📚 The s-Domain Transfer Function in Laplace Transforms | 拉普拉斯变换中的s域传递函数
The Laplace transform is one of the most powerful tools in applied mathematics and engineering. It converts a time-domain function, such as a signal or system response, into a complex-frequency-domain representation. In this article we focus on the transfer function H(s), which describes how a linear time-invariant system transforms an input into an output. Working in the s-domain allows us to replace calculus with algebra, solve differential equations more easily, and analyse stability, frequency response, and transient behaviour in a unified way.
拉普拉斯变换是应用数学与工程中最强大的工具之一。它将时间域中的函数（如信号或系统响应）转换为复频域表示。本文将重点讨论传递函数 H(s)，它描述线性时不变系统如何将输入变换为输出。在 s 域中求解，我们能用代数运算取代微积分，更方便地求解微分方程，并统一分析稳定性、频率响应与暂态行为。
1. The Laplace Transform as a Tool | 拉普拉斯变换：基本工具
For a function f(t) defined for t ≥ 0, the one-sided Laplace transform is defined by the following integral, provided the integral converges:
对于定义在 t ≥ 0 上的函数 f(t)，单边拉普拉斯变换由下列积分定义（若积分收敛）：
F(s) = ∫₀^∞ f(t)e⁻ˢᵗ dt
Here s = σ + jω is a complex variable, where j² = −1. The transform maps differentiation and integration in the time domain to multiplication and division by s in the s-domain, which is the key reason linear differential equations become algebraic equations in this domain.
其中 s = σ + jω 是复变量，j² = −1。该变换将时间域中的微分与积分分别映射为 s 域中的乘 s 与除以 s，这正是线性微分方程在 s 域中化为代数方程的关键原因。
2. Definition of the Transfer Function | 传递函数的定义
Consider a linear time-invariant (LTI) system with input x(t) and output y(t). Let X(s) = ℒ{x(t)} and Y(s) = ℒ{y(t)}. The transfer function H(s) is defined as the ratio of the output transform to the input transform under zero initial conditions:
考虑一个线性时不变（LTI）系统，输入为 x(t)，输出为 y(t)。令 X(s) = ℒ{x(t)}，Y(s) = ℒ{y(t)}。在零初始条件下，传递函数 H(s) 定义为输出变换与输入变换之比：
H(s) = Y(s) / X(s)
This ratio is independent of the particular input; it depends only on the system itself. Once H(s) is known, the response to any input can be obtained by multiplying X(s) by H(s) and then taking the inverse Laplace transform.
该比值不依赖于特定输入，而只取决于系统本身。一旦求得 H(s)，任何输入的系统响应都可以通过 X(s) 乘以 H(s) 再进行拉普拉斯逆变换得到。
3. From Differential Equations to the s-Domain | 从微分方程到s域
Suppose a system is described by an nth-order linear differential equation with constant coefficients:
假设某系统由常系数 n 阶线性微分方程描述：
aₙ y⁽ⁿ⁾ + aₙ₋₁ y⁽ⁿ⁻¹⁾ + … + a₀ y = bₘ x⁽ᵐ⁾ + bₘ₋₁ x⁽ᵐ⁻¹⁾ + … + b₀ x
Taking the Laplace transform of both sides and assuming all initial conditions are zero, we use the property ℒ{y'(t)} = sY(s), ℒ{y”(t)} = s²Y(s), and so on. The equation becomes an algebraic relationship between Y(s) and X(s). Therefore:
对等式两边取拉普拉斯变换，并假设所有初始条件为零，利用性质 ℒ{y'(t)} = sY(s)，ℒ{y”(t)} = s²Y(s) 等。方程化为 Y(s) 与 X(s) 之间的代数关系。因此：
H(s) = (bₘ sᵐ + bₘ₋₁ sᵐ⁻¹ + … + b₀) / (aₙ sⁿ + aₙ₋₁ sⁿ⁻¹ + … + a₀)
The transfer function is thus a rational function in s. Its numerator and denominator are polynomials whose coefficients come directly from the original differential equation.
因此传递函数是 s 的有理函数，其分子和分母都是多项式，系数直接来自原始微分方程。
4. Poles and Zeros | 极点和零点
In the s-domain, the zeros of H(s) are the roots of the numerator polynomial, and the poles are the roots of the denominator polynomial. Poles and zeros are usually plotted in the complex s-plane, with poles marked by × and zeros by ○.
