Comparison between an Amplifier and an Oscillator
As discussed in the previous module, an amplifier produces an output signal whose waveform is similar to the input signal but whose power level is generally high. This additional power is supplied by the external dc source. Hence, an amplifier is essentially an energy convertor i.e. it takes energy from the dc power source and converts it into ac energy at signal frequency. The process of energy conversion is controlled by the input signal. If there is no input signal, there is no energy conversion and hence there is no output signal.
Fig.1.1: Comparison between an amplifier and an oscillator An oscillator differs from an amplifier in one basic aspect: the oscillator does not require an external signal either to start or maintain energy conversion process (Fig. 1.1). It keeps producing an output signal so long as the dc power source is connected.
Moreover, the frequency of the output signal is determined by the passive components used in the oscillator and can be varied at will.
WAVE GENERATORS play a prominent role in the field of electronics.
They generate signals from a few hertz to several gigahertz. Modern wave generators use many different circuits and generate such outputs as SINUSOIDAL, SQUARE, RECTANGULAR, SAWTOOTH, and TRAPEZOIDAL wave-shapes. These wave-shapes serve many useful purposes in the electronic circuits you will be studying. For example, they are used extensively throughout the television receiver to reproduce both picture and sound.
One type of wave generator is known as an OSCILLATOR. An oscillator can be regarded as an amplifier which provides its own input signal. Oscillators are classified according to the wave-shapes they produce and the requirements needed for them to produce oscillations.
Classification of Oscillators
Electronic oscillator can be classified into two broad categories according to their output wave-shapes:
(i) Sinusoidal (or harmonic) oscillators—which produce an output having sine waveform
(ii) Non-sinusoidal (or relaxation) oscillators—they produce an output which has square, rectangular or saw-tooth waveform or is of pulse shape.
Sinusoidal Oscillators
A sinusoidal oscillator produces a sine-wave output signal. Ideally, the output signal is of constant amplitude with no variation in frequency.
Actually, something less than this is usually obtained. The degree to which the ideal is approached depends upon such factors as class of amplifier operation, amplifier characteristics, frequency stability, and amplitude stability.
Sinusoidal oscillators may be further subdivided into:
(a) Tuned-circuits or LC feedback oscillators such as Hartley, Colpitts and Clapp etc
(b) RC oscillators such as Wien-bridge oscillator
(c) Negative-resistance oscillators such as tunnel diode oscillator (d) Crystal oscillators such as Pierce oscillator
(e) Heterodyne or beat-frequency oscillator (BFO).
The active devices (bipolar, FETs or injunction transistors) in the above mentioned circuits may be biased class-A, B or C. Class-A operation is used in high-quality audio frequency oscillators. However, radio frequency oscillators are usually operated as class-C.
Sine-wave generators produce signals ranging from low audio frequencies to ultrahigh radio and microwave frequencies. Many low- frequency generators use resistors and capacitors to form their frequency-determining networks and are referred to as RC OSCILLATORS. They are widely used in the audio-frequency range.
Another type of sine-wave generator uses inductors and capacitors for its frequency-determining network. This type is known as the LC OSCILLATOR. LC oscillators, which use tank circuits, are commonly used for the higher radio frequencies. They are not suitable for use as extremely low-frequency oscillators because the inductors and capacitors would be large in size, heavy, and costly to manufacture.
A third type of sine-wave generator is the CRYSTAL-CONTROLLED OSCILLATOR.
The crystal-controlled oscillator provides excellent frequency stability and is used from the middle of the audio range through the radio frequency range.
Non-Sinusoidal Oscillators
Non-sinusoidal oscillators generate complex waveforms, such as square, rectangular, trigger, sawtooth, or trapezoidal. Because their outputs are generally characterized by a sudden change, or relaxation, they are often referred to as RELAXATION OSCILLATORS. The signal frequency of these oscillators is usually governed by the charge or discharge time of a capacitor in series with a resistor. Some types, however, contain inductors that affect the output frequency. Thus, like sinusoidal oscillators, both RC and LC networks are used for determining the frequency of oscillation. Within this category of non-sinusoidal oscillators are MULTIVIBRATORS, BLOCKING OSCILLATORS, SAWTOOTH GENERATORS, and TRAPEZOIDAL GENERATORS.
The Basic Oscillator
An oscillator can be thought of as an amplifier that provides itself (through feedback) with an input signal. By definition, it is a no rotating device for producing alternating current, the output frequency of which is determined by the characteristics of the device. The primary purpose of an oscillator is to generate a given waveform at a constant peak amplitude and specific frequency and to maintain this waveform within certain limits of amplitude and frequency.
