Complex Impedance

Use of Complex Impedance

复数阻抗的使用

The handling of the impedance of an AC circuit with multiple components quickly becomes unmanageable if sines and cosines are used to represent the voltages and currents. A mathematical construct which eases the difficulty is the use of complex exponential functions. The basic parts of the strategy are as follows:

处理包含多个元件的交流电路的阻抗变得难以管理，如果用正弦和余弦函数来表示电压和电流的话。一种数学构造，可以缓解这一困难，就是使用复指数函数。该策略的基本部分如下：

Math relationship underlying the technique

该技术背后的数学关系

Euler relation

欧拉关系

Polar form of
complex number

复数的极坐标形式

The real part of a complex exponential function can be used to represent an AC voltage or current.

复数指数函数的实部可以用来表示交流电压或电流。

The impedance can then be expressed as a complex exponential.

阻抗可以表示为复数指数形式。

Impedance combinations

阻抗组合

Phasor diagrams

相量图

The impedance of the individual circuit elements can then be expressed as pure real or imaginary numbers.

电路元件的阻抗可以表示为纯实数或纯虚数。

RL and RC combinations

RL和RC组合

Example for parallel elements

并行元素的例子

Complex Impedance for RL and RC

RL和RC的复阻抗

Using complex impedance is an important technique for handling multi-component AC circuits. If a complex plane is used with resistance along the real axis then the reactances of the capacitor and inductor are treated as imaginary numbers. For series combinations of components such as RL and RC combinations, the component values are added as if they were components of a vector. Shown here is the cartesian form of the complex impedance. They can also be written in polar form. Impedances in this form can be used as building blocks for calculating the impedances of combination circuits like the RLC parallel circuit.

使用复阻抗是处理多元件交流电路的重要技术。如果用复平面表示，电阻沿实轴方向，则电容器和电感器的电抗被当作虚数处理。对于如RL和RC组合这样的串联元件，其值被加在一起，如同将它们视为矢量的分量一样。这里显示的是复阻抗的直角形式。它们也可以用极坐标形式表示。这种形式的阻抗可以作为构建组合电路（如RLC并联电路）阻抗的积木。

Complex Impedance for RL and RC

RL和RC的复阻抗

This depicts the phasor diagrams and complex impedance expressions for RL and RC circuits in polar form. They can also be expressed in cartesian form.

这展示了RL和RC电路的相量图及极坐标形式的复阻抗表达式。它们也可以用直角坐标形式表示。