Q Multiplication in the Wien-bridge Oscillator

Article : Ramon Vargas Patron - Lima Peru

Email :ramonvargas15353@gmail.com

Q Multiplication in the Wien-bridge Oscillator

The Wien-bridge oscillator earns its name from the typical bridge arrangement of the feedback loops (fig.1). This configuration
is capable of delivering a clean output sine wave using a low-Q frequency-determining R-C network and some negative feedback.

We are interested in computing a figure of merit or Q for the oscillator that will account for harmonic rejection at the output, finding its
relationship with the R-C networks Q.

We shall start considering a signal V_{1} fed back from the output to the amplifiers inputs and resulting in a
differential input signal (V_{+ }- V_{-)}. We may write:

(1)

Here, it is assumed that the differential amplifiers input-impedance is very high. We can recognize the R-C frequency-sensitive
network as being a 2^{nd }order bandpass filter. This type of filter has a transfer function in the Laplace domain given by:

with a, b and c being circuit constants. For steady-state sinusoidal operation the above expression may be written as:

(2)

with s = jw .

The resonant frequency w
_{o} is given by:

Gain at resonance is:

The 3dB bandwidth can be shown to be:

The networks Q is:

(3)

Then, eq.(2) may be written as:

(4)

The amplitude-frequency response is described by:

The phase-angle response may be obtained from eq.(4):

We need now calculate the derivative of F (w ) with respect to w . From tables for derivatives we find that:

Then:

Evaluating F
^{'}(w
) at w
= w
_{o}:

or:

(5)

At this point we can verify, using eq.(3), that the Q of the frequency-sensitive network is 1/3. In the next section we will see how a Q multiplication takes place due to bridge operation in the oscillator.

Q Multiplication

Multiplying eq.(1) by A_{d} yields the condition that must be satisfied for oscillations to take place:

(6)

G(jw ) is the transfer function of the frequency-sensitive network.

K is the transfer function of the non-linear network.

A_{d} is the amplifier's open-loop gain.

At the oscillations frequency, G(jw
) 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 low-frequency pole, and its magnitude
rolls-off at 20dB 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:

where GBW is the gain-bandwidth product of the OP-AMP and f_{osc} is the oscillation's frequency in hertz.

Selectivity of the frequency-dependent feedback loop is given by its Q [eq.(5)]:

Total selectivity resulting from the action of the two feedback loops may be described by:

For small variations of frequency and phase angles:

From fig.2.b we may write:

and for small phase shifts:

We may deduce that:

(7)

At the oscillations frequency:

and:

Then:

(8)

Thus, the bridge is very nearly at balance and Q_{12} is many times Q_{1}.

Typical open-loop voltage gain variation with frequency is indicated in fig.3 for an OP-AMP with internal frequency compensation.
Here, G_{o} is the DC voltage gain expressed in decibels and f_{o} is the low-frequency pole. G is the voltage gain in decibels at
frequency f. f_{u} is the unity-gain frequency.

The following holds due to the 20dB per decade roll-off:

(9)

At a frequency f, the open-loop voltage gain is:

(10)

Substituting into eq. (8):

Then:

The effective Q then varies inversely with frequency.

A typical Q multiplication factor at 1kHz, with a 4MHz gain-bandwidth product OP-AMP is:

This would give a value of 444.44 for Q_{12}.

For the case of the modified Wien-bridge oscillator using a single variable resistor for frequency control:

(11)

(12)

Eq.(7) yields the Q multiplication factor:

(13)

Q_{12} is then given by:

(14)

If k_{1}(k_{2}+1)>>1:

(15)

Q_{12} is then approximately constant over one decade.

Using eq.(14) we may calculate the Q_{12} ratio when k_{1} varies between k_{1min} = 10 and
k_{1MAX} = 1000. Then:

(16)

Table I summarizes Q_{12 }Ratio and Q_{12 }values as given by eqs. (16) and (15), with k_{2} as a
parameter, for a Wien-bridge oscillator
designed for operation over the 1kHz to 10kHz decade.

TABLE I

k |
Q |
Q |

0.25 |
1.1645 |
40.47 |

0.5 |
1.1362 |
39.74 |

1 |
1.1014 |
31.61 |

2 |
1.0671 |
19.87 |

10 |
1.0181 |
3.30 |

Calculations for k

From the total selectivity point of view, low values for k

Eq. (15) indicates that higher decades exhibit smaller Q

Some THD measurements made on the modified Wien-bridge oscillator with k

1kHz to 10kHz decade

THD at: 1kHz 10kHz

is: 0.1% 0.03% 2kHz to 20kHz decadeTHD at: 2kHz 20kHz

is: 0.09% 0.06% 200Hz to 2kHz decadeTHD at: 200Hz 2kHz

is: 0.4% 0.017% 20Hz to 200Hz decadeTHD at: 20Hz 200Hz

is: 3.8% 0.04% (using 1 stabilising lamp) is: 2.47% 0.038% (using 2 stabilising lamps in series)Some comments

The lower end of the 20Hz to 200Hz 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 bulky). 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 1kHz, a high-pass filter may be switched-in for noise rejection. This may help lower the THD reading.

Ramon Vargas Patronrvargas@inictel.gob.pe

Lima-Peru, South America

June 2