for are preferred. We may also observe that given any frequency decade, selectivity at the lower end is slightly greater than that at the upper end ( Ratio>). Eq. (5) indicates that higher decades exhibit smaller values (the higher the decade, the smaller the RC product). Some THD measurements made on the modified Wien-bridge oscillator with and a 6-Volt pea-amplitude output sine wave are shown below. Measurements were conducted using a 334A Hewlett-Pacard Distortion Analyzer.
Hz to Hz decade THD at: Hz Hz is:.%.3% Hz to Hz decade THD at: Hz Hz is:.9%.6% Hz to Hz decade THD at: Hz Hz is:.4%.7% Hz to Hz decade THD at: Hz Hz is: 3.8%.4% (using stabilising lamp) is:.47%.38% (using stabilising lamps in series) Some comments The lower end of the Hz to Hz decade is adversely affected by environmental noise and non-linear distortion introduced by the stabilising lamp. Three or four of these lamps should be series-connected in order to reduce THD to acceptable levels. Miniature lamp types should be preferred (they are less buly). Also, the oscillator should be adequately shielded from external noise sources, such as fluorescent lamps, computers, switch-mode power supplies, etc. When conducting measurements with the Distortion Analyzer at frequencies above Hz, a high-pass filter may be switched-in for noise reection. This may help lower the THD reading. Ramon Vargas Patron rvargas@inictel.gob.pe Lima-Peru, South America June nd 4