Resonance

Resonance in AC circuits implies a special frequency determined by the values of the resistance , capacitance , and inductance . For series resonance the condition of resonance is straightforward and it is characterized by minimum impedance and zero phase. Parallel resonance , which is more common in electronic practice, requires a more careful definition.

Series Resonance

The resonance of a series RLC circuit occurs when the inductive and capacitive reactances are equal in magnitude but cancel each other because they are 180 degrees apart in phase. The sharp minimum in impedance which occurs is useful in tuning applications. The sharpness of the minimum depends on the value of R and is characterized by the "Q" of the circuit.

Detailed expressions 详细表达式 Calculation 计算

Selectivity and Q of a Circuit

Resonant circuits are used to respond selectively to signals of a given frequency while discriminating against signals of different frequencies. If the response of the circuit is more narrowly peaked around the chosen frequency, we say that the circuit has higher "selectivity". A "quality factor" Q, as described below, is a measure of that selectivity, and we speak of a circuit having a "high Q" if it is more narrowly selective.

An example of the application of resonant circuits is the selection of AM radio stations by the radio receiver. The selectivity of the tuning must be high enough to discriminate strongly against stations above and below in carrier frequency, but not so high as to discriminate against the "sidebands" created by the imposition of the signal by amplitude modulation. 共振电路的一个应用例子是收音机接收AM电台时的选择性调谐。调谐的选性必须足够高，以强烈地排斥载波频率上下两侧的其他电台，但又不能高到排斥由幅度调制产生的“边带”。

The selectivity of a circuit is dependent upon the amount of resistance in the circuit. The variations on a series resonant circuit at right follow an example in Serway & Beichner. The smaller the resistance, the higher the "Q" for given values of L and C. The parallel resonant circuit is more commonly used in electronics, but the algebra necessary to characterize the resonance is much more involved. 电路的选择性取决于电路中的电阻大小。系列共振电路的波动情况如Serway & Beichner中的例子所示。电阻越小，对于给定的L和C值，其“Q”值越高。虽然并联共振电路在电子学中更为常见，但用来描述共振的代数运算更为复杂。

Using the same circuit parameters, the illustration at left shows the power dissipated in the circuit as a function of frequency. Since this power depends upon the square of the current, these resonant curves appear steeper and narrower than the resonance peaks for current above. 使用相同的电路参数，左边的示意图显示了电路中功率随频率的变化情况。由于该功率与电流的平方成正比，因此这些共振曲线比电流高于共振峰时的曲线更加陡峭和狭窄。 The quality factor Q is defined by 品质因数 Q 定义为 where Δω is the width of the resonant power curve at half maximum. 其中Δω是共振功率曲线在半高处的宽度。

Since that width turns out to be Δω =R/L, the value of Q can also be expressed as

Power in a Series Resonant Circuit

The average power dissipated in a series resonant circuit can be expressed in terms of the rms voltage and current as follows:

Using the forms of the inductive reactance and capacitive reactance, the term involving them can be expressed in terms of the frequency.

This power distribution is plotted at left using the same circuit parameters as were used in the example on the Q factor of the series resonant circuit 这种功率分布如左图所示，使用了与串联谐振电路品质因数示例中相同的电路参数 The average power at resonance is just 共振时的平均功率只是 since at the resonant frequency ω0 the reactive parts cancel so that the circuit appears as just the resistance R. 在共振频率ω₀时，电抗部分相互抵消，因此电路表现为仅有的电阻R。