Wien Bridge Oscillator Circuit Explained with Calculations

In the RC Oscillator tutorial, we saw that by connecting resistors and capacitors together with an inverting amplifier we can create an oscillating circuit. One of the simplest ways to generate a sine wave is through a circuit known as the Wien Bridge Oscillator. This oscillator replaces the traditional LC (inductor-capacitor) tuned tank circuit with an RC (resistor-capacitor) network, allowing us to produce a smooth sinusoidal output.

The Wien Bridge Oscillator gets its name because it is based on a frequency-selective version of the Wheatstone bridge circuit. As we examine the circuit further we find that it is essentially a two-stage amplifier connected with RC components which makes it highly stable at its resonant frequency. This circuit has low distortion, meaning the output remains clean and consistent.

Plus it is easy for us to adjust or tune which is why it is such a popular choice when we need an audio frequency oscillator. However it is important to note that the way this circuit shifts the phase of the output signal is quite different from the phase shift in the basic RC Oscillator.

In this design we use a feedback loop consisting of a series RC circuit combined with a parallel RC circuit, both having the same component values. This setup allows us to create either a phase delay or phase advance depending on the frequency we are working with.

At the resonant frequency the phase shift is exactly 0°, meaning the input and output signals are perfectly aligned. Let us now take a closer look at the circuit below to understand this in more detail.

Analyzing an RC Phase Shift Network

The RC network I have shown in the above diagram is made up of two key parts, which are a series RC circuit and a parallel RC circuit. When we connect them together we actually create a High Pass Filter (which allows high frequencies to pass) connected to a Low Pass Filter (which allows low frequencies to pass).

This combination forms a second-order Band Pass Filter that is highly selective, which means it allows us to isolate a specific range of frequencies with precision. The filter has a high Q factor, that means it can focus on a narrow range of frequencies at the resonant frequency fr.

When we look at low frequencies we find that the series capacitor (C1) has a very high reactance, and it acts like an open circuit. This blocks most of the input signal from passing through and leaves us with almost no output signal (Vout). At high frequencies however, the parallel capacitor (C2) has a very low reactance, so it behaves like a short circuit across the output which again results in a very little to no output.

Therefore we know there must be a specific frequency somewhere between these two extremes where C1 stops behaving like an open circuit and C2 no longer acts as a short circuit. At this point the output voltage (Vout) reaches its peak. This frequency of the oscillator is the resonant frequency which is written as fr.

When the frequency reaches the resonance frequency, the reactance of the circuit becomes equivalent to its resistance, meaning Xc = R. As a result the phase difference between the input and output signals becomes 0 degrees, which means that they are now perfectly aligned. At this frequency we see that the output voltage reaches its maximum and equals one-third (1/3) of the input voltage.

Analyzing Oscillator Output Gain and Phase Shift

We see that at very low frequencies the phase angle, which is the difference between the input and output signals is “positive.” ⁤⁤This means that the output signal is ahead of the input signal in terms of timing, which is a condition called as “Phase Advanced.” ⁤⁤On the other hand at very high frequencies, the phase angle becomes “negative” indicating that the output signal lags behind the input signal which is called “Phase Delay.” ⁤

⁤In between these low and high frequencies we get a particular range of frequency at which the circuit reaches its resonant frequency denoted as (fr). ⁤⁤At this resonant frequency the input and output signals are perfectly aligned, meaning that there is no phase difference between them and the phase angle is exactly 0 degrees (0°). ⁤⁤This condition is called “in-phase.” ⁤

⁤We can describe this resonant frequency (fr) using the following formula.

Formula for the Wien Bridge Oscillator Frequency

fr = 1/(2πRC)

• In the above formula,
• fr  represents the Resonant Frequency, expressed in Hertz
• R  indicates the Resistance, expressed in Ohms
• C  denotes the Capacitance, expressed in Farads

As previously stated, when the output voltage Vout of the RC network equals one-third (1/3) of the input voltage Vin, it reaches its maximum value. The circuit is able to oscillate in this particular situation. However why is one third of the output selected rather than a different value? In order to properly understand this, we first examine the RC circuit’s complex impedance, denoted as Z = R ± jX, where “R” stands for resistance and “X” for reactance.

According to our earlier AC Theory, complex impedance is divided into two components. The imaginary component denotes the reactance (X), whereas the real part denotes the resistance (R). Because we are working with capacitors in this circuit, the reactance component is going to be the capacitive reactance (Xc).

To gain a better understanding the reason why the output of the RC circuit has to be one third or 0.333 times the input voltage we must take into consideration how these two RC circuits are connected and how their complex impedances interact with one another.

Analyzing the RC Network

When we draw the RC network once again as above, it allows us to see that it is made up of two coupled RC circuits, with the output being generated from the junction point where the RC components meet. The lower parallel branch is formed by resistor R₂ and capacitor C₂, whereas the upper series branch is formed by resistor R₁ and capacitor C₁.

