Chapter 7 Oscillators

---

Contents

Instabilities, Oscillations and Oscillators

Ideal Electronic Oscillator

Tunnel-Diode Oscillator

van der Pol Approximation

Simple Transistor Oscillator Circuits

Practical Consideration

Phase-shift Oscillator

Wien Bridge Oscillator

General Form of an Oscillator

LC-Tunable Oscillators

Crystal Oscillators

• If positive feedback is applied to an amplifier, the feedback signal is in phase with the input, a regenerative situation exists.
• If the magnitude of the feedback is large enough, an unstable circuit is obtained.
• To achieve the oscillator circuit function, we must ensure an unstable situation. In addition we need to develop the oscillatory power at a desired frequency, with a given amplitude and with excellent constancy of envelope amplitude and frequency.
• The design of good oscillators can be quite demanding because the governing equations of an oscillator are nonlinear, differential equations. Consequently oscillator analysis and design are not as advanced as that for linear circuits.
• Typical oscillator analysis involves reasonably simple approximate analyses of linearized or piecewise-linear-circuit models of the oscillator together with perturbations and power series techniques.
• There are a few oscillator circuits that can be solved exactly.

Ideal Electronic Oscillator

• An ideal harmonic oscillator can be modeled by a lossless L-C circuit. Due to its lossless nature energy is conserved and alternates between electrical and magnetic forms. The circuit equation is
• where
• The solution is

  

• where for the latter expression, the time origin is chosen to produce a zero phase angle. In the frequency domain, the characteristic equation of the circuit is
• The roots of the equation are which lie on the imaginary axis of the complex plane.
• A real oscillator will have loss leading to a damped sinusoid. In the frequency domain, the characteristic equation includes a linear term as shown
• The natural frequencies, which are the roots, lie in the left-hand plane

  

• If oscillation are to be maintained, energy must continuously be supplied to the circuit on a time average. This power is usually supplied by dc bias to the devices that convert the bias power into signal power in the form of a negative, nonlinear conductance or as regenerative feedback.

Tunnel-Diode Oscillator

• The model for a tunnel -diode oscillator is shown in Fig. 7-1
• If the diode is biased at V_DD the incremental input conductance presented to the passive resonant circuit is negative and can compensate for the positive losses of the inductance, capacitance, leads, etc. Such losses are modeled by a single conductance G=1/R.
• Shifting the axis to the Q-point, the new voltage variable is v'=V-V_DD. The original diode I-V characteristic is described functionally as I=f₁(V)
• The translated diode-current variable is i'=I-I₀=I-f₁(_DD)
• The circuit equation in the original variables is

  

• In terms of the new variables we have

  

• Both sides of equation are next differentiated and multiplied by L

  

• The combined first-order term in the equation can be viewed as the net nonlinearity of the oscillator.
• We need to determine if the equilibrium point of the circuit is unstable. At the Q-point, an incremental analysis is made. For the diode, the slope of the translated I-V characteristic at v¡¦= 0 is designated -a. The characteristic equation at the bias point and in the frequency domain is LCs²+L(G-a)s+1=0
• The natural frequencies of the linearized circuit about the bias point are

  

• If the magnitude of the slope of the diode characteristic at the bias point a, is greater than G, the loss conductance of the oscillator, the natural frequencies lie in the right-half-plane. Therefore, given an excitation, the oscillatory response grows exponentially. If the natural frequencies are complex, the response is an exponentially growing sinusoid
• This growth cannot continue indefinitely. As the diode voltage excursion gets large, the end points, p and q, of the negative slope region of the diode I-V characteristic are encountered the diode introduces more loss into the system. The voltage response can be illustrated as shown below in Fig. 7-2
• For very small excursions of the bias point, an exponential growth is produced. As the oscillation becomes larger, the total loss introduce a limiting condition, and a steady-state oscillation is produced. The output cannot be pure sinusoidal, the tips of the voltage output must be compressed, and the output waveform must contain harmonics. For the tunnel-diode oscillator, the output voltage will be quite nonsinusoidal in the best case since the non-linear I-V is not antisymmetrical about the bias point in the negative-conductance region
• It appears that by setting |a| = G, we may exactly cancel G and result in an ideal oscillator. However, the I-V is not linear at the bias point, or anywhere, thus the equality condition is not satisfied over the excursion of the variable. Moreover the diode characteristics are temperature and age dependent and therefore cancellation is only temporary

van der Pol Approximation

• About the bias point of the total I-V characteristic, he proposed a cubic polynomial approximation. The slope of the bias point is equal in magnitude to the total conductance. The cubic term of the approximation produces the essential limiting action
• For the tunnel-diode oscillator the nonlinear differential equation is repeated for easy reference
• in which the primes are dropped. The 1st order term combines the effects of the passive conductance and of the diode¡¦s I-V characteristic. It is convenient next to introduce a time scaling normalization. The dimensionless independent variable is
• Thus the differential equation becomes