An oscillator must provide amplification. Amplification of signal power occurs from input to output. In an oscillator, a portion of the output is fed back to sustain the input, as shown in figure 1.2.Enough power must be fed back to the input circuit for the oscillator to drive itself as does a signal generator. To cause the oscillator to be self-driven, the feedback signal must also be REGENERATIVE (positive). Regenerative signals must have enough power to compensate for circuit losses and to maintain oscillations.
Fig. 1.2: Basic oscillator block diagram.
Since a practical oscillator must oscillate at a predetermined frequency, a FREQUENCY-DETERMINING DEVICE ( ), sometimes referred to as a FREQUENCY-DETERMINING NETWORK ( ), is needed.
This device acts as a filter, allowing only the desired frequency to pass.
Without a frequency-determining device, the stage will oscillate in a random manner, and a constant frequency will not be maintained.
Before discussing oscillators further, let's review the requirements for an oscillator. First, amplification is required to provide the necessary gain for the signal. Second, sufficient regenerative feedback is required to sustain oscillations. Third, a frequency-determining device is needed to maintain the desired output frequency.
2.0 OBJECTIVES
At the end of this unit, you should be able to:
• understand the basic operations of an LC oscillator
• familiarize with the various types of oscillators that use the LC oscillatory circuit.
3.0 MAIN CONTENT 3.1 LC Oscillator
An LC circuit can store electrical energy vibrating at its resonant
plates, depending on the voltage across it, and an inductor stores energy in its magnetic field, depending on the current through it.
This oscillator consists of a capacitor and a coil connected in parallel. To understand how the LC oscillator basically works, let’s start off with the basics. Suppose a capacitor is charged by a battery. Once the capacitor is charged, one plate of the capacitor has more electrons than the other plate, thus it is charged. Now, when it is discharged through a wire, the electrons return to the positive plate, thus making the capacitor’s plates neutral, or discharged. However, this action works differently when you discharge a capacitor through a coil. When current is applied through a coil, a magnetic field is generated around the coil. This magnetic field generates a voltage across the coil that opposes the direction of electron flow. Because of this, the capacitor does not discharge right away. The smaller the coil, the faster the capacitor discharges. Now the interesting part happens. Once the capacitor is fully discharged through the coil, the magnetic field starts to collapse around the coil. The voltage induced from the collapsing magnetic field recharges the capacitor oppositely.
Then the capacitor begins to discharge through the coil again, generating a magnetic field. This process continues until the capacitor is completely discharged due to resistance.
Technically this basic LC circuit generates a sine wave that loses voltage in every cycle. To overcome this, additional voltage is applied to keep the oscillator from losing voltage. However, to keep this oscillator going well, a switching method is used. A vacuum tube (or a solid-state equivalent such as a FET) is used to keep this LC circuit oscillating. The advantage of using a vacuum tube is that they can oscillate at specified frequencies such as a thousand cycles per second.
3.2 Time Domain Solution
By Kirchhoff's voltage law, the voltage across the capacitor, VC, must equal the voltage across the inductor, :
From the constitutive relations for the circuit elements, we also know that
and
Rearranging and substituting gives the second order differential equation
The parameter ω, the radian frequency, can be defined as:
Using this can simplify the differential equation
The associated polynomial is , thus or
where is the imaginary unit.
Thus, the complete solution to the differential equation is
and can be solved for A and B by considering the initial conditions.
Since the exponential is complex, the solution represents a sinusoidal alternating current.
If the initial conditions are such that , then we can use Euler's formula to obtain a real sinusoid with amplitude 2A and angular frequency
Thus, the resulting solution becomes:
The initial conditions that would satisfy this result are:
and
3.3 Resonance Effect
The resonance effect occurs when inductive and capacitive reactance are equal in absolute value. (Notice that the LC circuit does not, by itself, resonate. The word resonance refers to a class of phenomena in which a small driving perturbation gives rise to a large effect in the system. The LC circuit must be driven, for example by an AC power supply, for resonance to occur (below).) The frequency at which this equality holds for the particular circuit is called the resonant frequency. The resonant frequency of the LC circuit is
Where L is the inductance in Henries, and C is the capacitance in Farads. The angular frequency has units of radians per second.
The equivalent frequency in units of hertz is
LC circuits are often used as filters; the L/C ratio is one of the factors that determines their "Q" and so selectivity. For a series resonant circuit with a given resistance, the higher the inductance and the lower the capacitance, the narrower the filter bandwidth. For a parallel resonant circuit the opposite applies. Positive feedback around the tuned circuit ("regeneration") can also increase selectivity.
