Sinewave Feedback L-C Oscillators - Sinewave Feedback L-C Oscillators

Popular sinewave feedback L-C tuned oscillators are listed below. These are principally used for generating RF oscillations.

Tuned Collector Oscillator
Tuned Based Oscillator
Hartley Oscillation
Colpitts Oscillator
Clapp Oscillator
Crystal Oscillator

Tuned Collector Oscillator

Figure 1 gives the basic circuit. A parallel tuned L-C circuit placed in the collector circuit constitutes the load impedance and determines the frequency of oscillation. The output voltage developed across the tuned circuit is inductively coupled to the basic circuit through the coil L₂. The winding direction of the two coils L₁ and L₂ are so chosen that positive feedback takes place from the collector circuit to the base circuit. The dots are placed on coils L₁ and L₂. When the dotted end of L₁ is positive, the dotted end of L₃ also become positive.

The tuned circuit in the collector circuit resonates at frequency $\omega_0 = \dfrac{1}{\sqrt{L_1C_1}}$ . At this frequency, the impedance of the tuned circuit is purely resistive and large. The ac voltage drop across the inductor L₁ from collector to ground is 180⁰ out of phase with the voltage from base to ground. The transformer winding directions are so chosen as to produce a phase shift of another 180⁰ assuming the secondary to be unloaded. Then the total loop phase shift is exactly 360⁰ or 0⁰ and oscillations are produced at the frequency of resonance $\omega_0$ . However, the secondary is always loaded to some extent. Hence the frequency of oscillation i.e. the frequency at which Barkhausen criterion is satisfied differs from the frequency of resonance of the tuned circuit.

Analysis using small signal equivalent circuit for the transistor and the Barkhausen criterion $(-A\beta = 1)$ yields the following results:

Condition for sustained oscillations:

$\dfrac{h_{fc}}{h_{ic}} = \dfrac{RC_1 + h_{oc}L_1}{M}$ …………………..(1)

Where M is the mutual inductance between coils L₁ and L₂ and R is the load resistance of the secondary coil L₂ reflected to the primary coil L₁.

Resistances of the coils L₁ and L₂ have been neglected.

Frequency of oscillation:

$\omega = \omega_0\sqrt{1 + h_{oc}R}$ ……………..(2)

Where $\omega_0$ is the frequency of resonance of L₁ and L₂ and equals $\dfrac{1}{\sqrt{L_1C_1}}$ .

From Eq. (2), we conclude that in tuned collector oscillator, the frequency of oscillation $\omega$ exceeds the frequency of resonance $\omega_0$ of the tuned circuit.

Tuned Based Oscillator

Figure 2 gives the basic circuit. A parallel tuned L-C circuit placed in the base-to-ground circuit determines the frequency of oscillation. The output developed across the tuned circuit is inductively coupled to the collector circuit through the coil L₂. The winding direction of the coils L₁ and L₂ are so selected that positive feedback takes place from the collector circuit to the base circuit resulting in oscillations. The tuned circuit in the base circuit resonates at frequency $\omega_0 = \dfrac{1}{\sqrt{L_1C_1}}$ . At this frequency, the impedance of the tuned circuit is purely resistive and large.

The voltage drop across the inductor L₂ is 180⁰ out of phase with the base to ground voltage. The transformer winding direction are so chosen as to produce phase shift of another 180⁰ assuming the oscillator to be non-loaded. Then the total phase shift is exactly 360⁰ or zero degree. However, the secondary is always loaded to some extent. Hence the frequency of oscillation differs slightly from the frequency of resonance of the tuned circuit.

Hartley Oscillator

Figure 3 gives thew circuit of Hartley oscillator using BJT. The collector supply voltage Vcc is applied to the collector through inductor L whose reactance is high compared with that of L₂ and may, therefore, be neglected. Capacitor C_b has a low reactance at the frequency of oscillation and may also be neglected in the equivalent circuit. At dc i.e. zero frequency, C_b, therefore, prevents collector supply Vcc from getting short circuited by L in series with L₂. The parallel combination of R_e and C_z in conjunction with= R₁ – R₂ combination provides stabilized self-bias. The circuit operates as class C amplifier with infinity gain. The tuned circuit constituted by L₁ – L₂ – C determines the frequency of resonance. Since the tuned circuit extends right from the input circuit, the feedback needed for sustained oscillation takes place through the tuned circuit itself.

Result of Analysis: Analysis making use of the equivalent circuit and the Barkhausen criterion yields the following results:

Condition for sustained oscillations $h_{cf}\approx \dfrac{L_1}{L_2}$ ………(3)

If there exists mutual inductance M between coils L₁ and L₂ then the condition for sustained oscillation is,

$h_{cf} = \dfrac{L_1 + M}{L_2 +M}$ …….(4)

Frequency of oscillation,

$\omega = \omega_2\sqrt{1-\dfrac{\omega_0^2 L_1L_2}{h_{ic}h_{ic}}} = \omega_0\sqrt{1-\dfrac{X_1X_2}{h_{ic}h_{oc}}}$ ….(5)

Where $X_1 = \omega L_1$ and $X_2 = \omega L_2$ ,

$\omega_0 = \dfrac{1}{\sqrt{(L_1+L_2)C}}$ …..(6)

From Eq. (5) we conclude that in Hartley oscillator, the frequency of oscillation is slightly smaller than the frequency of series resonance of L₁, L₂ and C.