在 s 域中，H(s) 的零点是分子多项式的根，极点是分母多项式的根。极点和零点通常绘制在复 s 平面中，极点用 × 标记，零点用 ○ 标记。
For example, consider:
例如，考虑：
H(s) = (s + 2) / (s² + 2s + 5)
The denominator is s² + 2s + 5 = (s + 1)² + 4, so the poles are s = −1 ± 2j. The zero is s = −2. These values provide an immediate qualitative picture of the system’s natural responses.
分母为 s² + 2s + 5 = (s + 1)² + 4，因此极点为 s = −1 ± 2j，零点为 s = −2。这些数值提供了系统自然响应的直观定性图像。
5. Stability and the s-Plane | 稳定性与s平面
For a causal LTI system, BIBO (bounded-input bounded-output) stability is determined entirely by the poles of H(s). The system is stable if and only if every pole has a negative real part, meaning all poles lie in the left half of the s-plane.
对于因果线性时不变系统，BIBO（有界输入有界输出）稳定性完全由 H(s) 的极点决定。系统稳定的充要条件是每个极点都具有负实部，即所有极点位于 s 平面的左半平面。
If a pole lies on the imaginary axis, the system may produce a sustained oscillation. If any pole lies in the right half-plane, the natural response grows without bound and the system is unstable.
若某个极点位于虚轴上，系统可能产生持续振荡。若任何极点位于右半平面，自然响应将无界增长，系统不稳定。
For example, H(s) = 1/(s + 5) is stable because its only pole is s = −5. In contrast, H(s) = 1/(s − 2) is unstable because the pole s = 2 lies in the right half-plane.
例如，H(s) = 1/(s + 5) 是稳定的，因为其唯一极点为 s = −5。相反，H(s) = 1/(s − 2) 不稳定，因为极点 s = 2 位于右半平面。
6. Partial Fraction Expansion | 部分分式展开
To recover the time-domain impulse response or system output from H(s), we often need the inverse Laplace transform. Partial fraction expansion decomposes a rational transfer function into simpler terms whose inverse transforms are known.
为从 H(s) 恢复时间域中的冲激响应或系统输出，通常需要进行拉普拉斯逆变换。部分分式展开将有理传递函数分解为若干简单项，这些项的逆变换是已知的。
For distinct real poles p₁, p₂, …, pₙ, we may write:
对于互异实极点 p₁, p₂, …, pₙ，可写成：
H(s) = A₁/(s − p₁) + A₂/(s − p₂) + … + Aₙ/(s − pₙ)
Each term has the inverse transform Aᵢe^(pᵢt)u(t), where u(t) is the unit step function. For example, if H(s) = 3/((s + 1)(s + 2)), then H(s) = 3/(s + 1) − 3/(s + 2), and the impulse response is h(t) = 3e^(−t) − 3e^(−2t) for t ≥ 0.
每一项的逆变换为 Aᵢe^(pᵢt)u(t)，其中 u(t) 是单位阶跃函数。例如，若 H(s) = 3/((s + 1)(s + 2))，则 H(s) = 3/(s + 1) − 3/(s + 2)，冲激响应为 h(t) = 3e^(−t) − 3e^(−2t)（t ≥ 0）。
7. Initial and Final Value Theorems | 初值与终值定理
The initial value theorem and final value theorem provide quick ways to determine f(0⁺) and f(∞) from F(s) without computing the full inverse transform.
初值定理和终值定理提供了一种快速方法，无需计算完整的逆变换，即可直接由 F(s) 确定 f(0⁺) 和 f(∞)。
f(0⁺) = limₛ→∞ sF(s)
f(∞) = limₛ→0 sF(s)
The initial value theorem holds if the limit exists and F(s) is proper. The final value theorem is valid only if all poles of sF(s) lie in the left half-plane, so that f(∞) actually converges. For example, if H(s) = 1/(s + a) with a > 0, then the corresponding h(t) has final value 0, and indeed limₛ→0 s/(s + a) = 0.
