Frequency characteristics of Wien bridge and double T bridge

Issues

So, you receive an analog signal, look at the results of the readings and see that the signal is completely far from ideal: noise, distortion, fluctuations. The figure shows the difference between the real and ideal signal. All because of the noise it receives. To isolate and equalize the desired signal and avoid endless hiss, it is important to understand how different filtering schemes can affect signal quality.

The blue signal is noisy, the orange signal is ideal, absolutely clean (Source: https://habr.com/ru/companies/selectel/articles/721558/)

Green - filtered signal (Source: https://habr.com/ru/companies/selectel/articles/721558/)

HPF vs. LPF

Frequency filters are an electrical circuit that effectively passes only one frequency range. The device allows you to “ignore” unnecessary frequencies. Thereby selecting and aligning signals of any shape.
Low Pass Filter – Passes frequencies below the cutoff frequency.
High Pass Filter – Passes frequencies above the cutoff frequency.
The cutoff frequency is the frequency at which filter attenuation is -3 dB on a logarithmic scale (in linear it is 0.707).

Operating principle of the low-pass filter:

$X_{c}=\frac{1}{2 \pi fC}$

At high frequencies, the capacitor begins to act as a conductor, since its reactance $X_{c}$ decreases with increasing frequency. This causes the high-frequency signal to be “shunted” through the capacitor to ground, and a weakened signal is sent to the output.
At low frequencies, the capacitor has high reactance and effectively blocks current, allowing low frequency components of the signal to pass through the resistor to the output.

Thus, we obtain the following frequency response and phase response of the low-pass filter. And also the cutoff frequency is $f_{avg}=\frac{1}{2 \pi RC}$ .

The principle of operation of the high-pass filter:

At low frequencies, the capacitor has high reactance $X_{c}$ so it does not allow low frequency signal to pass through, creating high circuit resistance.
At high frequencies, the reactance of the capacitor decreases and it begins to transmit the signal. This way, the high frequency signal can pass through the capacitor and reach the output.
The resistor helps create the correct voltage distribution at low and high frequencies, regulating signal transmission.

Thus, we obtain the following frequency response and phase response of the high-pass filter. The cutoff frequency is also $f_{avg}=\frac{1}{2 \pi RC}$ .

Frequency response and phase response of high frequency filter

Bridge of Wine

What if you combine HPF and LPF. We get a Wien bridge, which suppresses low and high frequencies, passing only signals in a certain range. The cutoff frequency is adjusted by selecting the values of resistors and capacitors, which allows you to precisely regulate the range of frequencies that the filter will pass through. Typically, resistor values are on the order of kOhms, and capacitors are nF, but more accurate readings are calculated using the cutoff frequency formula.

Wien Bridge (Passive RC Bandpass Filter)

Transmission ratio: $A=\frac{U_{out}}{U_{in}}=\frac{Z_{2}}{Z_{1}+Z_{2}}$ Where $Z_{1}$ And $Z_{2}$ – impedances.

We also introduce the dimensionless normalized frequency $\Omega=\omega RC$ . Then the transmission coefficient can be expressed as follows.

According to these formulas, we obtain the following frequency response and phase response. Maximum coefficient

transfers $A=\frac{1}{3}$ at quasi-resonant frequency $\omega=\frac{1}{RC}$ or $f_{avg}=\frac{1}{2 \pi RC}$ .

Frequency response and phase response of the Wine Bridge

Wien-Robinson Bridge

If the bandpass filter is supplemented with resistances R₁ and 2R₁then you get a Wien-Robinson bridge, which suppresses signals in a certain frequency region.

The output voltage is removed on the diagonal of the bridge between its two branches. The ohmic voltage divider provides a frequency-independent voltage equal to $\frac{1}{3}U_{e}$ . In this case, at the resonant frequency, the output voltage is zero, since the output voltage is removed on the diagonal of the bridge (see diagram). Unlike a bandpass filter, the amplitude-frequency characteristic of the gain at the resonant frequency has a minimum. The circuit is applicable for suppressing signals in a certain frequency region.

For high and low frequencies: $U_{a}=\frac{1}{3}U_{e}$ .
Resonance frequency $\omega=\frac{1}{RC}$ or $f_{avg}=\frac{1}{2 \pi RC}$ .

Frequency response and phase response of the Wien-Robinson Bridge

Double T-bridge

The double T filter has a frequency response identical to that of a Wien-Robinson bridge.

It is also suitable for suppressing a certain frequency region. Unlike the Wien-Robinson bridge, the output voltage is removed relative to a common point. For high and low frequencies $U_{a}=U_{e}$ . High frequency signals will be transmitted entirely through two capacitors C, and low frequency signals through resistors R.

Calculation of transmission coefficient and phase

For high and low frequencies: $U_{a}=U_{e}$ .
Resonance frequency $\omega=\frac{1}{RC}$ or $f_{avg}=\frac{1}{2 \pi RC}$ .

Frequency response and phase response of a double T-shaped bridge

Application of Wine Bridge

A generator with a Wien bridge is a sinusoidal oscillation generator.

The generator is an electronic amplifier covered by frequency-dependent positive feedback through a Wien bridge. By changing the parameters of the Wien bridge, the generator can generate voltage over a wide tunable frequency range and generates a sinusoidal voltage with small differences from an ideal sinusoidal signal.

Generator with a Wien bridge (Source: https://ru.wikipedia.org/wiki/Generator_with_a Wien_bridge)

In the above diagram, an example of an active amplifier element is an operational amplifier (op-amp), connected for the generated signal according to a non-inverting amplifier circuit. Voltage transfer coefficient of a non-verting amplifier on an op-amp: $K_{u}=1+\frac{R_{3}}{R_{4}}$

Thus, stable generation of a sinusoidal signal with low distortion and without amplitude fluctuations is ensured at K=3: $R_{3}=2R_{4}$ and the following conditions:

Conditions for stable operation of the generator

In this case, the frequency of the generated voltage will then be equal to the quasi-resonant frequency of the Wien bridge $f_{avg}=\frac{1}{2 \pi RC}$ .

Application of double T-bridge

Using a double T-bridge and an op-amp, you can create special selective amplifiers with a very narrow bandwidth, designed to isolate the “useful” signals of a certain frequency.

The gain of a selective double T-bridge amplifier in a negative feedback circuit is expressed in terms of the parameters of the amplifier and the feedback circuit. where β is the complex transmission coefficient of the feedback circuit (transmission coefficient of the T-shaped bridge).

Selective amplifiers with a double T-bridge in the feedback circuit work well at quasi-resonant frequencies from a few hertz to several megahertz. Their selective properties depend on the gain TO: The larger this coefficient, the better the desired signal is amplified compared to very low and very high frequencies.

The transmission coefficient of a given device can be calculated as follows:

Conclusions

Wien bridge and double T bridge are widely used in electronics for generating, filtering and processing signals.

The choice between a Wien bridge and a double T bridge depends on the specific requirements of the device. For general applications, a Wien bridge may be sufficient, while for more complex applications that require high signal stability and quality, a double T bridge may be better suited.

Sources

About filters: https://habr.com/ru/companies/selectel/articles/721558/
Selective amplifier: https://bstudy.net/806879/spravochnik/izbiratelnye_usiliteli
Generator with Wien Bridge: https://ru.wikipedia.org/wiki/Generator_with_bridge_Vina
Frequency characteristics: https://studfile.net/preview/16520089/page:7/