Colpitts Oscillator

Figure 4 gives the basic circuit. The collector supply voltage Vcc is applied to the collector through the inductor L_c whose reactance is high compared with the reactance of capacitor C₂. Hence L_c may be omitted from the equivalent circuit. Parallel combination of R_c and C_z along the resistor R₁ – R₂ provides the stabilized self-bias. The tuned circuit consists of capacitor C₁ and C₂ and inductor L. It extends from the collector circuit to the base circuit and basically determines the frequency of oscillation. The feedback is through the tank circuit itself.

Result of Analysis: Analysis making use of the small signal h-parameter model and the Barkhausen criterion yields the following results:

Condition for sustained oscillation $h_{oc} = \dfrac{C_2}{C_1}$ …..(7)
Frequency of oscillation $\omega = \dfrac{\omega_0}{\sqrt{1-\dfrac{1}{\omega_0^2C_1C_2h_{ic}h_{oe}}}} = \dfrac{\omega_0}{1-\dfrac{X_1X_2}{h_{ic}{h_oe}}}$ …….(8)

Where $X_1 = \dfrac{1}{\omega_0C_1}$ and $= \dfrac{1}{\omega_0C_2}$ and $\omega_0$ is the frequency of resonance of C₁, C₂ and L and is given by

$\omega_0 = \sqrt{\dfrac{1}{L}(\dfrac{1}{C_1}+\dfrac{1}{C_2})}$ …(9)

From Eq. (8) we conclude that in the case of Colpitts oscillator, the frequency of oscillation $\omega_0$ is slightly greater than the frequency of series resonance of C₁, C₂ and L.

Clapp Oscillator

Figure 5 gives the basic circuit of Clapp oscillator. It is modified version of Colpitts oscillator. The main difference between the two circuits lies in the introduction of another capacitance C₃ in series with the coil L.

The capacitance C₁ is made so large as to reduce the effect of collector capacitance variation due to collector voltage change. The emitter junction capacitance depends on the emitter current. In the circuit of Fig 5, resistor R₂ in the biasing network may be in the form of a thermistor so that the emitter current gets reduced at high temperatures.

The frequency of oscillation is given by,

$\omega = \dfrac{1}{\sqrt{L(\dfrac{C_1C_2C_3}{C_1C_2+C_2C_3+C_1C_3})}}$ ……(10)

Capacitors C₁ and C₂ are kept fixed while capacitor C₃ is used for tuning purpose.

Further $C_3 <<\dfrac{C_1C_2}{C_1+C_2}$

In that case, the frequency of oscillation is approximately given by,

$\omega = \dfrac{1}{\sqrt{LC_3}}$ ……(11)

Thus, the frequency of oscillation $\omega$ is nothing but the frequency of series resonance of L and C₃.

This condition for sustained oscillation is, $h_{fc}\geq \dfrac{C_2}{C_1}$ …..(12)

High frequency stability is obtained by (i) enclosing the entire circuit in a constant temperature chamber and (ii) by maintaining the operating voltage constant with the help of a zener diode.

Crystal Oscillator

Piezo-electric Quartz Crystal

For excellent stability of frequency of oscillation in the RF range, we may use a piezo-electric quartz crystal in place of the L-C tuned circuit in the oscillator. Fig 6 gives the electrical equivalent circuit of a piezo-electric quartz crystal.

In Fig 6, element R₁ L₂ C₁ from a series resonance circuit resonating at frequency f_s. The impedance of this series circuit at resonance is restive and equal to R₁. Complete circuit including C₂ along with R₁ L₂ C₁ forms a parallel tuned circuit resonating at frequency f_p. under parallel resonance condition at f_p, the impedance of the complete tuned circuit is again resistive but extremely large. Frequency f_p is greater than f_s but extremely close to it. When piezo-electric quartz crystal is used in place of conventional L-C tuned circuit in an oscillator, the frequency of oscillation is determined by the value of elements R₁ L₁ C₁ and C₂ but this frequency must lie in the frequency range from f_s to f_p. Since f_s and f_p are extremely close, an extremely high order of stability of frequency of oscillation is obtainable in a crystal oscillator i.e. an oscillator using quartz crystal.

Crystal Oscillator Circuit

A huge variety of crystal oscillator circuits are possible depending on (i) which active device is used, BJT or FET and (ii) part of the circuit where crystal is placed.

Tuned Gate-Tuned Drain Crystal Oscillator

Figure 1 shows an FET crystal oscillator of tuned gate-tuned drain type. Here drain-to-gate capacitance C_dg forms the feedback path from drain (output) to the gate (input). Stray wiring capacitance C_s also gets added to this circuit. For sustained oscillation, it is necessary that the L-C tuned circuit in the drain circuit and the quartz crystal equivalent crystal equivalent circuit in the gate circuit be inductive. Hence this circuit will oscillate at a frequency which lies between the frequencies f_s and f_p of the crystal. Hence high stability of frequency of oscillation results.

Colpitts Crystal Oscillator using BJT

Figure 2 shows crystal oscillator using Colpitts circuit using BJT. Here crystal is placed in place of inductor L of Colpitts tuned circuit. in this case also, the frequency od oscillation lies in the range determined by frequencies f_s and f_p of the quartz crystal.