**Indroduction**

An LC oscillator is actually a feedback oscillator which uses capacitors and inductors in its feedback network. It can be built from a transistor, an operational amplifier, a tube, or some other active (amplifying) device. Oscillation is brought about by applying a portion of the amplifiers’ output signal to its input. That feedback signal must be applied in phase with the original input signal. The amplifier is usually an inverter that provides 180^{o} of phase shift by itself, and an additional 180^{o} of phase shift must be provided through some other means.

In an LC oscillator circuit, the feedback network is a tuned circuit (often called a tank circuit). The tuned circuit is a resonator consisting of an inductor (L) and a capacitor (C) connected together. Charge flows back and forth between the capacitor's plates through the inductor, so the tuned circuit can store electrical energy oscillating at its resonant frequency. There are small losses in the tank circuit, but the amplifier compensates for those losses and supplies the power for the output signal. LC oscillators are often used at radio frequencies, when a tunable frequency source is necessary, such as in signal generators, tunable radio transmitters and the local oscillators in radio receivers. Typical LC oscillator circuits are the Hartley, Colpitts and Clapp circuits.

**Topology**

In an LC oscillator, the feedback network is mainly composed of three impedances (see figure 1). All three impedances Z1, Z2 and Z3 are pure reactance elements, implying that they can be either capacitive or inductive, depending on our design needs. In figure 1, for convenience, we have shown the amplifying element as an operational amp, but other devices could be also used.

By looking at figure 1, you’ll see that the feedback factor (the voltage gain of the feedback network), B(f), can be stated in a form that resembles the equation describing a resistive voltage divider.

Commonly Z1 and Z2 are the same type of reactance, and Z3 is of the “opposite” type. For example, if Z1 and Z2 were capacitors, then Z3 would be an inductor; if Z1 and Z2 were inductors, then Z3 would be a capacitor.

At this point, it is a good idea to remind that the necessary conditions for oscillation are:

and

Where A(f), B(f) are the amplifier’s gain and the feedback gain, respectively. In general, both the amplifier and the feedback network will alter the magnitude and the phase of the signal. So, both A and B are complex values, and each value has magnitude and argument (phase).

**The Colpitts oscillator**

A feedback oscillator which uses capacitors C1 and C2, for Z1 and Z2, and an inductor L1, for Z3, is known as a “Colpitts” oscillator (see figure 2). The Colpitts oscillator was invented in 1918 by the American engineer Edwin H. Colpitts. The Colpitts circuit, like other LC oscillators, consists of a gain device (such as a bipolar junction transistor, field effect transistor, operational amplifier, or vacuum tube) with its output connected to its input in a feedback loop containing a parallel LC circuit (tuned circuit) which functions as a bandpass filter to set the frequency of oscillation.

The frequency of oscillation is actually the resonance frequency. At resonance, total capacitive and total inductive reactances, are canceled out:

(SQR is the square root)

L is the inductance of L1, in Henrys, and C is the equivalent capacitance of elements C1 and C2, in Farads. The term C, is equivalent to the series combination of C1 and C2 in the standard manner:

The mathematical expression for B(f), can be derived from (3): B(f)=Z1/(Z1+Z3)

At resonance (see equation 4):

So, B(f)=-Z1/Z2 at resonance. Thus, the mathematical expression for B(fo) is given by:

Equation (7) states that, at resonance, the feedback factor is a purely negative number, which means that the feedback network provides 180^{o} (π rad) of phase shift by itself. By using an inverting amplifier (which also provides 180^{o} of phase shift), there is no doubt that the condition stated by equation (3), is well satisfied.

Now, let’s examine if the condition stated by (2) is also satisfied:

In our example, the amplifier is an inverting amplifier, based on an op-amp. So, its voltage gain is given by:

By combining (2), (7) and (8), we have:

Equation (9), states that the circuit will oscillate, if the gain of the amplifier is equal or greater than C1/C2. So C1/C2 is the minimum gain required. In practice, the C1/C2 ratio is a trade-off between amplitude and stability. By choosing the minimum gain, the total loop gain will be equal to 1 and the amplitude of the oscillations will remain constant. However, there are inevitable tolerances. So, in practice, we must choose a much greater gain, to ensure that oscillations will always start-up. At a much greater gain, the amplitude of the oscillations will rapidly grow up with time, until the amplifier will be saturated and it will start clipping.

**The Hartley oscillator**

The Hartley oscillator was invented in 1915 by American engineer Ralph Hartley. The distinguishing feature of the Hartley oscillator is that the tuned circuit consists of a single capacitor in parallel with two inductors in series (or a single tapped inductor), and the feedback signal needed for oscillation is taken from the center connection of the two inductors.

The classic Hartley oscillator is shown in Figure 3. The principal difference between the Hartley and the Colpitts oscillators is the type of reactance elements used for Z1, Z2, and Z3. In the Colpitts oscillator, Z1 and Z2 are capacitive, but in the Hartley version they’re inductive. A Hartley oscillator may use a tapped inductor, but is still a Hartley oscillator. Further, Z3 is inductive in the Colpitts, but capacitive in the Hartley. The circuit of figure 3 will oscillate (see equation 4) at a frequency, f_{o}, given by:

_{o}=1/2πSQR[(L1+L2)C1]

Like the Colpitts oscillator, the nominal gain of the Hartley oscillator is R2/R1, and the minimum gain is L1/L2.

