Application of Maximum Power Transfer Method


Speaker System

One of the most common applications of the maximum power transfer theorem introduced in this chapter is to speaker systems. An audio amplifier (amplifier with a frequency range matching the typical range of the human ear) with an output impedance of $8Ω$ is shown in [Fig. 1(a)].
Fig. 1: Components of a speaker system: (a) amplifier; (b) speaker; (c) commercially available unit.
Impedance is a term applied to opposition in ac networks- for the moment think of it as a resistance level. We can also think of impedance as the internal resistance of the source which is normally shown in series with the source voltage as shown in the same figure. Every speaker has an internal resistance that can be represented as shown in [Fig. 1(b)] for a standard 8Ω speaker. [Fig. 1(c)] is a photograph of a commercially available 8- woofer (for very low frequencies). The primary purpose of the following discussion is to shed some light on how the audio power can be distributed and which approach would be the most effective.
Fig. 2: Speaker connections: (a) single unit; (b) in series; (c) in parallel.
Since the maximum power theorem states that the load impedance should match the source impedance for maximum power transfer, let us first consider the case of a single 8Ω speaker as shown in [Fig. 2(a)] with an applied amplifier voltage of $12 V$. Since the applied voltage will split equally, the speaker voltage is $6 V$, and the power to the speaker is a maximum value of
$$P = V^2/R =(6 V)^2 /8Ω = 4.5 W$$
If we have two $8Ω$ speakers that we would like to hook up, we have the choice of hooking them up in series or parallel. For the series configuration of [Fig. 2(b)], the resulting current would be
$$I = E/R =12 V/24Ω =500 mA$$
and the power to each speaker would be
$$P =I^2R = (500 mA)^2(8 Ω) = 2 W$$
which is a drop of over 50% from the maximum output level of $4.5 W$. If the speakers are hooked up in parallel as shown in [Fig. 2(c)], the total resistance of the parallel combination is $4Ω$, and the voltage across each speaker as determined by the voltage divider rule will be $4 V$. The power to each speaker is
$$P =V^2/R = (4 V)^2/8Ω = 2 W $$
which, interestingly enough, is the same power delivered to each speaker whether in series or parallel. However, the parallel arrangement is normally chosen for a variety of reasons.
First, when the speakers are connected in parallel, if a wire should become disconnected from one of the speakers due simply to the vibration caused by the emitted sound, the other speakers will still be operating perhaps not at maximum efficiency, but they will still be operating. If in series they would all fail to operate. A second reason relates to the general wiring procedure. When all of the speakers are in parallel, from various parts of a room all the red wires can be connected together and all the black wires together. If the speakers are in series, and if you are presented with a bundle of red and black wires in the basement, you would first have to determine which wires go with which speakers.
Speakers are also available with input impedances of $4Ω$ and $16Ω$. If you know that the output impedance is 8Ω , purchasing either two $4Ω$ speakers or two $16Ω$ speakers would result in maximum power to the speakers as shown in [Fig. 3]. The $16Ω$ speakers would be connected in parallel and the $4Ω$ speakers in series to establish a total load impedance of $8Ω$.
Fig. 3: Applying 4Ω and 16Ω speakers to an amplifier with an output impedance of 8 Ω.
In any case, always try to match the total resistance of the speaker load to the output resistance of the supply. Yes, a $4Ω$ speaker can be placed in series with a parallel combination of 8- speakers for maximum power transfer from the supply since the total resistance will be $8Ω$. However, the power distribution will not be equal, with the $4Ω$ speaker receiving $2.25 W$ and the $8Ω$ speakers each $1.125 W$ for a total of $4.5 W$.
The $4Ω$ speaker is therefore receiving twice the audio power of the $8Ω$ speakers, and this difference may cause distortion or imbalance in the listening area. All speakers have maximum and minimum levels. A $50W$ speaker is rated for a maximum output power of $50 W$ and will provide that level on demand. However, in order to function properly, it will probably need to be operating at least at the 1- to 5-W level. A 100W speaker typically needs between 5 W and 10 W of power to operate properly. It is also important to realize that power levels less than the rated value (such as 40 W for the 50-W speaker) will not result in an increase in distortion, but simply in a loss of volume. However, distortion will result if you exceed the rated power level.
For example, if we apply 2.5 W to a 2-W speaker, we will definitely have distortion. However, applying 1.5 W will simply result in less volume. A rule of thumb regarding audio levels states that the human ear can sense changes in audio level only if you double the applied power [a 3-dB increase; decibels (dB) will be introduced in upcoming chapter]. The doubling effect is always with respect to the initial level. For instance, if the original level were 2 W, you would have to go to 4 W to notice the change. If starting at 10 W, you would have to go to 20 W to appreciate the increase in volume. An exception to the above is at very low power levels or very high power levels. For instance, a change from 1 W to 1.5 W may be discernible, just as a change from 50 W to 80 W may be noticeable.

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