Analog and Telecommunication Electronics

Prof. Dante Del Corso

Miniproject: Wien-Bridge Oscillators

Student Name: Emilio Orlandini

Student ID: 221773

A.Y. 2014/2015

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Index

INTRODUCTION

1. Oscillators generalities

1.1 Sinewave oscillator................................................................................pag. 3

1.2 Requirements for oscillations.................................................................pag. 3

1.3 Phase Shift in the oscillator....................................................................pag. 5

1.4 Gain in the oscillator..............................................................................pag. 6

2. The Wien-Bridge oscillator

2.1 Wien-Bridge oscillator...........................................................................pag. 7

2.2 Linear Analysis......................................................................................pag. 8

2.3 Introduction of non-linear elements.....................................................pag. 10

2.4 Lamp Stabilised Wien-Bridge Oscillator.............................................pag. 11

2.5 Diode stabilised Wien-Bridge Oscillator..............................................pag. 14

REFERENCES

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Introduction

Circuits that produce a specific, periodic waveform without any input signal are called

“Oscillators”. They can generate waveforms such as square, sinusoidal, triangular and

so on. They are generally composed by some active elements surrounded by passive

components such as resistors, capacitors and inductors. Basically, we divide oscillators

in two families: relaxation and sinusoidal. Relaxation oscillators generate waveforms

which are not sinusoidal, such as triangular, square, sawtooth, while sinusoidal

oscillators generate sinusoidal waveforms with a specific frequency (or period) and

amplitude. Oscillators may use amplifiers with external components, or crystals that

internally generate the oscillation. In this miniproject we are dealing with sine wave

oscillators, and in particular our attention is focused on Wien Bridge configuration,

which is based on an OpAmp.

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1 Oscillators generalities

1.1 Sine-wave oscillators

OpAmp oscillators are unstable circuits, but ones that are intentionally designed to

remain in an unstable or oscillatory state. Oscillators are useful for generating uniform

signals that are used as a reference in applications such as audio, function generators,

digital systems and communication systems. Sinewave oscillators base on an OpAmp

operate without any externally-applied input signal. Some combination of positive and

negative feedback is implemented in order to drive the OpAmp into an unstable state.

OpAmp oscillators are restricted to a low frequency spectrum because OpAmps do not

have the required bandwidth to achieve low phase shift at high frequencies. Voltage-

feedback OpAmps are usually limited to a low kHz range. Crystal oscillators, instead,

are used in high-frequency applications up to the hundreds of MHz range.

1.2 Requirements for oscillation

The fundamental characteristic of an oscillator is non linearity; in order to give rise to

oscillations the system must be initially unstable. A principle scheme is shown in figure

1.1.

Figure 1.1: canonical form of a feedback system

Since:

𝑉𝑜𝑢𝑡 = 𝐸 · 𝐴 (1)

𝐸 = 𝑉𝑖𝑛 + 𝛽 · 𝑉𝑜𝑢𝑡 (2)

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Substituing E in (1), we obtain:

𝑉𝑜𝑢𝑡

𝐴 = 𝑉𝑖𝑛 − 𝛽 · 𝑉𝑜𝑢𝑡 (3)

And collecting the terms in 𝑉𝑜𝑢𝑡 yelds:

𝑉𝑖𝑛 = 𝑉𝑜𝑢𝑡(1

𝐴+ 𝛽) (4)

Rearranging the elements produces equation 5, which is the classical form of a

feedback expression:

𝑉𝑜𝑢𝑡

𝑉𝑖𝑛 =

𝐴

1+𝐴𝛽 (5)

Oscillators does not need any input signal; they only use a fraction of the output signal

created by the feedback network as the input signal.

Oscillations occurs when the feedback system is not able to find a steady-state since its

transfer function cannot be satisfied. This happens when the denominator is equal to

zero, i.e. when 1 + 𝐴𝛽 = 0 or 𝐴𝛽 = -1. In the design of an oscillator, it must be ensured

that 𝐴𝛽 = -1. This is called the Barkhausen criterion. For a positive feedback system,

the expression is A = 1∠0° and the sign of the A term is negative in equation (5). This

criterion implies that the magnitude of the loop gain is unity, with a corresponding

phase shift of 180° (due to the minus sign).