Stagger tuning can provide an acceptably wide audio bandwidth, yet good selectivity.
3.3.1 Series LC Circuit
ResonanceHere L and C are in series in an ac circuit. Inductive reactance magnitude ( ) increases as frequency increases while capacitive reactance magnitude ( ) decreases with the increase in frequency. At a particular frequency these two reactances are equal in magnitude but opposite in sign. The frequency at which this happens is the resonant frequency ( ) for the given circuit.
Hence, at :
Converting angular frequency into hertz we get
Here f is the resonant frequency. Then rearranging,
In a series AC circuit, XC leads by 90 degrees while XL lags by 90.
Therefore, they cancel each other out. The only opposition to a current is coil resistance. Hence in series resonance the current is maximum at resonant frequency.
At , current is maximum. Circuit impedance is minimum. In this state a circuit is called an acceptor circuit.
• Below , . Hence circuit is capacitive.
• Above , . Hence circuit is inductive.
Impedance
First consider the impedance of the series LC circuit. The total impedance is given by the sum of the inductive and capacitive impedances:
Z = ZL + ZC
By writing the inductive impedance as ZL = jωL and capacitive
impedance as and substituting we have .
Writing this expression under a common denominator gives
.
The numerator implies that if ω2LC = 1 the total impedance Z will be zero and otherwise non-zero. Therefore the series LC circuit, when connected in series with a load, will act as a band-pass filter having zero impedance at the resonant frequency of the LC circuit.
3.3.2 Parallel LC Circuit
ResonanceHere a coil (L) and capacitor (C) are connected in parallel with an AC power supply. Let R be the internal resistance of the coil. When XL equals XC, the reactive branch currents are equal and opposite. Hence they cancel out each other to give minimum current in the main line.
Since total current is minimum, in this state the total impedance is maximum.
Resonant frequency given by: .
Note that any reactive branch current is not minimum at resonance, but each is given separately by dividing source voltage (V) by reactance (Z).
Hence I=V/Z, as per Ohm's law.
At fr, line current is minimum. Total impedance is maximum. In this state cct is called rejector circuit.
Below fr, circuit is inductive.
Above fr, circuit is capacitive.
Impedance
The same analysis may be applied to the parallel LC circuit. The total impedance is then given by:
and after substitution of ZL and ZC and simplification, gives
. Note that
but for all other values of ω2LC the impedance is finite (and therefore less than infinity). Hence the parallel LC circuit connected in series with a load will act as band-stop filter having infinite impedance at the resonant frequency of the LC circuit.
3.3.3 Applications of Resonance Effect
1. Most common application is tuning. For example, when we tune a radio to a particular station, the LC circuits are set at resonance for that particular carrier frequency.
2. A series resonant circuit provides voltage magnification.
3. A parallel resonant circuit provides current magnification.
4. A parallel resonant circuit can be used as load impedance in output circuits of RF amplifiers. Due to high impedance, the gain
of amplifier is maximum at resonant frequency
5. Both parallel and series resonant circuits are used in induction heating.
3.4 The LC OR Oscillatory Circuit
Fig. 3.1: The Oscillatory Circuit
As shown in Fig. 3.1 (a), suppose the capacitor has been fully-charged from a dc source.
Since S is open, it cannot discharge through L. Now, let us see what happens when S is closed.
When S is closed [Fig. 3.1 (b)] electrons move from plate A to plate B through coil L as shown by the arrow (or conventional current flows from B to A). This electron flow reduces the strength of the electric field and hence the amount of energy stored in it.
As electronic current starts flowing, the self-induced in the coil opposes the current flow. Hence, rate of discharge of electrons is somewhat slowed down.
Due to the flow of current, magnetic field is set up which stores the energy given out by the electric field [Fig. 3.1 (b)].
As plate A loses its electrons by discharge, the electron current has a tendency to die down and will actually reduce to zero when all excess electrons on Aare driven over to plate B so that both plates are reduced to the same potential. At that time, there is no electric field but the magnetic field has maximum value.
However, due to self-induction (or electrical inertia) of the coil, more electrons are transferred to plate B than are necessary to make up the electron deficiency there. It means that now plate B has more electrons than A. Hence, capacitor becomes charged again though in opposite direction as shown in Fig. 3.1 (c).
The magnetic field L collapses and the energy given out by it is stored in the electric field of the capacitor.
After this, the capacitor starts discharging in the opposite direction so that, now, the electrons move from plate B to plate A [Fig. 3.1 (d)]. The electric field starts collapsing whereas magnetic field starts building up again though in the opposite direction. Fig.3.1 (d) shows the condition when the capacitor becomes fully discharged once again.
