Oscillators
Oscillator: An oscillator is a circuit that generates a repetitive waveform of fixed amplitude and frequency without any external input signal [1, p. 279].
Function of Oscillator: The function of oscillator is to generate alternating current or voltage wave forms such as sinusoidal, square wave, triangular wave, sawtooth wave, etc. Used of Oscillator: Oscillators are used in radio, TV, computers, CRT, Oscillocope, and communications. Types of Oscillator: There are two main types of electronic oscillator:
(i) Harmonic oscillator and
(ii) Relaxation oscillator
Depending on the used elements of an oscillator, the types of oscillator are:
(i) RC oscillator,
(ii) LC oscillator and
(iii) Crystal oscillator.
Harmonic Oscillator: The harmonic oscillator produces a sinusoidal output. The basic form of a harmonic oscillator is an electronic amplifier with the output attached to a narrow-band electronic filter, and the output of the filter attached to the input of the amplifier. When the power supply to the amplifier is first switched on, the amplifier's output consists only of noise. The noise travels around the loop, being filtered and re-amplified until it increasingly resembles the desired signal.
Relaxation Oscillator: The relaxation oscillator is often used to produce a non-sinusoidal output, such as a square wave, sawtooth wave, and triangular wave. The oscillator contains a nonlinear component such as a transistor that periodically discharges the energy stored in a capacitor or inductor, causing abrupt changes in the output waveform.
Sinusoidal Oscillator: If the output signal of an oscillator circuit varies sinusodally, the circuit is referred to as a sinusoidal oscillator [3, p. 757].
Pulse or Square-Wave Oscillator: If the output voltages of an oscillator circuit rises quickly to one voltage level and later drops quickly to another voltage level, the circuit is generally referred to as a pulse or square-wave oscillator.
Oscillator Principles: An oscillator is a type of feedback amplifier in which part of the output is fed to the input via a feedback circuit. If the signal fed back is of proper magnitude and phase, the circuits produces alternating currents or voltages. To visualize the requirements of an oscillator, consider the block diagram of Figure 7-17 [1].
1
Fig. 7-17 Block diagram of oscillator.
This diagram looks identical to that of the feedback amplifiers. However, here the input voltage is zero (vin=0). Also, the feedback is positive because most oscillators use positive feedback. Finally, the closed-loop gain of the amplifier is denoted by Av.
In the block diagram of Fig. 7-17, vo = Av vd ; vd = v f + vin ; v f = β vo
Using these relationships, the following equation is obtained:
vo
Av
= vin 1 − βAv
However, vin=0 and vo≠0 implies that
βAv = 1
(7-20)
expressed in polar form,
βAv = 1 0 o or
360 o
(7-21)
Equation (7-21) gives the two requirements for oscillations:
(1) The magnitude of the loop gain Avβ must be at least 1, and
(2) The total phase shift of the loop gain Avβ must be equal to 0o or 360o.
For instance, as indicated in Figure 7-17, if the amplifier causes a phase shift of 180o, the feedback circuit must provide an additional phase shift of 180o, so that the total phase shift around the loop is 360o.
The waveforms shown in Fig. 7-17 are sinusoidal and are used to illustrate the circuit action. The type of waveform generated by an oscillator depends on the components in the circuit and hence may be sinusoidal, square, or triangular. In addition, the frequency of oscillation is determined by the components in the feedback circuit.
The Barkhausen Criterion:
(1) Oscillations will not be sustained if, at the oscillator frequency, the magnitude of the product of the transfer gain of the amplifier (Av) and the magnitude of the feedback factor (β) of the feedback network (the magnitude of the loop gain) are less than unity.
(2) The frequency at which a sinusoidal oscillator will operate is the frequency for which the total shift introduced, as a signal proceeds from the input terminals, through the amplifier and feedback network, and back again to the input, is precisely zero (or, of course, an integral multiple of 2π). Stated more simply, the frequency of a sinusoidal oscillator is determined by the condition that the loop-gain phase shift is zero.
2
The condition of unity loop gain –Avβ=1 is called the Barkhausen criterion. This condition implies, of course, both that |Avβ|=1 and that the phase of –Avβ is zero.
What happens to the output voltage?
If AB is less than 1, ABvin is less than vin and the output signal will die out. However AB is greater than 1, ABvin is greater than vin and the output signal build up.