As the phase shift reaches 180° and |𝐴𝛽| = 1, the output voltage of the unstable system

ideally tends to infinity but, in a real system it is limited to finite values of power

supply. At this point, one of three things can occur:

1) Nonlinearity in saturation causes the system to become stable and lock-up to the

current supply voltage;

2) The initial change causes the system to saturate and stay that way for a long time

before it becomes linear and heads for the opposite power rail;

3) The system stays linear and reverses direction, heading for the opposite power

rail.

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The second case produces highly distorted oscillations (usually quasi-square waves),

the resulting oscillators are called relaxation oscillators. The third case, instead,

produces a sine-wave oscillator.

1.3 Phase Shift in the Oscillator

The 180° phase shift required by the Barkhausen criterion is introduced by active and

passive components. Oscillators are made dependent on passive component phase shift

because it is very accurate. The phase shift contributed by active components is

minimized because it varies with temperature, has a wide initial tolerance, and is device

dependent.

A single RL or RC circuit contributed up to 90° phase shift per pole, and since 180° of

phase shift is required for oscillation, at least two poles must be used in the oscillator

design. An LC circuit has two poles, but LC and LR oscillators are not suitable for low-

frequency applications because inductors at low frequencies are expensive, heavy and

highly nonideal. LC oscillators are designed tipically in high frequency applications,

where the inductor size, weight and cost are less significant. When RC sections are

cascaded, the phase shift multiplies by the number of sections n (figure 1.2).

Figure 1.2: Phase plot of RC sections [1]

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In the region where phase shift is 180°, the frequency oscillation is very sensitive to

the phase shift. Thus, a tight frequency specification requires that the phase shift, dɸ,

must be kept within too much narrow limits for there to be only small variations in

frequency, dω, at 180°. Figure 1.2 shows that the value of dɸdω

⁄ at the oscillator

frequency is unacceptably small. Thus, in order to have a much higher dɸdω

⁄ , tipically

four RC sections are used.

Crystal or ceramic resonators make the most stable oscillators because resonators have

an extremely high dɸdω

⁄ as a result of their nonlinear properties. Anyway, OpAmps are

not generally used with crystal or ceramic resonator oscillators because of OpAmps’

low bandwidth. Experience shows that rather than using a low-frequency resonator for

low frequencies, it is more cost effective to build a high frequency crystal oscillator,

count the output down, and then fitler the output to obtain the low frequency [1].

1.4 Gain in the Oscillator

According to the Barkhausen criterion, the oscillator gain must be one in magnitude at

the oscillation frequency. When the gain exceeds unity with a phase shift of -180°, the

nonlinearity of the active device reduces the gain to unity and the circuits oscillates.

The nonlinearity becomes significant when the amplifiers get close to the supply

voltage level, since saturation reduces the active device (transistor) gain. The worst-

case design usually requires nominal gains exceeding unity, but excess gain causes

increased distortion of the output sinewave.

When the gain is too low, oscillations cease under worst case conditions, and when the

gain is too high, the output wave form looks more like a squarewave than a sinewave.

Distortion is a direct result of excessive gain overdriving the amplifier. Thus, gain must

be carefully controlled. The more stable the gain, the better the purity of the sinewave

output.

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2 The Wien-Bridge oscillator

2.1 Wien-Bridge Oscillator

There are many types of sinewave oscillator circuits. In an application, the choice

depends on the frequency and the desired monotonicity of the output waveform. This

section is focused on the “Wien Bridge configuration”.

The Wien Bridge is one of the simplest and best known oscillators and is largely used

in circuits for audio applications [1]. The basic configuration of a Wien-Bridge

oscillator is shown in figure 2.1.

Figure 2.1: Wien-Bridge circuit schematic [3]

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2.2 Linear Analysis

The first goal in the analysis of an oscillator regards the trigger of oscillations. A couple

of poles with positive real part is necessary in order to the system to oscillate. But, once

oscillations occurs, the positive real part poles must be shifted on the imaginary axis to

have a stable sinewave (this will be explained in details in the nonlinear analysis). [2]

Looking at the scheme to analyze the linear behaviour of the Wien-Bridge oscillator

(figure 2.2), we can identify the feedback block B(ω) and the gain block A.

