# Theory of Sinusoidal Oscillation | Loop Gain and Phase

The article discusses the theory and principles of sinusoidal oscillation, focusing on the necessity of positive feedback and loop gain in building oscillators. It explains how the feedback signal sustains oscillations, the role of thermal noise in initiating these oscillations, and the mechanisms that adjust loop gain to stabilize the output.

To construct a sinusoidal oscillator, an amplifier with positive feedback is required. The concept involves using the feedback signal as a substitute for the input signal. If the feedback signal is sufficiently large and has the correct phase, an output signal will be generated even without an external input signal.

Good To Know

In most oscillators, the feedback voltage is a fractional part of the output voltage. When this is the case, the voltage gain AV must be large enough to ensure that AVB = 1. In other words, the amplifier voltage gain must at least be large enough to overcome the losses in the feedback network. However, if an emitter follower is used as the amplifier, the feedback network must provide a slight amount of gain to ensure that AVB = 1. For example, if the voltage gain AV of an emitter follower equals 0.9, then B must equal 1/0.9 or 1.11. RF communication circuits sometimes use oscillators that contain an emitter follower for the amplifier.

## Loop Gain and Phase

Figure 1a shows an ac voltage source driving the input terminals of an amplifier. The amplified output voltage is:

$$v_{out}=A_v\left(v_{out}\right)$$

This voltage drives a feedback circuit that is usually a resonant circuit. Because of this, we get maximum feedback at one frequency. In Figure 1a, the feedback voltage returning to point x is given by:

$$v_f=A_vB\left(v_{in}\right)$$

where B is the feedback fraction.

(a)

(b)

(c)

(d)

(d)

Figure 1. (a) Feedback voltage returns to point x; (b) connecting points x and y; (c) sinusoidal oscillations die out; (d) sinusoidal oscillations increase; (e) sinusoidal oscillations are fixed in amplitude.

If the phase shift through the amplifier and feedback circuit is equivalent to 0°, AVB(vin) is in phase with vin.

Suppose we connect point x to point y and simultaneously remove voltage source vin, then the feedback voltage AVB(vin) drives the input of the amplifier, as shown in Figure 1b.

What happens to the output voltage? If AVB is less than 1, AVB(vin) is less than vin and the output signal will die out, as shown in Figure 1c. However, if AVB is greater than 1, AVB(vin) is greater than vin and the output voltage builds up (Figure 1d). If AVB equals 1, then AVB(vin) equals vin and the output voltage is a steady sine wave like the one in Figure 1e. In this case, the circuit supplies its own input signal.

In any oscillator, the loop gain AVB is greater than 1 when the power is first turned on. A small starting voltage is applied to the input terminals, and the output voltage builds up, as shown in Figure 1d. After the output voltage reaches a certain level, AVB automatically decreases to 1, and the peak-to-peak output becomes constant (Figure 1e).

## Starting Voltage Is Thermal Noise

Where does the starting voltage originate? Every resistor has some free electrons that, due to ambient temperature, move randomly in various directions, generating a noise voltage across the resistor. This motion is so random that it encompasses frequencies exceeding 1000 GHz. Essentially, each resistor can be viewed as a small AC voltage source producing a wide range of frequencies.

In Figure 1b, here’s what happens: When the power is first turned on, the only signals present in the system are the noise voltages generated by the resistors. These noise voltages are amplified and appear at the output terminals. The amplified noise, containing all frequencies, drives the resonant feedback circuit. By deliberate design, the loop gain is made greater than 1 and the loop phase shift is set to 0° at the resonant frequency. Above and below the resonant frequency, the phase shift deviates from 0°, causing oscillations to build up only at the resonant frequency of the feedback circuit.

## AVB Decreases to Unity

There are two ways in which AVB can decrease to 1: either AV can decrease, or B can decrease. In some oscillators, the signal builds up until clipping occurs due to saturation and cut-off, effectively reducing the voltage gain AV. In other oscillators, the signal builds up and causes B to decrease before clipping occurs. In both cases, the product AVB decreases until it equals 1.

Here are the key ideas behind any feedback oscillator:

1. Initially, loop gain AVB is greater than 1 at the frequency where the loop phase shift is 0°.
2. Once the desired output level is reached, AVB must decrease to 1 by reducing either AV or B.

### Sinusoidal Oscillation Key Takeaways

Understanding the principles of sinusoidal oscillation, such as positive feedback, loop gain, and the role of thermal noise, is crucial for designing and implementing oscillators in various applications. These oscillators are fundamental components in many electronic systems, including signal generators, communication devices, and clocks, where precise and stable waveforms are essential.