**The Clapp oscillator**

The Clapp oscillator is actually a modified version of the Colpitts oscillator. The Clapp oscillator is a Colpitts oscillator that has an additional capacitor placed in series with the inductor. It was published by James Kilton Clapp in 1948.

Referring to the notional circuit in figure 4, the network comprises a single inductor and three capacitors. Capacitors C1 and C2 form a voltage divider that determines the amount of feedback voltage applied to the amplifier imput. The oscillation frequency can be found from (4), and it is:

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 capacitor (C_{0}) in series with the inductor.

Like the Colpits oscillator, the nominal gain of the Clapp oscillator is R2/R1. And the minimum gain is C1/C2.

**Practical circuits**

A discrete-transistor implementation of the Colpitts oscillator is shown in Figure 5a. The split-capacitor voltage divider that provides feedback consists of C2 and C3. In general, C3 has a larger value than C2. Resistor R3 biases the transistor, and R1 provides stability. The output signal is developed across R1. The frequency of oscillation is determined by the resonant tank circuit Z, which is connected to the base of the transitor via C4 as shown.

The tank circuit is composed of a coil and a capacitor, which may be connected in series, as shown in Figure 5b, or in parallel as shown in Figure 5c. The series circuit is a Clapp oscillator.

A Hartley oscillator may be built as shown in Figure 6. Again, a bipolar transistor is the gain-producing element. Resistors R1 and R2 bias Q1, and serve no other purpose. The oscillating frequency is determined by the resonant tank circuit that is made up of L1 and C1, and the output signal can be taken either from the emitter of the transitor via C5, or from coil L2, which is coupled to L1. The position of L1’s tap is a trade-off between the stability and the amplitude of the output signal.

Both Hartley and Colpitts oscillators are common-collector circuits, because the collector of the transistor is at AC ground by virtue of the presence of a bypass capacitor – that capacitor is C5 in figure 5a, and C4 in Fihure 6.

**The Armstrong oscillator**

There are many additional types of LC oscillators, but two are particularly interesting, historically speaking. The Armstrong oscillator is named after Major Edwin Armstrong who invented the regenerative detector, the super-heterodyne radio, frequency modulation, and other things. A practical circuit of an Armstrong oscillator is shown on figure 7. That circuit uses a FET as the amplifying element; the original Armstrong circuit used a tube. Another popular name of the Armstrong oscillator is the “tickler oscillator”, we’ll see why in a moment.

The Armstrong circuit’s frequency of oscillation is determined by the values of the components in the parallel-resonant tank circuit composed of C1 and L1. A feedback coil or “tickler” coil, L2, is closely coupled with L1, and the tickler serves to feed part of the output signal back to the input. Care must be taken to be sure that the mutual inductance between L2 and L1 is of the proper “polarity”. Otherwise, feedback will occur with the wrong pahse, and the circuit won’t oscillate.

The amount of feedback, or regeneration, is controlled by a variable resistor R3; that resistor controls the amount of current flowing in the tickler coil. In older circuits, the tickler coil was wound on a mechanism that permitted the position of the coil to be varied with respect to L1.When coupling was maximum, so was feedback. The frequency of oscillation of the Armstrong oscillator may be calculated by plugging the values of L1 and C1 into the resonance equation (4), that we indroduced earlier in this article.

**The TITO oscillator **

Our final LC oscillator has been called by several different names in the past. In older texts that illustrate the use of vacuum-tube circuits, the circuit is called a Tuned-Grid-Tuned-plate (TGTP) oscillator. More recently it has been called a Tuned-Base-Tuned-Collector (TBTC) oscillator. Perhaps now that we have so many types of amplifying devices, the circuit should be called a Tuned-Input-Tuned-output (TITO) oscillator.

Whatever we call it, it looks something like the circuit shown in figure 8.The frequency of oscillation of the circuit is set by both the L1/C1 and L2/C2 tank circuits. Most textbooks recommend that the two tank circuits resonate at almost equal but slightly different frequencies.

Feedback in the TITO oscillator is brought about through inter-element capacitance in the transistor. In other words, Cbe (the capacitance between base and the emitter) and Ccb (the capacitance between collector and base) are not external capacitors; they are part of the structure of the transistor. That is why they have been shown in dashed lined in the illustration.

The TITO oscillator is rarely used these days. That sort of circuit may appear to have no capacitors and only inductors but don’t be fooled. The designers of the circuit indented for stray device and circuit capacitance to be sufficient to produce oscillation at the desired frequency.

**Conclusion **

We’ve covered quite bit of basics in this article. We have covered the theoretical background of LC oscillators. Actually, the theoretical background of LC oscillators is based on the general concepts of the Feedback Oscillator. Then, we went to look at Colpitts, Hartley, and Clapp oscillators. After examining them theoretically, we discussed several practical circuit implementations. Finally we discussed several historically interesting circuits: The Armstrong oscillator and the Tuned-Input-Tuned-output (TITO) oscillator. When we continue in a future article, we’ll discuss RC oscillators that may be built from op-amps.