Figure 2.2: Scheme of the Wien-Bridge oscillator for the linear analysis [2]

Let us evaluate the expression of B(ω):

B(ω) = 𝑉𝑓

𝑉𝑜 =

𝑍𝑝

𝑍𝑝+ 𝑍𝑓 (5)

Since 𝑍𝑝 = 𝑅𝑝

1+𝑗𝜔𝜏𝑝 and 𝑍𝑓 =

1+ 𝑗𝜔𝜏𝑠

𝑗𝜔𝐶𝑠, substituting these values in (5) we obtain

(6):

B(ω) = 𝑗𝜔𝐶𝑠𝑅𝑝

𝑗𝜔𝐶𝑠𝑅𝑝+1+𝑗𝜔(𝜏𝑝+𝜏𝑠)+(𝑗𝜔)2𝜏𝑝𝜏𝑠 (6)

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Then, imposing 𝑅𝑠 = 𝑅𝑝 = R, 𝐶𝑠 = 𝐶𝑝 = C, 𝜏𝑠 = 𝜏𝑝 = τ, the final expression

(7) is:

B(ω) = 1

3+𝑗(𝜔𝜏− 1

𝜔𝜏) (7)

Now, since A = 1+𝑅2

𝑅1 (see figure 2.1, A is the gain of an OpAmp non-inverting

amplifier), and 𝛽(𝑗𝜔) = 𝐵(𝑗𝜔) we get:

A𝛽(𝑗𝜔) = 𝑗𝜔𝑅𝐶𝐴

(1−𝜔2𝑅2𝐶2)+3𝑗𝜔𝑅𝐶 (8)

Now, according to Barkhausen criterion (for a positive feedback system), we must

satisfy:

∠A𝛽(𝑗𝜔) = 0 ⇒ 𝜔2𝑅2𝐶2 − 1 = 0 ⇒ 𝜔 =1

𝑅𝐶= 𝜔0

at 𝜔 = 𝜔0 we have:

𝐴𝛽(𝑗𝜔0) =𝐴

3 ⇒ |𝐴𝛽(𝑗𝜔0)| = 1 ⇒ 𝑅2 = 2𝑅1.

Summarizing, we have found two conditions to be satisfied in order to have a sinewave

at the output at frequency 𝑓𝑜𝑢𝑡:

𝑓𝑜𝑢𝑡 =1

2𝜋𝑅𝐶 (9)

𝑅2 = 2𝑅1 (10)

In figure 2.3, the output waveform is shown for different conditions of the A gain:

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Figure 2.4: output waveforms for different values of A [2]

2.3 Introduction of non-linear elements

As we can see in figure 2.4, what we want is to force the system to oscillate (A>3), and

then to get stable oscillations (A=3). This behaviour can only be obtained by

introducing some nonlinearity in our circuit. Different ways can be exploited in order

to have nonlinearity, such as a lamp, diodes, in general something which resistance

varies with increasing of potential drop across it. This is straightforward to understand:

at first we want A>3, so 𝑅2 > 2𝑅1, but, when the magnitude of the output voltage

reaches a desired value, we need 𝑅2 = 2𝑅1.

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2.4 Lamp Stabilised Wien-Bridge Oscillator

The first Wien-Bridge stable oscillator was developed by Bill Hewlett in the late 1930s,

as his Stanford’s Master thesis. He introduced a light bulb in order to stabilize the gain

without producing distortion. It was then commercialized as HP200A audio oscillator

[4]. It can still be built using similar principles to the early Hewlett Packard versions,

but with modern components [5] (figure 2.6).

Figure 2.6: Basic lamp stabilized Wien Bridge oscillator [5]

It is a property of filament lamps that the resistance of the tungsten filament increases

in a non-linear manner as the filament heats up. The gain of the amplifier is set by

𝐴 =𝑅3+𝑅𝐿𝑎𝑚𝑝

𝑅𝐿𝑎𝑚𝑝 (11)

Therefore the greater the resistance of the lamp, the lower the amplifier gain. By

choosing a suitable lamp, the gain of the amplifier can be automatically controlled over

an appropriate range. Usually, a lamp with a maximum current flow of around 50mA

or less is used, to give an initial gain of more than 3 as the oscillator starts, falling

quickly to 3 as the lamp heats up. In figure 2.7 there is a graph showing how positive

temperature coefficient resistance varies with voltage.

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Figure 2.7: Non-linear resistance of a filament lamp [5]

The useful area of the lamp characteristic, where the largest change in resistance occurs

is shaded green, the oscillators amplitude is stabilised by making use of this area. To

find a value for 𝑅3 that will give the correct amount of gain at start up and during

oscillation, its value should be slightly greater than twice the value of the lamps

resistance at the lower end of the green shaded area (i.e. 2 x 30Ω = 60Ω) to give a gain

>3, but no greater than twice the value of the lamp’s resistance at the upper limit of its

useful slope (i.e. 2 x 75 = 150 Ω), the gain should then stabilise at 3, with the oscillator

providing an undistorted sinewave output. The realization and output waveform of the

lamp-stabilised circuit is shown in figure 2.8:

Figure 2.8: realization of a lamp-stabilised Wien Bridge oscillator (left) and its output waveform

(right) [5].

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The design formulae are the following:

𝑓𝑜𝑠𝑐 =1

2𝜋𝑅𝐶 (12)

𝑅 =1

𝑓𝑜𝑠𝑐2𝜋𝐶 (13)

𝐶 = 1/(𝑓𝑜𝑠𝑐2𝜋𝑅) (14)

In order to obtain an output frequency of 159Hz (fig. 2.8), the component list to build

the circuit is the following:

𝐼𝐶1 = 𝐿𝑀324 (OpAmp)

𝐶1 = 𝐶2 = 100𝑛𝐹

𝑅1 = 𝑅2 = 10𝑘Ω

𝑅3 = 68Ω

𝐿𝑎𝑚𝑝 = 𝑇1 5𝑉 45𝑚𝐴 𝑤𝑖𝑟𝑒 𝑒𝑛𝑑𝑒𝑑 𝑙𝑎𝑚𝑝 (suitable from Rapid Electronics UK,

May 2015)

𝑆𝑢𝑝𝑝𝑙𝑦 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 ±9V

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2.5 Diode stabilised Wien-Bridge Oscillator

An alternative to lamp stabilization is provided by using a pair of diodes in parallel

with the feedback resistor as shown in figure 2.9.

Figure 2.9: Basic diode stabilized Wien-Bridge oscillator

In this circuit, the ratio of the non-inverting amplifier is controlled by the ratio between

𝑅3 and 𝑅4. Initially, it will be slightly greater than 3, in order to allow oscillations to

start. Once the feedback signal from the output produces a waveform across 𝑅3 that

approaches 0.6𝑉𝑝𝑝, the diodes begin to conduct. Their forward resistance reduces, and

since they are in parallel with 𝑅3, this reduces the value of 𝑅3 and the amplifier gain

as well. A design example, for a 𝑓𝑜𝑠𝑐 = 132𝐻𝑧, has been carried out with the following

components:

IC LM224N (OpAmp)

𝐶1 = 𝐶2 = 100𝑛𝐹

𝑅1 = 𝑅2 = 12𝑘Ω

𝑅3 = 80Ω

𝑅4 = 22Ω

𝑆𝑢𝑝𝑝𝑙𝑦 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 ±9V

The gain in controlled by the 𝑅3

𝑅4 ratio, which is 3.63. The output waveform for this

configuration is shown in figure 2.10:

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Figure 2.10: Output waveform (left) and frequency spectrum (right) for a diode stabilised oscillator

with large gain [6]

As it can be seen from the output waveform, the main problem of the diode stabilised

configuration is that the output is more prone to distortion that in the lamp stabilised

circuit, as the diodes will tends to distort the waveform peaks if the gain set by 𝑅3 and

𝑅4 is slightly larger than needed. This also means that the oscillator’s output amplitude

is somewhat restricted if distortion is to be minimized. Adding a lowpass filter could

also improve the quality of the signal in terms of harmonics.

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References

[1] Ron Mancini and Richard Palmer, Design of OpAmp sinewave oscillators,

SLOA060, March 2001.

[2] “Elettronica delle telecomunicazioni”, course material by Prof. Giovanni Breglio

from “Università degli studi di Napoli Federico II”, a.y. 2013/2014.

[3] The Wien Bridge oscillator, website tutorial, Electronics Tutorial.

[4] HP Model 200A audio oscillator, 1939, HP website Model 200A.

[5] Wien Bridge Oscillators, website tutorial, Learn About Electronics, Wien Bridge

Oscillator.

[6] Measurements carried out at Politecnico di Torino - Sede Centrale - Cittadella

Politecnica, 2nd floor, LED2.