More about Open Loop Gain

In Figure 3, the loop is open at the point just before where just the sound from the loudspeaker is added back to the microphone. The locations of the input and the output of this open loop condition are as shown in Figure 4.

Although opening the loop at this point is almost impossible to implement, opening the loop enables the study of this open loop characteristics. The ratio of the output over the input across the frequency range of interest defines this open loop function. Understanding this open loop function is essential in predicting the behavior of the system when the loop is closed in actual application.

In actual close loop application, as to how the input to the microphone adds up upon itself after going through the loop is dependent on both the magnitude and phase of the open loop function. In earlier illustrations and explanations, phase has been intentionally ignored for simplicity. The importance of phase when adding two sine waves will be discussed in the next section.

Addition of 2 Sine Waves of Same Frequency but Different Amplitude and Phase

The output of the open loop function can have different magnitude and phase from the input. When the loop is closed as in actual application, the output will be added to the input. Hence, it is important to understand in general how two signals at the same frequency but with different magnitudes and phase add up.

Consider two different signals at the same frequency but with different magnitudes and a phase angle between them. For illustration purpose, let Signal A and Signal B have amplitude of 1 and 0.5 respectively and with a phase angle of 45° between them. The equivalent time diagram and the resultant addition of these two sine waves are shown in Figure 5. The resultant addition can be obtained by adding the respective amplitudes of Signal A and Signal B at each small time interval.

It is apparent that the phase between the 2 signals affects the resultant addition of the two signals. If the phase is zero, then the two sine waves are lined up and the resultant magnitude is simply the linear addition of the magnitudes of the two sine waves. As in Figure 5, if there is a phase angle between the two sine waves, then the addition is more complex.

Instead of performing mathematical computation, addition of two sine waves can be visualized by using vectors. As shown in Figure 6, a vector can represent the position of a rotating wheel with a constant rotating speed but frozen in time. A resulting sine wave is obtained when the vertical amplitude of the vector is plotted out against the phase angle as shown in the graph at the right portion of Figure 6.

A sinusoidal signal at a fixed frequency can thus be represented by a single vector. Two signals at the same frequency but with different magnitude and phase can then be represented by two vectors as shown in Figure 7a. The length of the vector represents the magnitude and the phase between the two signals is represented by the angle ɸ between the two vectors.

For the addition of two sine waves, since each vector represent the entire respective sine wave, the addition can be visualized by lining up the vector of one signal (e.g. Signal B) on top of the other signal (e.g. Signal A). This is shown in Figure 7b which demonstrates the vectorial addition of the two individual vectors. Hence, this is a convenient shortcut way of visualizing the addition of 2 sine waves of the same frequency.

If the two signals are in phase i.e. ɸ = 0, the two vectors overlap each other and the summation of these two signals is the linear addition of the respective amplitudes of the two signals. If the phase angle is 180°, the vectors are in direct opposite direction and the summation resultant will be a linear subtraction between the two amplitudes.

Mathematically, if Signal A has a magnitude of “a” and Signal B has a magnitude of “b”, and there is a phase angle of ɸ between these 2 signals, the magnitude of the summation can be computed as follows: Summed magnitude = (a2 + b2 + 2ab.cosɸ)1/2

This approach in calculating the summation of two sine waves of the same frequency is useful in analysing the summation of acoustical or voltage signals where it is often encountered in audio systems. Examples include the summation of sound sources such as in-between two loudspeaker in an array, combination of direct and reflected sound of a loudspeaker or subwoofer placed next to a wall boundary and the determination of stability of a sound system with given open loop magnitude and phase response.

Part 2 of this article will deal with an interesting methodology in estimating the acoustic feedback margin of a typical sound reinforcement system.