So now, Z_S may be used to denote the whole DC impedance of the series branch made up of R₁ and C₁, and Z_P can denote the total impedance of the parallel branch formed by R₂ and C₂.

Z_S and Z_P act as a voltage divider because they have been connected in series across the input voltage, V_IN. As seen in the figure, the output voltage can be measured across Z_P.

Now, suppose we have the R₁ and R₂ resistors both of them as: 10 kΩ, likewise we have both the capacitors C₁ and C₂ values identical at: 4.7 nF and we have the value of the frequency f as 4 kHz. Then we can implement the calculations in the following manner:

Calculating the Series Circuit

We can calculate the total impedance of the series combination built using the resistor R₁ and the capacitor C₁ as shown below:

R = 8.5 kΩ but X_C = 1/(2πfC)

∴ X_C = 1/(2π * 4 * 4.7) = 8.5 kΩ

Z_S = √(R² + X_C²)= √(8500² + 8500²) = 12020 Ω = 12 kΩ

As per the given data above, at 4 kHz supply frequency, the capacitor’s reactance matched the resistor’s resistance of 8.5 kΩ. This interesting similarity yields an upper series impedance (Z_S) of 12 kΩ.

Turning our focus to the lower parallel impedance indicated as Z_P, we have to take a different approach due to the unique nature of the parallel configuration. In this circuit arrangement the two components, which are the capacitor and the resistor are connected in parallel to one another.

This configuration shows significant influence on the overall impedance of the circuit, and therefore it becomes important for us to analyze the combined effects of the parallel components, so that we can accurately determine the resultant impedance.

Calculating the Parallel Circuit Network

The total impedance of the lower parallel combination network which includes the resistor R₂ and capacitor C₂ are provided to us as:

R = 8.5 kΩ, and X_C = 8.5 kΩ

1/Z = 1/R + 1/X_C = (1/8500 + 1/8500)

∴ Z = 4250 Ω, or 4.2 kΩ

With the supply frequency set at 4000 Hz, same as 4 kHz, the total DC impedance for the parallel RC circuit will be at 4.2 kΩ (R || Xc). To find this parallel impedance, we must do a vector sum calculation, which can be represented as follows:

R = 4.2 kΩ and X_C = 4.2 kΩ (Parallel)

Z_p = √(R² + X_C²) = √(4200² + 4200²)

∴ Z_p = 5940 Ω or 6 kΩ

Now we have calculated the value for the vector sum of the series impedance which is equal to 12 kΩ, ( Z_S = 12 kΩ ) and we have also determined the same for the parallel impedance, which is equal to 6 kΩ, ( Z_P = 6 kΩ ). So now it is easy for us to calculate the total output impedance of the voltage divider network Zout at the given frequency, as indicated below:

Z_out = Z_p/(Z_p + Z_s) = 6/(6 + 12) = 0.3333 or 1/3

At the oscillation frequency we find that the magnitude of the output voltage Vout, will be equal to the product of the output impedance Zout and the input voltage Vin.

This relationship can be expressed mathematically and it becomes particularly interesting because it reveals that Vout is in fact one-third (1/3) of the input voltage Vin.

This specific feature of the frequency-selective RC network is important to the operation of the Wien Bridge Oscillator circuit.

Now if we position this RC network across a non-inverting op-amp amplifier having a gain calculated using the formula 1+ R1/R2, we can effectively create the foundational circuit of the basic Wien Bridge Oscillator.

This combination of components and their interactions is what allows the oscillator to work perfectly as per its specifications.

Understanding a Wien Bridge Oscillator Circuit

As shown above, we feed the output signal from the op-amp back to both of its inputs. One part of this feedback signal goes to the inverting input through a resistor divider network made up of R1 and R2. This setup allows us to adjust the amplifiers voltage gain within specific limits using negative feedback.

The other part of the feedback is more interesting. It creates a network with resistors and capacitors that connects to the non-inverting input. This is where the positive feedback happens, thanks to the RC Wien Bridge network. It is this positive feedback that helps to generate the oscillation in the circuit.

You may be already aware of the fact that when the RC network is in the positive feedback path, it exhibits zero phase shift at one specific frequency called the resonant frequency.

When this resonant frequency denoted as fr is reached, the voltages at the inverting and non-inverting inputs become equal and in-phase. This causes the positive feedback to effectively cancel out the negative feedback, initiating the oscillation process in the circuit output.

For oscillations to begin we must ensure the voltage gain of the circuit is at least three. This is because the input is one-third of the output. The gain condition (Av ≥ 3) is set by the feedback resistors R1 and R2, which we can calculate using the formula 1 + (R1/R2) for a non-inverting amplifier.

Please also keep in mind that operational amplifiers have limitations when it comes to open-loop gain. If you want to work at frequencies above 1 MHz, you need to use special high-frequency op-amps to make it work.

Solving a Wien Bridge Oscillator Problem #1

We have a Wien Bridge Oscillator circuit which includes a resistor of 12 kΩ and a variable capacitor which can be adjusted between 1nF to 600 nF, we want to calculate the maximum and minimum frequency of oscillations of the circuit.