Where Does the Starting Voltage Come From?
Every conductive wire or resistor contains free electrons. Because of ambient temperature, these free electrons move randomly in different directions and generate a noise voltage over 1000 GHz. So the conductive wire or resistor acts as a small ac voltage source producing all frequencies.
When the power is turned on, the only signals in the system are noise voltages generated by the conductive wire or resistor. These nose voltages are amplified and appear at the output terminals. The amplified nose, which contains all frequencies, drives the resonant feedback circuit. According to the design of an oscillator, the loop gain is greater 1 and the loop phase shift is equal to 0o or 360o at the resonant frequency.
Above or bellow the resonant frequency the phase shift is different from 0o or 360o.
As a result oscillation will build up only at the resonant frequency of the feedback circuit. Phase-Shift Oscillator: An oscillator circuit that follows the basic development of a feedback circuit is the phase-shift oscillator.
Figure 7-18 (and Fig. 14-29) shows a phase shift oscillator, which consists of an opamp as the amplifying stage and three RC cascaded networks as the feedback circuit. The feedback circuit provides feedback voltage from the output back to the input of the amplifier. The op-amp is used in the inverting mode; therefore, any signal that appears at the inverting terminal is shifted 180o at the output.
An additional 180o phase shift required for oscillation is provided by the cascaded RC networks. Thus the total phase shift around the loop is 360o (or 0o). At some specific frequency when the phase shift of the cascaded RC networks is exactly 180o and the gain of the amplifier is sufficiently large, the circuit will oscillate at that frequency. This frequency is called the frequency of oscillation, fo, and is given by
1
fo =
2π R1 R2 C1C 2 + R2 R3 (C1C 2 + C 2 C 3 + C 3C1 ) + R2 R1 (C1C 2 + C 3C1 )
fo =
1
0.065
=
RC
2π 6 RC
(7-22a)
At this frequency, the gain Av must be at least 29. That is,
R
Av = − F = 29
R1
Or
RF = 29R1
(7-22b)
3
Thus the circuit will produce a sinusoidal waveform of frequency fo if the gain is
29 and the total phase shift around the circuit is exactly 360o.
Fig. 7-18 Phase-shift oscillator.
For a desired frequency of oscillation, choose a capacitor C, and then calculate the value of R from equation (7-22a). A desired output, however, can be obtained with back-to-back zeners connected at the output terminal.
7-13 WIEN BRIDGE OSCILLATOR [R. A. Gayakward]
Because of it simplicity and stability, one of the most commonly used audio-frequency oscillators is the Wien Bridge. Figure 7-19 shows the Wien Bridge Oscillator in which the
Wien Bridge circuit is connected between the amplifier input terminals and the output terminal.
The bridge has a series RC network in one arm and a parallel RC network in the adjoining arm. In the remaining two arms of the bridge, resistors R1 and RF are connected (see Figure 7-19).
The phase angle criterion for oscillation is that the total phase shift around the circuit must be 0o. This condition occurs only when the bridge is balanced, that is, at resonance. The frequency of the oscillation fo is exactly the resonant frequency of the balanced Wien bridge and is given by
fo =
1
0.159
=
RC
2πRC
(7-23a)
assuming that the resistors are equal in value, and the capacitors are equal in value in the reactive leg of the Wien bridge. At this frequency the gin required for sustained oscillation is given by
4
Av =
1
=3
B
1+
That is,
RF
= 3;
R1
R F = 2 R1
(7-23b)
Figure 7-19 Wien bridge oscillator
For the derivation of Equations (7-23a) and (7-23b), refer to Appendix C. The Wien bridge oscillator is designed using Equations (7-23a) and (7-23b), as illustrated in Example 7-13.
Example 7-13: Design the Wien bridge oscillator of Figure 7-19 so that fo=965Hz.
Solution: Let C= 0.05 µF. Therefore, from Equation (7-23a),
R=
0.159
0.159
=
= 3.3 KΩ f oC
965 × 5 × 10 −8
Now let R1=12KΩ. Then, from Equation (7-23b),
R F = 2 × 12KΩ = 24KΩ
7-14 Quadrature Oscillator [R. A. Gayakward]
As its name implies, the quadrature oscillartor generates two signals (sine and cosine) that are in quadrature, that is, out of phase by 90o. Although the actual location of the sine and cosine is arbitrary, in the quadrature oscillator of Figure 7-20 the output of A1 is labeled a sine and the output of A2 is a cosine.
