Impedance in the s Domain | s域中的阻抗

📚 Impedance in the s Domain | s域中的阻抗

In circuit analysis, moving from the time domain to the s‑domain transforms differential equations into algebraic ones, making it easier to study transient and steady‑state behaviour. Impedance in the s‑domain generalises the familiar complex impedance and allows us to model the energy‑storage elements—inductors and capacitors—as simple “resistors” with s‑dependent values.

在电路分析中，将时域转换到 s 域会把微分方程变为代数方程，从而更容易研究瞬态和稳态行为。s 域中的阻抗推广了熟悉的复阻抗，并允许我们将储能元件（电感和电容）建模为具有依赖于 s 的简单“电阻”。

1. Introduction to the s‑Domain and Laplace Transform | s域与拉普拉斯变换导论

0) into a function F(s) of a complex variable s. It is defined by F(s) = L{f(t)} = ∫₀∞ f(t)·e^−st a; has both a real part σ (neper frequency) and an imaginary part ω (angular frequency). This transformation replaces differentiation with multiplication by s and integration with division by s, provided initial conditions are included.

拉普拉斯变换将时间函数 f(t) (t ≥ 0) 转换为复变量 s 的函数 F(s)。其定义为 F(s) = L{f(t)} = ∫₀∞ f(t)·e^−st dt。参数 s = σ + jω 同时具有实部 σ（奈培频率）和虚部 ω（角频率）。在计入初始条件的前提下，该变换用乘以 s 替代微分，用除以 s 替代积分。

F(s) = ∫₀∞ f(t)·e^−st dt, s = σ + jω

2. Impedance Concept in the s‑Domain | s域中的阻抗概念

In the s‑domain, the impedance Z(s) of a two‑ l initial conditions are zero: Z(s) = V(s) / I(s). This algebraic relationship mirrors Ohm' rk analysis. Conversely, admittance Y(s) is the reciprocal, Y(s) = 1 / Z(s) = I(s) / V(s).

在 s 域中，二端元件的阻抗 Z(s) 定义为所有初始条件为零时电压与电流拉普拉斯变换的比值：Z(s) = V(s) / I(s)。这种代数关系与欧姆定律形式相同，极大地简化了网络分析。相应地，导纳 Y(s) 是阻抗的倒数，Y(s) = 1 / Z(s) = I(s) / V(s)。

Z(s) = V(s) / I(s) (zero initial conditions)

3. Resistor in the s‑Domain | s域中的电阻

A resistor obeys the time‑ is always satisfied for a memoryless element) gives V(s) = R·I(s). Therefore, its s‑domain impedance is simply Z_R(s) = R.

电阻遵守时域关系 v(t) = R·i(t)。对该式进行拉普拉斯变换（无记忆元件恒满足零初始条件）可得 V(s) = R·I(s)。因此其 s 域阻抗就是 Z_R(s) = R。

Z_R(s) = R

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4. Inductor in the s‑Domain | s域中的电感

The voltage‑ his with the Laplace derivative property yields V(s) = sL·I(s) − L·i(0⁻), where i(0⁻) is the initial current through the inductor. If we assume zero initial condition (i(0⁻) = 0), the relation reduces to V(s) = sL·I(s). Hence the s‑domain impedance of an inductor is Z_L(s) = sL.

电感 L 的电压‑电流关系为 v(t) = L·(di/dt)。利用拉普拉斯变换的微分性质得到 V(s) = sL·I(s) − L·i(0⁻)，其中 i(0⁻) 是电感中的初始电流。若假设零初始条件 (i(0⁻) = 0)，则关系简化为 V(s) = sL·I(s)。因此电感的 s 域阻抗为 Z_L(s) = sL。

Z_L(s) = sL (zero initial current)

If an initial current exists, the inductor can be modelled in the s‑domain as an impedance sL in series with a voltage source L·i(0⁻), allowing transient analysis without removing the initial condition.

如果存在初始电流，可以在 s 域中将电感建模为阻抗 sL 与一个电压源 L·i(0⁻) 的串联，这样无需去除初始条件即可进行瞬态分析。

5. Capacitor in the s‑Domain | s域中的电容

or C, the terminal relation is i(t) = C·(dv/dt), or equivalently v(t) = (1/C)∫₀^t i(τ)dτ + v(0⁻ Laplace transform (using the integral property) leads to V(s) = (1/(sC))·I(s) + v(0⁻ With zero initial voltage, V(s) = (1/(sC))·I(s), giving the capacitive impedance Z_C(s) = 1/(sC).

对于电容 C，端部关系为 i(t) = C·(dv/dt)，或等价地 v(t) = (1/C)∫₀^t i(τ)dτ + v(0⁻)。应用拉普拉斯变换（利用积分性质）可得 V(s) = (1/(sC))·I(s) + v(0⁻)/s。在零初始电压下，V(s) = (1/(sC))·I(s)，因此电容阻抗 Z_C(s) = 1/(sC)。

Z_C(s) = 1 / (sC) (zero initial voltage)

An initial capacitor voltage can be modelled as an impedance 1/(sC) in series with a voltage source v(0⁻)/s, or equivalently as a Norton equivalent with a current source.

初始电容电压可以建模为阻抗 1/(sC) 与一个电压源 v(0⁻)/s 串联，或者等效为带有电流源的诺顿等效电路。

6. General Impedance and Admittance | 广义阻抗与导纳

Collectively, passive linear elements are described by their s‑ . The term reactance commonly refers to the imaginary part when s = jω, but in the s‑ Admittance Y(s) = 1/Z(s) allows parallel combinations to be handled by simple addition: Y_eq = Y₁ + Y₂ + … .

概括而言，无源线性元件由其 s 域阻抗描述：电阻 R、电感抗 sL、电容抗 1/(sC)。术语“电抗”通常指 s = jω 时的虚部，但在 s 域中阻抗是广义传递函数。导纳 Y(s) = 1/Z(s) 使得并联组合可通过简单相加处理：Y_eq = Y₁ + Y₂ + … 。

Element	Time domain	s‑domain impedance Z(s)
Resistor	v = R i	R
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