We already know that the frequency of oscillations for a Wien Bridge Oscillator can be expressed using the formula:

fr = 1 / (2 * π * R * C)

The Minimum Frequency of the Wien Bridge Oscillator can be calculated as:

f_min = 1/(2 * π * 12000 * 600 * 10^-9) = 22 Hz

The Maximum Frequency of the Wien Bridge Oscillator can be calculated as:

f_max = 1/(2 * π * 12000 * 1 * 10^-9) = 13263 Hz

Solving a Wien Bridge Oscillator Problem #2

We have a Wien Bridge Oscillator circuit and we want it to generate an output of sinusoidal waveform at 4700 Hertz (or 47kHz). Let us Calculate the values of the resistors R1 and R2, along with the two capacitors C1, C2 which together determine the frequency of the oscillator.

Another thing is that if we assume our oscillator circuit is designed using a non-inverting operational amplifier configuration. So, now we want to calculate the minimum values for the gain resistors which may generate the expected magnitude of oscillation frequency. At the end we would also draw the resultant oscillator circuit having the specific calculated values.

We begin by using our pervious formula for the frequency of oscillations:

fr = 1 / (2 * π * R * C) = 4700

From the given data we know that frequency of oscillations for the Wien Bridge Oscillator was given as 4700 Hertz. Now for this configuration the resistors R₁ = R₂ and capacitors C₁ = C₂ . Now, for the feedback capacitors let us take to be a value of 5 nF, then the subsequent value of the feedback resistors can be calculated as per the following calculations:

fr = 1/(2 * π * R * C)

∴ R = 1/(2 * π * fr * C)

= 1/(2 * π * 4700 * 5 * 10^-3)

= 6772.55 Ω or simply 6.8 kΩ

To initiate sinusoidal oscillations in a Wien Bridge circuit, we must ensure that the voltage gain reaches at least 3 (Av ≥ 3). In a non-inverting op-amp setup, this gain is fixed by the feedback resistor network consisting of R3 and R4 and we can express this as follows:

A_V = V_OUT/V_IN = 1 + R₃/R₄ = 3 or higher

Suppose we take the resistor R₃ to be, let’s say 82 kΩ, so now we can calculate the values of the resistor R4 through the following calculations:

1 + R₃/R₄ = 3

∴ R₄ = R₃/(3 – 1) = R₃/2

Since we assumed the value for R₃ to be 82 kΩ, so we get the value of R₄ as shown below:

R₄ = 82/2 = 41 kΩ

It is true that, for the oscillations to start, the op-amp circuit should have a gain of at least 3, but in real-world applications, a somewhat larger number is usually required. So, possibly you can perhaps recalculate the R₄ using a gain of 3.1, which may yield a result of 39 kΩ . So with these values in hand we can finally draw our calculated Wien Bridge Oscillator circuit as shown below:

Wien Bridge Oscillator Circuit Diagram with Calculated Part Values

Conclusions

To create oscillations in a Wien Bridge Oscillator circuit, we have to ensure certain crucial criteria are satisfied while designing it.

First and foremost, even without the presence of an input signal, a Wien Bridge Oscillator should be able to produce continuous output oscillation. This inherent property is critical to make sure the circuit keeps oscillating non-stop

Additionally the Wien Bridge Oscillator is very flexible in terms of frequency generation because it can generate a wide range of frequencies suitable for a variety of applications.

Another important criterion is that the voltage gain of the amplifier used in the circuit exceed 3. This condition is required to ensure that the system can sustain its oscillatory nature.

The RC network used in conjunction with the oscillator can operate well with a non-inverting amplifier design. This configuration creates the essential phase shift for persistent oscillations.

Furthermore the amplifier’s input resistance should be much higher than the circuit’s resistance (𝑅). This condition assures that the amplifier does not change the RC network, retaining the required operational characteristics.

In addition, the amplifiers output resistance must remain low, which is critical for reducing the effects of external loads which would otherwise destroy the oscillations’ stability and stability.

Another primary concern is the necessity for a system to control the amplitude of the oscillations. If the amplifier’s voltage gain is insufficient, the intended oscillatory output will progressively decrease and eventually come to a stop On the other hand, in the event the gain is set too high, the output will get close to saturation levels, which are restricted by the supply rails and cause waveform distortion.

To allow continuous oscillations from the Wien Bridge Oscillator, we should opt for amplitude stabilizing methods in the circuit, for example we can make use of feedback diodes. These components can be used to sustain the oscillatory output enabling it to oscillate endlessly.

As we conclude our investigation of oscillators, we will turn our attention to the Crystal Oscillator, in our next upcoming tutorial. This specific kind of oscillator uses a quartz crystal as its tank circuit, which allows it to produce a high-frequency and extremely stable sinusoidal waveform, demonstrating yet another unique aspect of oscillatory systems.

References: Wien bridge oscillator

Design of op amp sine wave oscillators