5
Figure 7-19 Quadrature Oscillator. A1 and A2 dual op-amp: 1458/353.
This oscillator requires a dual op-amp and three RC combinations. The first op-amp A1 is operating in the noninverting mode and appears as a noninverting integrator. The second op-amp
A2 is working as a pure integrator.
Furthermore, A2 is followed by a voltage divider consisting of R3 and C3. The divider network forms a feedback circuit, whereas A1 and A2 form the amplifier stage.
The total phase shift of 360o around thee loop required for oscillation is obtained in the following way. The op-amp A2 is a pure integrator and inverter. Hence it contributes -270o (or
90o) of phase shift. The remaining -90o (or 270o) of phase shift needed are obtained at the voltage divider R3C3 and the op-amp A1. The total phase shift of 360o, however, is obtained at only one frequency fo, called the frequency of oscillation. This frequency is given by
fo =
1
0.159
=
RC
2πRC
(7-24a)
where R1C1= R2C2= R3C3=RC. At this frequency,
Av =
1
= 1.414
B
(7-24b)
which is the second condition for oscillation.
Thus, to design a quadrature oscillator for a desired frequency fo, choose a value of C; then, from Equation (7-24a), calculate the value of R. To simplify design calculations, choose
C1=C2=C3 and R1= R2= R3. In addition, R1 may be a potentiometer in order to eliminate any possible distortion in the output wave forms.
Example: Design the quadrature oscillator of Figure 7-20 so that fo=159 Hz.
Solution: Let C=0.01 µF. Then, from equation (7-24a),
6
R=
0.159
0.159
=
= 100 KΩ f o C 159 × 10 −8
Thus C1=C2=C3=0.01 µF and R1= R2= R3=100 KΩ. However, R1 may potentiometer, which can be adjusted for undistorted output wave forms.
be
a
200
KΩ
23.3 Twin-T Oscillator [3]
Figure Twin-T oscillator
An oscillator configuration that uses a Twin-T filter in the feedback path is shown in the following Figure 23.12. The Twin-T filter consists of two Tee-shaped networks connected in parallel. These Twin-T filters are also known as band-reject filters or notch filters. In the Twin-T filter shown in Figure 1.38, the elements connected to ground have values of 2C and R/2, respectively. The twin-T circuits acts as a lead-lag circuit with a changing phase angle. There is a frequency fo at which the voltage gain drops to 0. The equation of the frequency is
1
fo =
2πRC
The positive feedback to the noninverting input is through a voltage divider. The negative feedback is through the Twin-T filter.
To ensure that the oscillation frequency is close to the notch frequency fo, the voltage divider should have R2 much larger than R1.
The Twin-T oscillator is not popular circuity because it works well only at one frequency.
14-18 A General Form of Oscillator Circuit [J. Millman, C. C. Halkias]
Many oscillator circuits fall into the general form shown in Figure 14-32a.
7
Figure 14-32 (a) The basic configuration for many resonant-circuit oscillators. (b) The linear equivalent circuit using an operational amplifier.
The active device may be a bipolar transistor, an operational amplifier or an FET. In the analysis that follows we assume an active device with infinite input resistance such as an FET, or an operational amplifier.
Figure 14-32b shows the linear equivalent circuit of Figure 14-32a, using the amplifier with negative gain –Av and output resistance Ro. Clearly the topology of Fig. 14-32 is that of voltage-series feedback.
