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Audio Pre-Amplifier Hard Limiting (Electrical & Audio Engineering) Solution

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Problem Overview

We are asked to evaluate the asymptotic steady-state limit of the composite output voltage function:

Vout(v)=5⋅tan⁡−1(ln⁡(1+ev))V_{\text{out}}(v) = 5 \cdot \tan^{-1}\left(\ln\left(1 + e^v\right)\right)
as the input voltage surges toward infinity (v→∞v \to \infty).

Step-by-Step Mathematical Solution

Step 1: Evaluate the innermost exponential boundary layer

Consider the innermost expression g(v)=1+evg(v) = 1 + e^v. As v→∞v \to \infty, the exponential term grows without bound:

lim⁡v→∞(1+ev)=∞\lim_{v \to \infty} \left(1 + e^v\right) = \infty

Step 2: Evaluate the intermediate natural logarithm layer

Let u=1+evu = 1 + e^v. Since u→∞u \to \infty, evaluate the limit of the natural logarithm as its argument grows infinitely large:

lim⁡u→∞ln⁡(u)=∞\lim_{u \to \infty} \ln(u) = \infty

Step 3: Evaluate the outer inverse tangent saturation layer

Let w=ln⁡(u)=ln⁡(1+ev)w = \ln(u) = \ln(1 + e^v). Since w→∞w \to \infty, evaluate the outer scaled arctangent function:

lim⁡w→∞5⋅tan⁡−1(w)\lim_{w \to \infty} 5 \cdot \tan^{-1}(w)

Recall the horizontal asymptote of the inverse tangent function: lim⁡w→∞tan⁡−1(w)=π2\lim_{w \to \infty} \tan^{-1}(w) = \frac{\pi}{2}. Multiplying by the amplitude scale factor 55:

lim⁡w→∞5⋅tan⁡−1(w)=5⋅(π2)=5π2≈7.854 Volts\lim_{w \to \infty} 5 \cdot \tan^{-1}(w) = 5 \cdot \left(\frac{\pi}{2}\right) = \frac{5\pi}{2} \approx 7.854\text{ Volts}

Engineering Conclusion

As the raw microphone input voltage surges infinitely high (v→∞v \to \infty), the composite limiting circuit safely saturates and clamps the output signal at exactly 5π2 V≈7.85 V\frac{5\pi}{2}\text{ V} \approx 7.85\text{ V}. This verified mathematical upper bound guarantees that downstream power amplifiers and loudspeaker components will not experience catastrophic electrical burnout.