Wein Bridge Oscillators, how do they work?

[Azelketh]

Homework Statement

Hi, I am revising for an exam, so this isn't really a homework question, but not sure which sub-forum it really fits on so its here.

Anyway, I am trying to understand how a simple wein brige oscillator circuit works, as shown in the diagram
http://www.electronics-tutorials.ws/oscillator/osc27.gifWhat I am finding difficult is the phase change of the circuit, apparently at resonance there is no phase change over the whole circuit, so the imaginary component of the impedances of the capacitors has to be 0 in total, but how do you mathmatically prove that for this circuit the oscillatory frequecny that this happens at is

$$\omega = \frac{1}{CR}$$
?

Homework Equations

The Attempt at a Solution

I've tried adding the impedance of the capacitor and resistor in series with that of the capacitor and resistor in parallel.
resistance of capacitor and resistor in series = $$z_u$$
resistance of capacitor and resistor in parallel = $$z_L$$

$$z_u = R - \frac{\imath}{ C \omega}$$

$$\frac{1}{z_L} = \frac{1}{R} + \imath C \omega$$
thus
$$z_L = \frac{R}{1 + \imath C \omega R}$$

multiplying by the complex conjugate of the denominator to get an easier to work with number

$$z_L = \frac{R - \imath \omega C R^2}{1 + \omega^2 C^2 R^2 }$$

total impedance= $$z_T= z_u + z_L$$

$$z_T = \frac{R - \imath \omega C R^2}{1 + \omega^2 C^2 R^2 } + R - \frac{\imath}{ C \omega}$$

rearranging into real and imaginary sections
$$z_T = \frac{R}{1+ \omega^2 C^2 R^2} + R -\imath (\frac{\omega C R^2}{1+ \omega^2 C^2 R^2} + \frac{1}{\omega C})$$

then as the imaginary component has to be 0 for no overall phase change of the output, only considering the imaginary part;

$$0 = \frac{\omega C R^2}{1+ \omega^2 C^2 R^2} + \frac{1}{\omega C}$$

which if $$\omega = \frac{1}{CR}$$ gives

$$\frac{1}{2} = -1$$
which is impossible, so i don't understand how the resonant frequency can be that.

Can anyone help point out where I've gone wrong? Or explain how the phase change at resonant frequency is 0?
Thanks for your time if you've read this.

 

[KataKoniK]

Hi there,

First of all, great job on your attempt at solving this problem! I can see that you have a good understanding of complex impedance and how to calculate it for different components in a circuit.

To answer your question, let's take a closer look at the Wein Bridge oscillator circuit. At resonance, the output voltage is in phase with the input voltage, meaning there is no phase difference between them. This can be seen by the fact that the output voltage is taken from the midpoint of the two capacitors, and this voltage is the same as the input voltage.

Now, let's consider the impedance of the two capacitors in the circuit. At resonance, the two capacitors are in parallel, and their combined impedance is given by:

Z = (1/jωC) || (1/jωC) = (1/jωC)/2 = -j/(ωC)

This means that at resonance, the capacitors have an equivalent impedance of -j/(ωC), which is purely imaginary. This is where your calculation went wrong - you assumed that the imaginary component of the capacitor impedance is zero, when in fact it is purely imaginary at resonance.

Now, let's substitute this value for the capacitive impedance into your equation for the total impedance:

ZT = R + (-j/(ωC))

= R - j/(ωC)

= R(1 - j/(ωCR))

= R(1 - j/1)

= R(1 - j)

= R - jR

As you can see, the imaginary component of the total impedance is indeed zero at resonance, as the output voltage is in phase with the input voltage. This also means that the phase change over the whole circuit is zero, as you correctly mentioned in your post.

I hope this helps to clarify things for you. Good luck with your exam!