The Loop gain: The value of –AB will be obtained by considering the circuit of Fig. 14-32a to be a feedback amplifier with output taken from terminals 2 and 3 and terminals 1 and 3. The load impedance ZL consists of Z2 in parallel with series combination of Z1 and Z3 that means
ZL=Z2(Z1+Z3)/(Z1+Z2+Z3). The gain without feedback is A=-AvZL/(ZL+Ro). The feedback factor is
B=-Z1/(Z1+Z3). The loop gain is found to be
− AB =
− Av Z 1 Z 2
Ro ( Z 1 + Z 2 + Z 3 ) + Z 2 ( Z 1 + Z 3 )
(14-65)
Reactive Elements Z1, Z2, and Z3: If the impedances are pure reactances (either inductive
or capacitive), then Z1=jX1, Z2=jX2, and Z3=jX3. For an inductor X=ωL, and for a capacitor X=1/ωC. Then
− AB =
Av X 1 X 2 jRo ( X 1 + X 2 + X 3 ) + X 2 ( X 1 + X 3 )
(14-66)
For the loop gain to be real (zero phase shift)
X1 + X 2 + X 3 = 0
(14-67)
and
8
− AB =
Av X 1 X 2
− Av X 1
=
− X 2 (X1 + X 3 ) X1 + X 3
(14-68)
From Eq. (14-67) we see that the circuit will oscillate at the resonant frequency of the series combination of X1, X2, and X3.
Using Eq. (14-67) in Eq. (14-68) yields
− AB =
Av X 1
X2
(14-69)
Since –AB must be positive and at least unity in magnitude, then X1 and X2 must have the same sign (Av is positive). In other words, they must be the same kind of reactance, either both inductive or both capacitive. Then, from Eq. (14-67), X3=-(X1+X2) must be inductive if X1 and X2 are capacitive, or vice versa.
If X1 and X2 are capacitors and X3 is an inductor, the circuit is called a Colpits oscillator.
If X1 and X2 are inductors and X3 is an capacitor, the circuit is called a Hartley oscillator.
In this latter case, there may be mutual coupling between X1 and X2 (and the above equations will then not apply).
Transistor versions of above types of LC oscillators are possible. As an example, a transistor Colpits Oscillator is indicated in Fig 14-33a. Qualitatively, this circuit operates in the manner described above. However, the detailed analysis of a transistor oscillator circuit is more difficult, for two fundamental reasons.
First, the low impedance of the transistor shunts Z1 in Fig. 14-32a, and hence complicates the expressions for the loop gain given above.
Second, if the oscillation frequency is beyond the audio range, the simple low-frequency h-parameter model is no longer valid. Under these circumstances the more complicated highfrequency hybrid-Π model of Fig. 11-5 must be used. A transistor Hartley oscillator is shown in
Fig. 14-33b.
Figure. (a) An Op-amp Colpits Oscillator. (b) An op-amp Hartley Oscillator.
9
Z1
X1
X
=−
= 1 Q X1 + X 3 = −X 2
Z1 + Z 3
X1 + X 3 X 2
AX
R
− AB = v 1 = Av B ∴ A = − Av = F
R1
X2
1 R
A= = F
B R1
B=−
From (14.67)
For Hartley: X 1 + X 2 + X 3 = 0 ; jX L1 + jX L 2 − jX c = 0 ;
Clapp Oscillator:
The Clapp oscillator is a Colpitts oscillator with an additional capacitor placed in series with the inductor. The oscillation frequency in hertz (cycles per second) for the circuit of
Clapp oscillator, is
1 1 1
1
1 fo =
( +
+ )
2π L C0 C1 C 2
A Clapp circuit is often preferred over a
Colpitts circuit for constructing a variable frequency oscillator (VFO). In a Colpitts
VFO, the voltage divider contains the variable capacitor (either C1 or C2). This causes the feedback voltage to be variable as well, sometimes making the Colpitts circuit less likely to achieve oscillation over a portion of the desired frequency range. This problem is avoided in the Clapp circuit by using fixed capacitors in the voltage divider and a variable Figure. An Op-amp Clapp Oscillator. capacitor (C0) in series with the inductor.
Crystal Oscillators
18-9 [Robert Boylestad, Louis Nashelsky]
A crystal oscillator is basically a tuned-circuit oscillator using piezoelectric crystal as a resonant tank circuit. The crystal (usually quartz) has a greater stability in holding constant at whatever frequency the crystal is originally cut to operate. Crystal oscillators are used whenever great stability is required, for example, in communication transmitters and receivers.
Characteristics of a Quartz Crystal
A quartz crystal (one of a number of crystal types) exhibits the property that If a piezoelectrical crystal, usually quartz, has electrodes plated on opposite faces when mechanical stress is applied across the faces of the crystal, a difference of potential develops across opposite faces of the crystal. This property of a crystal is called the piezoelectric effect. Similarly, a voltage applied across one set of faces of the crystal causes mechanical distortion in the crystal shape. When alternating voltage is applied to a crystal, mechanical vibrations are set up-these vibrations having a natural resonant frequency dependent on the crystal. Although the crystal has electromechanical resonance, we can represent the crystal action by an equivalent resonant circuit as shown in Fig. 14-35.
11
(a)
(c)
(b)
Fig. 14-35 A piezoelectric crystal. (a) symbol; (b) electrical model; (c) the reactance function
(if R=0)
The inductor L and capacitor C represent electrical equivalents of a crystal mass and compliance while R is an electrical equivalent of the crystal structure’s internal friction. The shunt capacitance CM represents the capacitance due to the mechanical mounting of the crystal.
Because the losses of crystal, represented by R, are small, the equivalent crystal Q (quality factor) is high. Values of Q up to almost 106 can be achieved by using crystals.
The crystal as represented by the equivalent electrical circuit of Fig. 14-35 can have two resonant frequencies. One resonant condition occurs when the reactances of series
RLC leg are equal (and opposite). For this condition the series-resonant impedance is very low (equal to R).
Figure 18.32 Crystal impedance versus frequency. The other resonant condition occurs at a high frequency when the reactance of the seriesresonant leg equals the reactance of capacitor CM. This is a parallel resonance or anti-resonance condition of the crystal. At this frequency the crystal offers a very high impedance to the external circuit. The impedance versus frequency of the crystal is shown in Fig. 18-32.
In order to use he crystal properly it must be connected in a circuit so that its low impedance in the series operating mode or high impedance in the anti-resonant operating mode is selected. Series-Resonant Circuits
To excite a crystal for operation in the series-resonant mode it may be connected as a series element in a feedback path. At the series-resonant frequency to the crystal its impedance is
12
smallest and the amount of (positive) feedback is largest. A typical transistor circuit is shown in
Fig. 18.33. Resistors R1, R2, and RE provide a voltage-divider stabilized dc bias circuit.
Capacitor CE provides ac bypass of the emitter resistor and the RFC coil provides for dc bias while decoupling any ac signal on the power lines from affecting the output signal. The voltage feedback from collector to base is a maximum when the crystal impedance is minimum (in seriesresonant). The coupling capacitor CC has negligible impedance at the circuit operating frequency but blocks any dc between collector and base.
Figure 18.33 Crystal-controlled oscillator using crystal in series-feedback path.
The resulting circuit frequency of oscillation is set, then, by the series-resonant frequency of the crystal. Changing in supply voltage, transistor device parameters, and so on, have no effect on the circuit operating frequency which is held stabilized by the crystal. The circuit frequency stability is set by the crystal frequency stability, which is good.
Parallel-Resonant Circuits
Since the parallel-resonant impedance of a crystal is a maximum value, it is connected in shunt. At the parallel-resonant operating frequency a crystal appears as an inductive reactance of largest value. Figure 18-34 shows a crystal connected as the inductor element in a modified
Colpits circuit. The basic dc bias circuit should be evident. Maximum voltage is developed across the crystal at its parallel-resonant frequency. The voltage is coupled to the emitter by a capacitor voltage divider-capacitors C1 and C2.
Figure 18.34 Crystal-controlled oscillator operating in parallel-resonant mode.
13
A Miller crystal-controlled oscillator is shown in Fig. 18.35. A tuned LC circuit in the drain section is adjusted near the crystal parallel-resonant frequency. The maximum gate-source signal occurs at the crystal anti-resonant frequency controlling the circuit operating frequency.
Figure 18-35 Miller crystal-controlled oscillator.
Crystal Oscillator
An op-amp can be used in a crystal oscillator as shown in Fig. 18.36. The crystal is connected in the series-resonant path and operates at the crystal series-resonant frequency. The present circuit has a high gain so that an output square-wave signal results as shown in the figure.
A pair of Zener diodes is shown at the output to provide output amplitude at exactly the Zener voltage (VZ).
Figure 18.36 Crystal oscillator using op-amp.
14-20 [J. Millman, C. C. Halkias]
If a piezoelectrical crystal, usually quartz, has electrodes plated on opposite faces and if a potential is applied between these electrodes, forces will be exerted on the bound charges within
14
the crystal. If this device is properly mounted, deformations take place within the crystal, and an electromechanical system is formed which will vibrate when properly excited.
