The asymptotic magnitude Bode plot of a linear, minimum-phase dynamic system has the...

For a minimum-phase system, zeros create +20 dB/decade slope increases and poles create -20 dB/decade slope decreases on the Bode plot. A +40 dB/decade slope change at ω₁ = 1 rad/s indicates two zeros at s = -1; a -40 dB/decade change at ω₂ = 100 rad/s indicates two poles at s = -100. The DC gain of 0 dB with final magnitude 80 dB confirms the gain factor 10⁴. Poles must have negative real parts (left half-plane) for stability in a causal, minimum-phase system.

Step-by-step Derivation:
Step 1: Interpret slope changes.

• Slope = 0 dB/decade below 1 rad/s → 0 poles and 0 zeros in the essential part.
• Slope increases by +40 dB/decade at ω₁ = 1 rad/s → 2 zeros at s = -1 (each contributes +20 dB/decade).
• Slope decreases by -40 dB/decade at ω₂ = 100 rad/s → 2 poles at s = -100 (each contributes -20 dB/decade).
• Final slope returns to 0 dB/decade: 2 zeros - 2 poles = 0. ✓

Step 2: High-frequency behavior.

• At high frequencies (ω ≫ 100 rad/s), |G(jω)| → |K| where K is the gain constant.
• Given: magnitude levels at 80 dB = 20 log₁₀(|K|) → |K| = 10⁴. ✓

Step 3: Construct transfer function.

• G(s) = 10⁴(s + 1)² / (s + 100)²

Step 4: Verification.

• DC response (s → 0): G(0) = 10⁴(1)² / (100)² = 10⁴ / 10⁴ = 1 → 0 dB. ✓
• High-frequency response: For s = jω, ω ≫ 100: |G(jω)| ≈ |10⁴(jω)² / (jω)²| = 10⁴ → 80 dB. ✓
• Poles at s = -100 (stable). ✓
• Zeros at s = -1 (minimum-phase). ✓