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Impedance matching is a key concept in electrical engineering, particularly when dealing with the transmission of signals or power. In simple terms, it refers to making the impedance (opposition to current flow) of two circuits equal to ensure that power is transferred efficiently between them. This is important when you have a source like an amplifier and a load like a loudspeaker. If the impedances are not matched, only some of the power gets to the loudspeaker, and the rest is wasted, primarily as heat.

This is why transformers are often used: they allow engineers to adjust the impedance levels, so they match perfectly. By using the correct turns ratio, power loss is minimized, and the device operates more efficiently. This principle of impedance matching plays a crucial role in various fields, including audio equipment, telecommunications, and especially when dealing with sensitive medical devices like an electrocardiogram amplifier.

An electrocardiogram (ECG or EKG) amplifier is an essential component in medical diagnostics, particularly for monitoring heart activity. ECG amplifiers must be highly sensitive, as they pick up electrical signals from the heart which are very small compared to electrical noise from other sources.

The main role of an ECG amplifier is to amplify these weak signals for further processing, display, or storage. In our problem, the ECG amplifier has an output impedance of 45 kΩ. This high impedance is typical for such amplifiers, which need to maintain fidelity while amplifying small signals.

However, when you connect an ECG amplifier to a loudspeaker or other output device with significantly different impedance, you often need a transformer to ensure the amplifier output is correctly delivered to the loudspeaker, maintaining signal strength and quality.

The Maximum Power Transfer Theorem is a fundamental principle in electrical circuits, stating that maximum power is transferred to the load when the load resistance equals the source resistance—this includes both real and imaginary parts in AC circuits. When both the source and load are not matched, power delivered to the load is less than the maximum possible.

Applying this theorem in the context of our problem, we use impedance matching via a transformer to adjust the load impedance (in this case, the loudspeaker's 8 Ω) to be equal to the source impedance (the amplifier's 45 kΩ). By setting the turns ratio of the transformer, we effectively "transform" the loudspeaker’s impedance to be seen by the amplifier as 45 kΩ, achieving maximum power transfer.

In mathematical terms, the turns ratio formula was given as \( n^2 = \frac{Z_{primary}}{Z_{secondary}} \). By plugging in our values, we calculated \( n = 75 \), indicating that the primary winding of the transformer needed 75 times more turns than the secondary to achieve the impedance match necessary for maximum power transfer.