The resonant frequency and the Q depend upon the crystal dimensions, how the surfaces are oriented with respect to its axes, and how the device is mounted.
Frequency ranging from a few kilohertz (kHz) to a few megahertz (MHz), and Q’s in the range from several thousand to several hundred thousand, are commercially available.
These extraordinarily high values of Q and the fact that the characteristics of quartz are extremely stable with respect to time and temperature account for the exceptional frequency stability of oscillators incorporating crystals.
The electrical equivalent circuit of a crystal is indicated in Fig. 14-35. The inductor L, capacitor C, and resistor R are the analogs of the mass, the compliance (the reciprocal of the spring constant), and thee viscous-damping factor of the mechanical system. CM is mounting capacitance. Typical values for a 90-kHz crystal are L=137 H, C=0.235 pF, and R= 15 K, corresponding to Q=5,500. The dimensions of such a crystal are 30 by 4 by 1.5 mm. Since CM represents the electrostatic capacitance between electrodes with the crystal as a dielectric, its magnitude (∼3.5 pF) is very much larger than C.
If we neglect the resistance R, the impedance of the crystal is a reactance jX whose dependence upon frequency is given by
jX =
j ω 2 − ω s2 ωC M ω 2 − ω 2 p (14-75)
1 is the series resonant frequency (the zero impedance frequency), and
LC
1 1
1
ω2 = ( +
) is the parallel resonant frequency (the infinite impedance frequency). p L C CM
Since CM>>C, then ω p ≈ ω s . For the crystal whose parameters are specified above, the
where ω s2 =
parallel frequency is only three-tenth of 1 percent higher than the series frequency. For ω s < ω < ω p , the reactance is inductive, and outside this range it is capacitive, as indicated in
Fig. 14-35.
A variety of crystal-oscillator circuit is possible. If in the basic configuration of Fig. 1432a a crystal is used for Z1, a tuned LC combination for Z2, and the capacitance Cdg between drain and gate for Z3, the resulting circuit is as indicated in Fig. 14-36.
From the theory given in the preceding section, the crystal reactance, as well as that of the
LC network, must be inductive. For the loop gain to be greater than unity, we see from Eq. (1469) that X1 cannot be too small. Hence the circuit will oscillate at a frequency which lies between ωs and ωp but close to the parallel-resonance value. Since ωp≈ωs, the oscillator frequency is essentially determined by the crystal, and not by the rest of the circuit.
References:
15
[1] Ramakant A. Gayakward, “Op-Amps and Linear Integrated Circuits (Fourth Edition)”,
Pearson Education, Inc., 2000. pp. 279
[2] Jacob Millman, and Christos C. Halkias, “Integrated Electronics: Analog and Digital Circuits and Systems”, Tata McGrew-Hill Publishing Conpamy Ltd., 1972. pp. 486
[3] Albert Paul Malvino, “Electronic Principles”, Tata McGraw-Hill, 1999
[4] Robert Boylestad, and Louis Nashlsky, “Electronic Devices and Circuit Theory”, PrenticeHall inc., 1994.
First consider the feedback circuit consisting of RC combinations of the phase shift oscillator. For simplicity we use the Laplace transform. Thus, the circuit is represented in the S domain as shown in Figure C-8. Let us determine Vf(S)/Vo(S) for the circuit.
Writing Kirchhof’s current law (KCL) at node V1(S), we get
I 1 (S ) = I 2 (S ) + I 3 (S )
V o ( S ) − V1 ( S ) V1 ( S ) V1 ( S ) − V 2 ( S )
=
+
1 / SC
R
1 / SC
Solving for V1(S), we have
V1 ( S ) =
[V o ( S ) − V 2 ( S )]RCS
2 RCS + 1
(C-8)
Writing KCL at node V2(S),
I 3 (S ) = I 4 (S ) + I 5 (S )
16
To show: f o =
1
0.159
; RF = 2R1 for Wien bridge oscillator
=
2πRC
RC
First consider the feedback circuit of the Wien bridge oscillator of Figure 7-19. The circuit is transformed in the S domain and redrawn in Figure C-10.
