# 11. Chapter: Amplitude stabilization of the harmonic oscillator

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1 Punčochář, Mohylová: TELO, Chapter Chapter: Amplitude stabilization of the harmonic oscillator Time of study: 3 hours Goals: the student should be able to define basic principles of oscillator amplitude stabilization describe low distortion oscillator (spot sinus) Text Two basic oscillator structures you can see in the 6.Chapter. A positive-feedback loop is formed by an amplifier A and a frequency-selective network β. The amplifier produces a 0 or 180 voltage phase shift, as does the feedback network. This results in a combined 0 voltage phase shift around the loop, which is the same thing as a 360 phase shift. In order to oscillate, the loop gain (return ration) Aβ must be equal to unity. Because circuit components and transistors change characteristics (drift) with age, temperature, voltage, etc., the Aβ =1 condition could not be permanently satisfied. The product will become either less or larger than unity. In the former case the oscillation simply stops, and in the latter case nonlinearity is required in order to limit the amplitude. An oscillator in which the loop gain is exactly unity is an abstraction completely unrealizable in practice. In every practical oscillator the loop gain is slightly larger than unity, and the amplitude of the oscillations is limited by the onset of nonlinearity. The distortion is low if the amplitude of oscillation remains within the linear region of the amplifier. It must not be allowed to go into a full-swing oscillation. An amplitude-limiting mechanism is basically an automatic gain control (AGC) circuit that forces the amplifier gain to decrease when the amplitude of the oscillation increases. Wien bridge oscillator The positive feedback network (RC + RC) is frequency selective, and at the most favored frequency passes a maximum of 1/3 of the output swing back to the + input Fig.1. The negative feedback (560Ω + lamp; R lamp grows with current - positive temperature coefficient (PTC) thermistor) adjusts the gain (noninverting amplifier, 1+560/R lamp ). The incan-

2 Punčochář, Mohylová: TELO, Chapter 10 2 descent lamp is used as a variable-resistance element (with a long time-constant of response; the lamp is rated at 14 ma and 10V). The initial gain (not yet oscillation, the lamp is cold, R lamp is small) is greater than 3 the oscillator begins to oscillate. As the output level rises, the lamp heats slightly, reducing the gain to 3. Fig.1: Wien bridge oscillator amplitude stabilization (LAMP) Figure 2 shows Wien-bridge oscillators with diode amplitude limiting mechanisms; when the diodes are off, the gain is 1 + R2 R1; and when a diode is on, the gain is reduced to 1 + (R2 R3 )/R1. The start up condition requires a gain slightly greater than 3 or The inequality above can be satisfied by making it equal to a value between 2.1 to 2.2. When a diode is on, the gain should be slightly less than 3, or The inequality above can be satisfied by making it equal to a value between 1.8 to 1.9. When a diode is conducting, the amplitude of the output voltage is limited. Since v + = v = v o /3, a nodal equation gives or

3 Punčochář, Mohylová: TELO, Chapter 10 3 where V D» 0.5V for an actual diode. Fig.2: Wien bridge oscillator amplitude stabilization (diodes) Fig.3: Wien bridge oscillator another amplitude stabilization (diodes) Another Wien-bridge oscillator with a diode amplitude-limiting circuit is shown in Figure 3. In this oscillator the amplitude-limiting circuit consists of the diodes D1 and D2, and the resistors R3, R4, R5, and R6. To understand the operation of the amplitude-limiting circuit, observe that as v o increases, the voltage at node v y will exceed the voltage v 1, forcing D2 to conduct. When D2 conducts, the value of v y is v y = v , and v o is clamped at the value v o(max), given by (superposition theorem)

4 Punčochář, Mohylová: TELO, Chapter 10 4 Since v 1 is approximately v o /3, it follows from equation above that Similarly, as v o decreases, the voltage v x will drop below v 1, forcing D1 to conduct. When D1 conducts, the voltage v x is v x = v 1 0.7, and v o is clamped at the value v o(min), given by The simultaneous solution of equations gives the value of the resistors that limit the output voltage to v o(min) < v o < v o(max). In order to obtain a symmetrical sinusoidal voltage, the selection R3 = R6 and R4 = R5 is usually made. Example 1 -Design the Wien-bridge oscillator shown in Fig. 3 to oscillate at 5 khz. - Design an amplitude-limiting circuit. The amplitude of the sinusoidal output voltage is to be limited to v o = 5V. Solution A practical value of 0.01 μ F for the capacitors can be selected. Then, using the value of R is A practical value of 3 kω can be used in series with a trimming potentiometer to set the frequency of oscillation at 5 khz. To start the oscillation, a value of A vo = 3.2 is used. From a gain of 3.2 is obtained with R 2 = 22 kω and R 1 = 10 kω. The supply voltages of the op amp can be selected as 12V and 12V.Tthe output voltage will reach saturation producing a clipping in the output waveform and, therefore, a significant amount of distortion. This occurs because the starting condition requires A vo > 3, and the gain of the amplifier changes when its output reaches saturation. Some sort of amplitude-limiting mechanism is needed to reduce the harmonic distortion. With v o(max) = 5V and v o(min) = -5V, it follows from

5 Punčochář, Mohylová: TELO, Chapter 10 5 that R3 = R6 = 10.9 kω and R4 = R5 = 2 kω. In the circuit in Fig.4, an amplitude discriminator consisting of the diodes and RC adjusts the AC gain by varying the resistance of the JFET, which behaves like a voltagevariable resistance for small voltages. The LM103 is a two-terminal monolithic reference diode electrically equivalent to a breakdown diode. The long time constant is used (2s) to avoid distortion, since fast feedback will distort the wave by attempting to control the amplitude within the time of one cycle. 1 μf 2N5457 A 1N914 LM103 6,8 k 2,2 μf Fig.4: Wien bridge oscillator JFET amplitude stabilization The small amplitude generates a small voltage (negative) JFET is open (small resistor in parallel with 6,8 kω) small JFET resistance determines the maximum gain of more than 3. As the amplitude grows, the voltage (negative, on RC) grows too and JFET closes the minimum amplifier gain is about 1+ 10/6,8 = 2,47. We can add a buffer (voltage gain of 1) to isolate the non-linear effects of rectifier (diodes) from the oscillator output Fig. 5 and thus reduce the distortion of the oscillator.

6 Punčochář, Mohylová: TELO, Chapter 10 6 A output BUFFER 1 to 1N914 Fig.5: Wien bridge oscillator JFET amplitude stabilization; buffer reduces distortion of the oscillator Phase shift oscillator Phase-shift oscillators usually use RC networks in the feedback path. The op amp is used in an inverting configuration with a gain of -R 2 /R 1. Thus, the signal experiences a phase shift of -180 through the amplifier, and the phase shift from each RC section is 60 at the frequency of oscillation, for a total phase shift in the feedback path of 180. In this oscillator the RC sections are connected without isolation and, therefore, there is loading. In the last stage the resistors R and R 1 appear in parallel. The loading of R 1 can be neglected if R 1 R» R, or in some cases by removing R in the third stage and letting R1 = R. To summarize, the phase-shift oscillator in Fig. 6 will oscillate at the frequency ω o given by if the gain is A vo = R 2 /R 1 > 29. The loading of the op amp is minimized by making R1 > 10R. Fig.6: Phase shift oscillator amplitude stabilization (diodes)

7 Punčochář, Mohylová: TELO, Chapter 10 7 The harmonic distortion can be significantly reduced with an amplitude-limiting circuit. The amplitude-limiting circuit is designed using (see Fig. 3) and with v 1 set equal to zero (i.e., v y = 0.7V and v x = -0.7V). Example 2 -Design the phase shift oscillator shown in Fig. 3 to oscillate at 1 khz. - Design an amplitude-limiting circuit. The amplitude of the sinusoidal output voltage is to be limited to v o = 5V. Solution A practical value of 0.01 μ F for the capacitors can be selected. Then, using The resistors R 1 and R 2 must provide the gain A vo > 29 in order to prevent loading R 1 R R. Letting R 1 = 15 kω, then R 2 = 29R 1 = 435 kω. A 495-kΩ resistor was used to implement R 2. This will allow for some extra gain to satisfy the start of oscillation condition (i.e., A vo > 29). From (supply voltage ± 12 V) with v 1 = 0 and v o(max) = 5V, we obtain which can be satisfied with R 5 = 2 kω and R 6 = 5.9 kω. From for symmetry, we obtain R 3 = R 6 = 5.9 kω and R 4 = R 5 = 2 kω.

8 Punčochář, Mohylová: TELO, Chapter 10 8 Band pass filter comparator oscillator (spot sinus) The basic idea of low THD BPF-based oscillators is to incorporate a bandpass filter (BPF) along with a limiter and a comparator, in a positive feedback loop Fig.7. The oscillation frequency is set by the center frequency of the filter while the amplitude is set by the limiter (this filter has independent control of frequency, amplitude and distortion of the output). Fig.7: Bandpass filter comparator oscillator Input of BPF is roughly a square wave. According to its Fourier series, a 50% duty-cycle square wave consists of odd order harmonic sine waves with the fundamental at the same frequency as the square wave. Fourier Series for a Square Wave where k = peak amplitude of the square wave. Thus THD is dominated by lower order harmonics. The THD is directly proportional to the quality factor of the loop filter Fig.8. Fig.9 shows the THD of the oscillator versus the quality factor (Q) of a second-order filter. Achieving linearity better than 62 db requires very high-q filter (Q > 70). Implementing such high-q filter requires large op-amp gain bandwidth product as well as a large spread of the capacitor values, and will end up with larger silicon area. Fig.8: The Fourier series versus Q of BPF

9 Punčochář, Mohylová: TELO, Chapter 10 9 Fig.9: The third harmonic versus Q Notice that although the filter has unity gain, the amplitude of the sine wave output signal is greater than that of the square wave. This is because the fundamental has an amplitude of 4/π times that of the square wave as shown by the Fourier series. The bandpass filter will also filter out any DC component of the square wave input. The very simple (only limiter is used) circuit is shown in Fig. 10, for example. The limiter is a pair of diodes (and R 1 ) to have a squarewave at v 2. The active filter (other circuit elements) selects the fundamental frequency and provides the sinus output at v 1 (any filter circuit with positive gain can be used to implement the bandpass filter). Fig.10: Bandpass filter the limiter used only

10 Punčochář, Mohylová: TELO, Chapter Basic texts Other text Questions Answers you find in this text 1. Why isn t an input signal (to the oscillator) needed to obtain an output voltage signal? 2. Compare the operation of the described oscillator circuits. 3. Why does a harmonic oscillator need an amplitude control circuit? 4. Explain the function of the buffer on Fig.4 5. How can we get the sinusoidal voltage from the square wave voltage? 6. Why we need high-q band pass filter on Fig.7? Problems 1. Redesign the circuit of Fig. 3 for operation at 1 khz. The amplitude of the sinusoidal output voltage is to be limited to v o = 2V. 2. Redesign the circuit of Fig. 3 for operation at 5 khz. The amplitude of the sinusoidal output voltage is to be limited to v o = 2V. 3. Determine the needed Q of BPF (Fig.7) if we need HD 3-40 DB. Problems key See example 1 and example 2 and Fig

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