However, these discharging electrons overshoot and again an excess amount of electrons flow to plate A, thereby charging the capacitor once more.
This sequence of charging and discharging continues. The to and fro motion of electrons between the two plates of the capacitor constitutes an oscillatory current.
It may be also noted that during this process, the electric energy of the capacitor is converted into magnetic energy of the coil and vice versa.
These oscillations of the capacitor discharge are damped because energy is dissipated away gradually so that their amplitude becomes zero after sometime. There are two reasons for the loss of the energy:
• Some energy is lost in the form of heat produced in the resistance of the coil and connecting wires
• and some energy is lost in the form of electromagnetic (EM) waves that are radiated out from the circuit through which an oscillatory current is passing.
3.4.1 Frequency of LC Circuit or Oscillatory Current
The frequency of time-period of the oscillatory current depends on two factors:
(a) Capacitance of the Capacitor
Larger the capacitor, greater the time required for the reversal of the discharge current i.e. lower its frequency.
(b) Self-inductance of the Coil
Larger the self-inductance, greater the internal effect and hence longer the time required by the current to stop flowing during discharge of the capacitor. The frequency of this oscillatory discharge current is given by
where L = self-inductance in and C = capacitance in
It may, however, be pointed out here that damped oscillations so produced are not good for radio transmission purpose because of their limited range and excessive distortion. For good radio transmission, we need un damped oscillations which can be produced if some additional energy is supplied in correct phase and correct direction to the LC circuit for making up the losses continually occurring in the circuit.
3.5 Frequency Stability of an Oscillator
The ability of an oscillator to maintain a constant frequency of oscillation is called its frequency stability. Following factors affect the frequency stability:
Operating Point of the Active Device
The Q-point of the active device (i.e. transistor) is so chosen as to confine the circuit operation on the linear portion of its characteristic.
Operation on non-linear portion varies the parameters of the transistor which, in turn, affects the frequency stability of the oscillator.
Inter-element Capacitances
Any changes in the inter-element capacitances of a transistor particularly the collector- to emitter capacitance cause changes in the oscillator output frequency, thus affecting its frequency stability.
The effect of changes in inter-element capacitances can be neutralized by adding a swamping capacitor across the offending elements—the added capacitance being made part of the tank circuit.
Power Supply
Changes in the dc operating voltages applied to the active device shift the oscillator frequency.
This problem can be avoided by using regulated power supply.
Temperature Variations
Variations in temperature cause changes in transistor parameters and also change the values of resistors, capacitors and inductors used in the circuit. Since such changes take place slowly, they cause a slow change (called drift) in the oscillator output frequency.
Output Load
A change in the output load may cause a change in the Q-factor of the LC tuned circuit thereby affecting the oscillator output frequency.
Mechanical Vibrations
Since such vibrations change the values of circuit elements, they result in changes of oscillator frequency. This instability factor can be eliminated by isolating the oscillator from the source of mechanical vibrations.
3.6 Essentials of a Feedback LC Oscillator
The essential components of a feedback LC oscillator are shown in Fig.
3.2 overleaf:
Fig. 3.2: Feedback LC Oscillator
• A resonator which consists of an LC circuit. It is also known as frequency-determining network (FDN)or tank circuit.
• An amplifier whose function is to amplify the oscillations produced by the resonator.
A positive feedback network (PFN) whose function is to transfer part of the output energy to the resonant LC circuit in proper phase. The amount of energy fed back is sufficient to meet losses in the LC circuit.
The essential condition for maintaining oscillations and for finding the value of frequency is
It means that
The feedback factor or loop gain ,
The net phase shift around the loop is 0° (or an integral multiple of 360°). In other words, feedback should be positive.
The above conditions form Barkhausen criterion for maintaining a steady level of oscillation at a specific frequency.
Majority of the oscillators used in radio receivers and transmitters use tuned circuits with positive feedback. Variations in oscillator circuits are due to the different way by which the feedback is applied. Some of the basic circuits are:
• Armstrong or Tickler or Tuned-base Oscillator — it employs inductive feedback from collector to the tuned LC circuit in the base of a transistor.
• Tuned Collector Oscillator—it also employs inductive coupling but the LC tuned circuit is in the collector circuit.
• Hartley Oscillator—Here feedback is supplied inductively.
• Colpitts Oscillator—Here feedback is supplied capacitively.
• Clapp Oscillator—it is a slight modification of the Colpitts oscillator.
3.7 Tuned Base Oscillator
Such an oscillator using a transistor in CE configuration is shown in Fig.
3.3. Resistors determine the dc bias of the circuit.