Using the voltage-divider rule:
Z P ( S )Vo ( S )
Z P (S ) + Z S (S )
1
R
RSC + 1
1
; Z S (S ) = R +
=
= where Z P ( S ) = R
SC RSC + 1
SC
SC
V f (S ) =
Therefore, substituting ZP(S) and ZS(S) values, we get
V f (S ) =
( RCS )Vo ( S )
( RCS + 1) 2 + RCS
V f (S )
RCS
= 2 2 2
B=
Vo ( S ) R C S + 3RCS + 1
Or
(C-15)
Figure C-10 Feedback circuit of the Wien bridge oscillator of Figure 7-19 represented in the S domain. Next, consider the op-amp part of the Wien bridge oscillator. The circuit is redrawn in Figure C11.
Figure C-11 Op-amp part of the Wien bridge oscillator.
The voltage gain Av of the op-amp is
Av =
Vo ( S )
R
= 1+ F
V f (S )
R1
(C-16)
Finally, the requirement for oscillation is:
Av B = 1
17
Therefore using Equations (C-15) and (C-16), we have
RF
RCS
) 2 2 2
=1
R1 R C S + 3RCS + 1
Substituting S=jω in this equation and then equating the real and imaginary parts, we get the
(1 +
frequency of the oscillation fo and the gain required for oscillation, as follows:
(1 +
RF
) jRCω = (− RCω ) 2 + j 3RCω + 1
R1
ω2 =
1
R C2
fo =
1
0.159
=
2πRC
RC
(real part)
2
or
(7-23a)
and
(1 +
1+
RF
) RCω = 3RCω
R1
RF
= 3;
R1
RF = 2 R1
To show: f o =
Vo = (1 +
1 / sC
)Vi ;
R
(imaginary part)
(7-23b)
1
0.159
=
; Av=1/B=1.414 for Quadrature oscillator
RC
2πRC
1 / sC
Vo1 = −
Vo ;
R
Vf = −
1 / sC
V
R + 1 / sC o1
j ω 2 − ω s2
To show jX = for crystal oscillator ωC M ω 2 − ω 2 p The impedance of series branch is: Z s = R + j ( X L − X C )
Neglecting R, Z s = j ( X L − X C )
The resonant occurs at: X L − X C = 0;
XL = XC; ωL = (1 / ωC ) ;
ω 2 = (1 / LC ) if we denote the resonant frequency for series branch by ωs then ω s = 1 / LC .
The impedance of overall circuit is:
Zp =
− Z s jX M
− j ( X L − X C ) jX M
(X L − X C )X M
=
=−j
Z s − jX M j ( X L − X C ) − jX M
XL − XC − XM
18
1
1
ω 2 LC − 1 1
)
ωC ωC M ωC ωC M
1
ω 2 LC − 1
=−j 2
=−j
Zp = −j
1
1 ω ω 2 LCC M − (C + C M ) ω LCC M − (C + C M ) ωL −
−
ωC ωC M ωCC M
(ωL −
Zp = −j
LC (ω 2 − 1 / LC )
(ω 2 − 1 / LC )
1
=−j ω CC M [ω 2 L − (C + C M ) / CC M ] ωC M [ω 2 − (1 / L){(1 / C M ) + (1 / C )}]
Zp =−j
2
2
1 ω − ωs
;
2 ωC M ω 2 − ω p
1
2 ω p = (1 / L){(1 / C M ) + (1 / C )}
OR
1
s = jω sC R s 2 + s + ω s2
L
=
R
2
2
s[ s + s + ω p ]
L
The impedance of series branch is: Z s = R + sC +
Zs = Zs
1
1
= [ R + sC +
]
sC M sC 1 sC M
The following steps should be followed to discuss or explain about the operation an oscillator:
Step 1: Draw the circuit diagram and indicate the amplifier part and feedback network part
Step 2: Discuss or explain, what type of amplifier is used, what is the expression of amplifier gain, and how much phase shift is occurred in the amplifier circuit.
Step 3: Discuss or explain, what type of feedback network is used, how much phase shift is required to obtain from the feedback network.
Step 4: Discuss or explain, what is the expression of frequency to satisfy the condition that the angle of loop gain is 0 or 360 degree.
Step 5: Discuss or explain, what is the requirement of the parameters of amplifier gain expression to satisfy the condition that the amplitude or magnitude of loop gain is greater than or equal 1.