  

• where the total device and conductance function is defined as F(v)=Gv+f(v)
• A plot of this total function is the addition of f(v) and Gv. The slope of this total function at the origin is -(a-G) where -a is the slope of the diode characteristic alone at the bias point. The van der Pol approximation for F(v) is labeled F_v(v)

  

• for v=0, the slope of the approximation is set equal to the slope of the actual total characteristic a₁=-(a-G)
• To determine an appropriate value for b1 for the van der Pol approximation are

  

• This leads to where are the crossover points on the voltage axis
• The oscillator equation containing the cubic approximation is transformed into a standard form, from which an approximate closed-form solution for the equation is obtained and the output is nearly sinusoidal
• Fig. 7-3
• To obtain th evan der Pol Equation we introduce the parameter which is called the van der Pol parameter e. This is the negative of the slope of the normalized total nonlinear function at the origin, multiplied by (L/C)^1/2. Thus
• Next, a scaling of the voltage variable is introduced

  

• using these parameters, we obtain the standard form of the van der Pol equation
• About the equilibrium (bias) point the differential equation of the system becomes p²-ep+1=0 where p is the normalized complex frequency variable, p=(LC)^0.5s. The natural frequencies of the linearized, normalized system are
• For positive e, the natural frequencies are in the RHP and complex if e< 2. For e> 0 but very small, the buildup of oscillation has the form
• We expect that after a steady state has been reached, the form of the response should be u(T)=Constxcostvan der Pol proposed a solution which after making several reasonable assumptions and simplifications, which are valid for near-sinusoidal oscillation, the solution in terms of the original variables becomes

  

• In this equation, a phase angle f₀ is introduced to provide the proper phase for the cosine term in relation to the choice of the constant t₀. For small values of time, the envelope of the oscillation grows exponentially. After a long period of time, the zero-to-peak amplitude reaches a maximum value
• Since
• Since the value of the voltage at the negative peak of the total characteristic is
• The maximum amplitude is twice the value of the voltage at which the negative maximum occurs as shown in Fig, 7-4

Simple Transistor Oscillator Circuits

• Simply put the condition for oscillation is that the total phase shift around the loop must be 360¢X at the frequency of oscillation and the magnitude of the open-loop gain must be unity at that frequency.
• If a single-stage common-emitter (or common source) amplifier is used with feedback from the collector to base as shown, then the feedback network must supply 180¢X phase shift between the base and the collector signals
• If a common-bse (or common gate) amplifier is used there is no phase shift between the emitter and collector signals, then a necessary condition for oscillation is that there be no phase shift between the input and output of the feedback network.
• Fig. 7.5
• If a small phase shift occurs in the forward loop, this must be compensated for by an equal and opposite phase in the feedback network.
• The primary purpose of the feedback network is to control the frequency of oscillation, the network is designed so that the Nyquist criteria are satisfied at only a single frequency.
• The following circuit, in Fig. 7-6, is analyzed to determine the conditions for oscillation.
• The equivalent circuit is shown in the next diagram, in which the transistor output resistance r_o is ignored, as is the large biasing resistor R_B. Also, the capacitor connected to the base is assumed to be so large that the base is at ground potential for the small-signal analysis. Note that the transistor is connected in the common-base configuration and has no voltage phase inversion.
• The conditions for oscillation are and because the amplifier is noninverting, the phase shift of the network must also be
• Fig. 7-7
• The loop gain is calculated by opening the feedback loop, applying a signal, and measuring the return difference. It is necessary that when the loop is opened, the impedance seen at any point be the same as it is with the loop closed. In this case it is convenient to open the loop at the transistor emitter. The impedance shunting the capacitor C₂ is the resistor R_E in parallel with the input impedance ri of the common-base amplifier. For common-base amplifier the input resistance is so the equivalent circuit of Fig. 7-4a is redrawn in Fig. 7-4b.
• The circuit analysis can be simplified by assuming that

  

• and that the Q of the load impedance is high. In this case the circuit reduces to Fig. 7-8 which is the small signal equivalent of the open-loop circuit with the loop or feedback network opened at the emitter and the circuit terminated in the correct impedance. The network could be opened at other points, but opening it at the emitter is very convenient because it is relatively easy to determine the terminating impedance at this point. The feedback voltage is given by and the equivalent resistance reflected across the coil is

  The forward loop gain is where

• The feedback ratio is
• A necessary condition for oscillation is that
• Such H does not depend on frequency in this example, if arg(GH) is to be zero, the phase shift of the load impedance Z_L must be zero. This occurs only at the resonant frequency of the circuit where
• At this frequency

