Transfer Functions in the Frequency Domain | 频域中的传递函数

📚 Transfer Functions in the Frequency Domain | 频域中的传递函数

In many areas of applied mathematics, engineering and even IB higher level analysis, we model systems using transfer functions. These functions describe how an input signal is transformed into an output signal. In the frequency domain, the transfer function becomes a powerful tool to analyse stability, filtering behaviour, and resonance without solving differential equations directly. This article revisits the key concepts of transfer functions in the frequency domain, tailored for IB Mathematics students who are comfortable with complex numbers and exponential functions.

在应用数学、工程学乃至 IB 高级课程中，我们常用传递函数来建模系统。传递函数描述输入信号如何被转换成输出信号。在频域中，传递函数成为分析稳定性、滤波特性和谐振的强大工具，而无需直接求解微分方程。本文梳理频域中传递函数的核心概念，适合熟悉复数和指数函数的 IB 数学学生。

1. Transfer Functions: Definition and Basic Form | 传递函数：定义与基本形式

A transfer function H(s) is the ratio of the Laplace transform of the output y(t) to the Laplace transform of the input x(t), assuming zero initial conditions. In block diagram form we write Y(s) = H(s) X(s). Here s is a complex variable, s = σ + jω. The transfer function captures all the dynamics of a linear time-invariant (LTI) system using polynomials in s.

传递函数 H(s) 是输出 y(t) 的拉普拉斯变换与输入 x(t) 的拉普拉斯变换之比，并假设初始条件为零。用框图形式可写成 Y(s) = H(s) X(s)。这里 s 是复变量，s = σ + jω。传递函数用 s 的多项式捕获线性时不变 (LTI) 系统的所有动态特性。

For example, an RC low-pass filter has transfer function H(s) = 1/(1 + sRC). In the frequency domain we are particularly interested in the behaviour when σ = 0, i.e. s = jω, because this corresponds to sinusoidal steady-state excitation.

例如，一个 RC 低通滤波器的传递函数为 H(s) = 1/(1 + sRC)。在频域中，我们特别关注 σ = 0 即 s = jω 时的行为，因为这对应于正弦稳态激励。

The order of the transfer function is the highest power of s in the denominator, which indicates the number of independent energy-storage elements in the system.

传递函数的阶数是分母中 s 的最高次幂，它表明系统中独立储能元件的个数。

2. The Complex Frequency s and s-Plane | 复频率 s 与 s 平面

In IB mathematics, complex numbers are represented as z = x + jy. Extending this idea, the complex frequency s = σ + jω combines a real exponential decay/growth factor (σ) with an angular frequency (ω). All possible values of s form the s-plane, where the vertical axis (jω) is the frequency axis and the horizontal axis (σ) indicates transient behaviour.

在 IB 数学中，复数表示为 z = x + jy。推广这一概念，复频率 s = σ + jω 将实指数衰减/增长因子 (σ) 与角频率 (ω) 结合在一起。s 的所有可能值构成 s 平面，其中纵轴 (jω) 为频率轴，横轴 (σ) 表示暂态行为。

Placing s = jω means we are moving along the imaginary axis, purely oscillatory. The frequency response H(jω) is therefore the transfer function evaluated on the imaginary axis.

令 s = jω 意味着我们沿虚轴移动，仅有振荡。因此频率响应 H(jω) 就是在虚轴上求值的传递函数。

3. Steady-State Frequency Response: H(jω) | 稳态频率响应：H(jω)

By substituting s = jω into H(s), we obtain the frequency response function H(jω), a complex-valued function of real frequency ω. It tells us how the system modifies the amplitude and phase of a sinusoidal input. If x(t) = A sin(ωt), the steady-state output is y(t) = A |H(jω)| sin(ωt + ∠H(jω)).

将 s = jω 代入 H(s)，我们得到频率响应函数 H(jω)，它是一个关于实频率 ω 的复值函数。它告诉我们系统如何改变正弦输入的幅值和相位。如果 x(t) = A sin(ωt)，则稳态输出为 y(t) = A |H(jω)| sin(ωt + ∠H(jω))。

It is crucial to separate H(jω) into its magnitude (gain) and phase components. The magnitude |H| indicates amplification or attenuation; the phase φ tells the time shift relative to

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