品质因数是谐振电路的重要的参数,关于谐振电路品质因数的定义为:
![eb4c34d0186a4aa2a82ec3d0baad89b3.jpg eb4c34d0186a4aa2a82ec3d0baad89b3.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500l8fzfff18fdf2ay8.jpg)
假设电容两端电压为
![e28a1c9b4c93472fa2006abf0cbfb199?from=pc.jpg e28a1c9b4c93472fa2006abf0cbfb199?from=pc.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500y44dzihhbm0bi4z4.jpg)
,电容瞬时功率为
![e4bb725cbd5d4bb28a362107980ae656.jpg e4bb725cbd5d4bb28a362107980ae656.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500s2i8agig73j8igk2.jpg)
,则它在一个谐振周期交换的能量为:
![c0d43c4c33de4801b924ca46098eeabe.jpg c0d43c4c33de4801b924ca46098eeabe.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500axoxwozb4l48c8u8.jpg)
假设流过电感的电流为
![57fdf651f7634161812a8c83080cc6c7?from=pc.jpg 57fdf651f7634161812a8c83080cc6c7?from=pc.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500dg7ptvuqh12pwm1u.jpg)
,电感两端的瞬时功率为
![c8bede3b77844606a576f1abe2241cb3.jpg c8bede3b77844606a576f1abe2241cb3.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500d959rc8r8lz8nckg.jpg)
,则它在一个周期内交换的能量为:
![29bdddbdfde5483988d98f891d0944be.jpg 29bdddbdfde5483988d98f891d0944be.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500fdrr27swr3g5nino.jpg)
可以看出,品质因数的定义可以理解为,在一个谐振周期内,电容或电感交换的能量与消耗在电路中的能量之比,这也可以理解为电路的无功功率与有功功率之比。
对于RLC串联谐振电路,如下图,其品质因数为:
![87637e386c42418ebfcd107ecb358a9a?from=pc.jpg 87637e386c42418ebfcd107ecb358a9a?from=pc.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500ad92w2fndf1z76w2.jpg)
![a0ed3a9fd8fe4a47a09262ad2eaa0a73.jpg a0ed3a9fd8fe4a47a09262ad2eaa0a73.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500zk4d65886sc582nd.jpg)
对于RLC并联谐振电路,其品质因数Q为:
![767eb3f829e049de89a7ba770fcaf501.jpg 767eb3f829e049de89a7ba770fcaf501.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500go4xztfh912jogao.jpg)
![e0d3cbd032b1483fa79d9935a10fb785.jpg e0d3cbd032b1483fa79d9935a10fb785.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500mrlj75j2e5070sri.jpg)
以上两种电路的谐振频率均为:
![7d7975496c044f778dbb6a901d4cb008.jpg 7d7975496c044f778dbb6a901d4cb008.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500bwtqr4i44b4ssj46.jpg)
串联谐振电路和并联谐振电路有很多相似的特性,对于串联谐振电路:
![ae9ef65865bd4823a77efee108577541?from=pc.jpg ae9ef65865bd4823a77efee108577541?from=pc.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500btl2e55enypiny9z.jpg)
并联谐振与串联谐振是对偶的关系,其电流特性也呈现出上面的特点:
![4c11b77a4fff4a30b9286e1d5fea1f44.jpg 4c11b77a4fff4a30b9286e1d5fea1f44.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500vn1zqnglipy593nk.jpg)
而实际电路中R、L串联,然后与C并联的电路更常见,如电感线圈与电容并联,
![827cc61d271f43008248484d6a74fa3a.jpg 827cc61d271f43008248484d6a74fa3a.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500rkkqqksgyext46yk.jpg)
这种电路的谐振与上述两种有所不同,电路的阻抗为:
![87193a59c69f45daa362e1a9a65e97cf.jpg 87193a59c69f45daa362e1a9a65e97cf.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500h4av4jx8210g000g.jpg)
当电路发生谐振时,阻抗虚部为0,谐振频率为:
![3c1312c4980345eb94c8ac28bb7f973f.jpg 3c1312c4980345eb94c8ac28bb7f973f.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500b9qp3cp3gkhkvump.jpg)
可见只有当1-CR2/L>0,即
![1460a99c4fa745cc9fbba71a3ee6b17c?from=pc.jpg 1460a99c4fa745cc9fbba71a3ee6b17c?from=pc.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500toey2948qee2iiys.jpg)
时电路才会谐振,否则电路不会发生谐振,如果远小于则近似于并联谐振的特性。如果电路能够发生谐振,则品质因数为下式(也可将电路等效为串联或并联谐振电路进行求解)。
![ba24cd6148064164b6e13cf7acf27494.jpg ba24cd6148064164b6e13cf7acf27494.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500pzlb6j6f6tslinbz.jpg)
这里当电路阻抗呈现纯阻性时(与信号频率无关),即当
![400ee846ec13495698861d6d76788644.jpg 400ee846ec13495698861d6d76788644.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500sh44f048tn4e48gt.jpg)
时,此时的电阻阻值又称为电路的特征阻抗,此时电路的阻抗为:
![e41a4dab4f5f43c8b361534401e92316.jpg e41a4dab4f5f43c8b361534401e92316.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500frrhv3573qez0hur.jpg)
对于R、C并联然后与L串联的电路也很常见,如下图所示,
![9a274b36c3144e3b9d3920118efe262d?from=pc.jpg 9a274b36c3144e3b9d3920118efe262d?from=pc.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500n0emubqizluque0c.jpg)
电路的阻抗为:
![47d77ec1f6fa43dd894d97b3b508429a.jpg 47d77ec1f6fa43dd894d97b3b508429a.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500yzr1wgmnanjpz1m6.jpg)
当电路发生谐振时,虚部阻抗为0,谐振频率为:
![91e26a1264624a40962c97a1cd24dc14.jpg 91e26a1264624a40962c97a1cd24dc14.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500iij5g38987953ajg.jpg)
可见与R-L//C电路呈现对偶关系。只有当1-L/CR2>0,即
![eb82a8c5fcd1414ea5e923af4a2b79c4.jpg eb82a8c5fcd1414ea5e923af4a2b79c4.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500cfwlka6uiji9zkt9.jpg)
时,电路才会谐振,否则电路不会发生谐振,如果远大于则近似于串联谐振的特性。同样,该电路的品质因数为:
![a11bca360602420a90d393fba4bc9a55.jpg a11bca360602420a90d393fba4bc9a55.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500duq00hgsun0izw5u.jpg)
如果电路呈现纯阻性(与信号频率无关),即满足
![b2c2dd9a9b124423ab0d679072738186.jpg b2c2dd9a9b124423ab0d679072738186.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500uz8q1z1ww158wz3c.jpg)
时,
![d74c33c6a68c4dc093f0749551847ee0.jpg d74c33c6a68c4dc093f0749551847ee0.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500sre3t5l591ztzil1.jpg)
即为该电路的特征阻抗,电路的阻抗为:
![d0c57b80be67460494de518ec825aef2?from=pc.jpg d0c57b80be67460494de518ec825aef2?from=pc.jpg](https://static.assets-stash.eet-china.com/forum/202107/29/115500vg79np1pgkktztgn.jpg)
可见以上两种电路当负载阻抗等于电路特征阻抗时,电路呈现电阻特性,且特征阻抗的表达式是相同的,电路所表现出的阻抗即为负载阻抗。
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