  

• The other condition for oscillation is the magnitude constraint that

  

• Example
• Design a common-base sinusoidal oscillator using a transistor with a minimum b of 100.
• Solution: Oscillator design consists of a trial-and-error procedure using the above equations. We assume a bias current I_C=1mA. The common base input resistance

  

• Since this is so small, we can safely assume that the emitter bias resistor RE is much larger than ri so that
• Also, the above equations were based on the assumption that If we choose , then and the above inequality is satisfied so the assumption was justified.
• Practically, a factor of 10 difference will usually satisfy the the ¡§<<¡¨ condition. In addition, for oscillations to occur, the loop gain must be at least 1. In oscillator design the loop gain is typically selected to about 3, which allows for some error in the approximation. With a loop gain > 1, the system is unstable and the oscillations increase in amplitude until the transistor current begins to saturate. When this occurs, the b of the transistor is reduced, and thus gm is also reduced. This reduces the loop gain and stabilizes the amplitude of oscillation. In this example the loop gain is
• So C₁=C₂/2=500pF. The value of the inductance is found to be L=0.19 mH. also, R_eq=234W. A load resistor R_L can be shunted across the inductor without affecting the calculations if it is much larger than R_eq. In this case a resistance of 1500W could be safely added. The complete design would require selecting a supply voltage and bias resistors so that the Quiescent collector current is 1 mA. A completed circuit diagram is given in Fig. 7-9.

Practical Consideration

• The condition may not be met due to the aging of the transistors and the circuit. It is therefore necessary to have |T| somewhat larger than unity (say 5%) ensure |T| does not fall below unity in case of incidental changes of circuit parameters. In the case where the magnitude of oscillation is limited by the onset of nonlinearity.

Phase-shift Oscillator

• A typical phase-shift oscillator is shown below. Here a discrete-component JFET amplifier is followed by three cascaded arrangements of a capacitor C and a resistor R, the output of the last RC combination being returned to the gate. If the loading of the phase-shift network on the amplifier can be neglected that is R>> R_L, the amplifier shifts by 180° the phase of any voltage which appears on the gate, and the network of resistors and capacitors shifts by an additional amount. At some frequency the phase shift introduced by the RC network will be precisely 180°, and at this frequency the total phase shift from the gate around the circuit and back to the gate will be exactly zero.
• Fig. 7-10
• This particular frequency will be the one at which the circuit oscillates, provided that the magnitude of the amplification is sufficiently large.
• where . For the circuit to oscillate T should be real and
• . At
• and g_mR_L=29 to sustain oscillation.
• It is possible to use an Op-Amp in place of the transistor as shown. In this case R₂/R₁=29.

Wien Bridge Oscillator

• An oscillator circuit in which a balanced bridge is used as the feedback network is the Wien bridge oscillator as shown in Fig. 7-11
• he four arms of the bridge are Z₁, Z₂, R₁, R₂. The input to the bridge is the output Vo of the Op-Amp, and the output of the bridge between nodes 1 and 2 supplies the differential input to the Op-Amp. there are two feedback paths in the figure: positive feedback through Z₁ and Z₂, whose components determine the frequency of oscillation, and negative feedback through R₁ and R₂, whose elements affect the amplitude of oscillation and set the gain of the Op-Amp. The loop gain is given by
• For
• The equivalent circuit is as shown in Fig. 7-12

General Form of an Oscillator

• Many oscillator fall into the general form as shown. in Fig. 7-13
• In this case the return ratio is

  

LC-Tunable Oscillators

• The oscillators described so far are RC tunable circuits which are often limited to operations in the range of a few hundred kHz. For higher frequencies LC tunable circuits are needed. Two types of LC-tunable oscillators are commonly used.
• Fig. 7-14
• From
• where Z₁=jX₁, Z₂=jX₂, Z₃=jX₃

  X=wL for inductance and -1/wC for capacitors

Crystal Oscillators

• If a piezoelectric crystal, usually quartz, has electrodes plated on opposite faces and a potential is applied between these electrodes, forces will be exerted on the bound charges within the crystal. If this device is properly mounted, deformations takes place within the crystal, and an electromechanical system is formed which will vibrate when properly excited. Frequencies ranging from a few kHz to a few hundred MHz. Q values range from several thousand to several hundred thousand. The equivalent circuit of a crystal is shown below L,C and R are analogs of mass, compliance (reciprocal of spring constant) and viscous-damping factor of the mechanical system. If we neglect R the impedance of the crystal is
• where
• Fig. 7-15
• A variety of crystal-oscillator circuits is possible. If a crystal is used for Z₁ in the basic configuration shown before, a tuned LC combination for Z₂ and capacitance C_gd between the gate and drain for Z₃, the resulting circuit is as indicated