Question 22.6: Impedance matching transformers are also quite evident in pu...
Impedance matching transformers are also quite evident in public address systems, such as the one appearing in the 70.7 V system in Fig. 22.10. Although the system has only one set of output terminals, up to four speakers can be connected to this system (the number is a function of the chosen system). Each 8Ω speaker is connected to the 70.7 V line through a 10 W audio-matching transformer (defining the frequency range of linear operation).
a. If each speaker in Fig. 22.10 can receive 10 W of power, what is the maximum power drain on the source?
b. For each speaker, determine the impedance seen at the input side of the transformer if each is operating under its full 10 W of power.
c. Determine the turns ratio of the transformers.
d. At 10 W, what are the speaker voltage and current?
e. What is the load seen by the source with one, two, three, or four speakers connected?
a. Ideally, the primary power equals the power delivered to the load, resulting in a maximum of 40 W from the supply.
b. The power at the primary:
P_{p}=V_{p} I_{p}=(70.7 V ) I_{p}=10 W.
and I_{p}=\frac{10 W }{70.7 V }=141.4 mA.
so that Z_{p}=\frac{V_{p}}{I_{p}}=\frac{70.7 V }{141.4 mA }= 500 \Omega.
\text { c. } Z_{p}=a^{2} Z_{L} \Rightarrow a=\sqrt{\frac{Z_{p}}{Z_{L}}}=\sqrt{\frac{500 \Omega}{8 \Omega}}=\sqrt{62.5}=7.91 \cong 8: 1.
\text { d. } V_{s}=V_{L}=\frac{V_{p}}{a}=\frac{70.7 V }{7.91}=8.94 V \cong 9 V.
e. All the speakers are in parallel. Therefore,
One speaker: R_{T}=500 \Omega.
Two speakers: R_{T}=\frac{500 \Omega}{2}=250 \Omega.
Three speakers: R_{T}=\frac{500 \Omega}{3}=167 \Omega.
Four speakers: R_{T}=\frac{500 \Omega}{4}=125 \Omega.
Even though the load seen by the source varies with the number of speakers connected, the source impedance is so low (compared to the lowest load of 125 ) that the terminal voltage of 70.7 V is essentially constant. This is not the case where the desired result is to match the load to the input impedance; rather, it was to ensure 70.7 V at each primary, no matter how many speakers were connected, and to limit the current drawn from the supply.