Bode Magnitude and Phase Analysis Simulation

SYSTEM DYNAMICS: BODE MAGNITUDE & PHASE ANALYSIS

Precision Frequency Domain Simulation with Exact Transfer Functions

System Transfer Function $H(s)$ Low-Pass Filter (1 / (1 + s/ωc)ⁿ) High-Pass Filter ((s/ωc)ⁿ / (1 + s/ωc)ⁿ) Pure Integrator (1 / (s/ωc)ⁿ)

Filter Order / Poles ($n$)

Corner Frequency ($f_c$): 1000 Hz

Display Overlays Show Asymptotic Lines Show Actual Physical Curve

THEORETICAL SLOPES:

Decade Rate: -20.0 dB/dec

Octave Rate: -6.02 dB/oct

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At $2f_c$ (+1 Octave): -3.01 dB actual
At $10f_c$ (+1 Decade): -10.04 dB actual

MAGNITUDE RESPONSE $|H(j\omega)|_{\text{dB}}$

PHASE RESPONSE $\angle H(j\omega)$

Analytical Foundations of Frequency Domain Analysis

Frequency response analysis translates time-domain differential equations into the complex frequency domain ($s = \sigma + j\omega$). The Bode Plot provides a dual representation of a system's steady-state response to sinusoidal inputs across a logarithmic frequency spectrum.

1. Exact Mathematical Definitions

For an $n^{\text{th}}$-order continuous-time low-pass prototype system, the transfer function $H(s)$ is expressed as:

$$\text{Transfer Function: } H(s) = \left( \frac{1}{1 + \frac{s}{\omega_c}} \right)^n \implies H(j\omega) = \left( \frac{1}{1 + j\frac{\omega}{\omega_c}} \right)^n$$

Evaluating the continuous magnitude and phase functions yields:

$$\text{Magnitude (dB): } |H(j\omega)|_{\text{dB}} = 20 \log_{10} |H(j\omega)| = -10 \cdot n \cdot \log_{10}\left(1 + \left(\frac{f}{f_c}\right)^2\right)$$ $$\text{Phase (Degrees): } \theta(f) = \angle H(j\omega) = -n \cdot \arctan\left(\frac{f}{f_c}\right) \times \frac{180^\circ}{\pi}$$

2. Asymptotic Approximations vs. Physical Reality

Bode plots are traditionally drawn using straight-line piecewise linear approximations. The true magnitude response deviates predictably from these asymptotes:

• At Corner Frequency ($f = f_c$): The asymptotic magnitude prediction is $0\text{ dB}$. The actual physical response is down by $-3.0103 \times n \text{ dB}$ ($1/\sqrt{2}$ voltage ratio per pole).
• At $+1\text{ Octave}$ ($f = 2f_c$): The asymptotic slope predicts a drop of $-6.0206 \times n\text{ dB}$. The actual continuous drop from $DC$ is $-10 \log_{10}(1 + 2^2) \cdot n \approx -6.9897 \times n\text{ dB}$.
• At $+1\text{ Decade}$ ($f = 10f_c$): The asymptotic slope predicts $-20 \times n\text{ dB}$. The actual drop is $-10 \log_{10}(1 + 10^2) \cdot n \approx -20.043 \times n\text{ dB}$.

3. The Precise Relationship Between Octaves and Decades

An octave represents a factor-of-2 frequency ratio ($f_2/f_1 = 2$), whereas a decade represents a factor-of-10 ratio ($f_2/f_1 = 10$). The exact mathematical conversion between decibel roll-off rates is derived from the properties of logarithms:

$$\text{Ratio} = \frac{\log_{10}(2)}{\log_{10}(10)} = \log_{10}(2) \approx 0.30102999$$ $$\text{Slope}_{\text{octave}} = \text{Slope}_{\text{decade}} \times \log_{10}(2) \implies -20\text{ dB/dec} \times 0.30103 = \mathbf{-6.0206\text{ dB/oct}}$$

4. Phase Response Dynamics

Phase changes begin roughly one decade before the corner frequency and extend one decade after it:

• For $f \ll f_c$, phase is $0^\circ$.
• At $f = f_c$, phase is precisely $-45^\circ \times n$.
• For $f \gg f_c$, phase approaches the asymptotic limit of $-90^\circ \times n$.

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