初值定理在极限存在且 F(s) 为真分式时成立。终值定理仅在 sF(s) 的所有极点位于左半平面时有效，以保证 f(∞) 确实收敛。例如，若 H(s) = 1/(s + a) 且 a > 0，则对应 h(t) 的终值为 0，且确实有 limₛ→0 s/(s + a) = 0。
8. Application to Electrical Circuits | 电路应用
In circuit analysis, the Laplace transform replaces time-domain elements by s-domain impedances under zero initial conditions. A resistor of resistance R has impedance R, an inductor of inductance L has impedance sL, and a capacitor of capacitance C has impedance 1/(sC).
在电路分析中，零初始条件下，拉普拉斯变换将时域元件替换为 s 域阻抗：电阻 R 的阻抗为 R，电感 L 的阻抗为 sL，电容 C 的阻抗为 1/(sC)。
For a simple RC low-pass filter with input voltage vᵢₙ(t) and output voltage vₒᵤₜ(t) across the capacitor, the transfer function is:
对于输入电压为 vᵢₙ(t)、电容两端输出电压为 vₒᵤₜ(t) 的简单 RC 低通滤波器，传递函数为：
H(s) = Vₒᵤₜ(s) / Vᵢₙ(s) = 1 / (1 + sRC)
The impulse response is h(t) = (1/RC)e^(−t/(RC))u(t). This result is obtained directly from the s-domain model without solving the differential equation in the time domain step by step.
其冲激响应为 h(t) = (1/RC)e^(−t/(RC))u(t)。这一结果直接来自 s 域模型，无需在时间域逐步求解微分方程。
9. Frequency Response and the Substitution s = jω | 频率响应与 s = jω
The frequency response of a system is obtained by evaluating the transfer function on the imaginary axis of the s-plane. Setting s = jω gives H(jω), a complex function of the real frequency ω.
系统的频率响应通过在 s 平面的虚轴上评估传递函数获得。令 s = jω 得到 H(jω)，它是实频率 ω 的复函数。
For the RC low-pass filter H(s) = 1/(1 + sRC), substituting s = jω gives:
对于 RC 低通滤波器 H(s) = 1/(1 + sRC)，代入 s = jω 得：
H(jω) = 1 / (1 + jωRC)
The magnitude and phase are:
其幅值与相位为：
|H(jω)| = 1 / √(1 + (ωRC)²)
∠H(jω) = −tan⁻¹(ωRC)
The cutoff frequency occurs at ω = 1/(RC), where |H(jω)| = 1/√2. This is a standard result in control theory and signal processing.
截止频率出现在 ω = 1/(RC)，此时 |H(jω)| = 1/√2。这是控制理论与信号处理中的标准结果。
10. Convolution and the Transfer Function | 卷积与传递函数
In the time domain, the output of an LTI system is the convolution of the input with the impulse response h(t):
在时间域中，LTI 系统的输出等于输入与冲激响应 h(t) 的卷积：
y(t) = ∫₀^t h(τ)x(t − τ)dτ
The convolution theorem states that convolution in the time domain corresponds to multiplication in the s-domain:
卷积定理指出，时间域中的卷积对应 s 域中的乘法：
Y(s) = H(s)X(s)
Therefore H(s) fully characterises the impulse response of the system: h(t) = ℒ⁻¹{H(s)}. This is why the transfer function is so widely used in modelling and simulation.
因此 H(s) 完全刻画了系统的冲激响应：h(t) = ℒ⁻¹{H(s)}。这也是传递函数在建模与仿真中被广泛使用的原因。
11. Limitations and Practical Considerations | 局限性与实际考虑
The transfer function framework applies strictly to linear, time-invariant systems with zero initial conditions. If the system is nonlinear, time-varying, or starts from nonzero states, the direct use of H(s) is not valid without appropriate modifications.
传递函数框架严格适用于零初始条件下的线性时不变系统。若系统是非线性、时变的，或从非零状态开始，则不能直接使用 H(s)，除非进行适当修改。
Additionally, physically realisable systems usually require that the degree of the numerator polynomial be no greater than the degree of the denominator polynomial, i.e. H(s) must be a proper rational function. Systems with time delays introduce terms such as e^(−sτ), which make H(s) non-rational and require special handling.
此外，物理可实现系统通常要求分子多项式的次数不大于分母多项式的次数，即 H(s) 必须是真有理函数。带时间延迟的系统会出现 e^(−sτ) 项，使 H(s)
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply