正弦波
正弦波信号是常用的电路波形信号, 正弦波信号具有周而复始周期性变化的特性.
数学表达式为
u(t)=Upsinωt u(t) = U_p \sin \omega t u(t)=Upsinωt
式中, UpU_pUp是幅值VVV, ω\omegaω是角频率.
周期为
T=2πωT = \frac{2\pi}{\omega}T=ω2π
频率为
f=1T=ω2π f = \frac{1}{T} = \frac{\omega}{2 \pi}f=T1=2πω
峰峰值为
Upp=2UpU_{pp} = 2 U_p Upp=2Up
平均值为
Uave=∫02πUpsinωtdt=0 U_{ave} = \int_0^{2\pi} U_p \sin \omega t \mathrm{d}t = 0 Uave=∫02πUpsinωtdt=0
有效值为
Urms=1T∫tt+T(Upsinωt)2dt U_{rms} = \sqrt{\frac{1}{T}\int_{t}^{t+T}(U_p \sin \omega t)^2 \mathrm{d}t}Urms=T1∫tt+T(Upsinωt)2dt
推导过程如下
Urms=1T∫tt+T(Upsinωt)2dt=Up2T∫tt+Tsin2ωt dt=Up12T∫tt+T(1−cos2ωt)dt=Up12T[∫tt+Tdt−12∫tt+Td(sin2ωt)]=Up12−14T[sin(2ωt+2ωT)−sin2ωt]=Up12−14T(sin2ωt−sin2ωt)=Up12=Up2 \begin{align*} U_{\text{rms}} &= \sqrt{\frac{1}{T} \int_{t}^{t+T} \left( U_{\text{p}} \sin\omega t \right)^2 dt} \\[6pt] &= \sqrt{\frac{U_{\text{p}}^2}{T} \int_{t}^{t+T} \sin^2\omega t \, dt} \\[6pt] &= U_{\text{p}} \sqrt{\frac{1}{2T} \int_{t}^{t+T} \left( 1 - \cos2\omega t \right) dt} \\[6pt] &= U_{\text{p}} \sqrt{\frac{1}{2T} \left[ \int_{t}^{t+T} dt - \frac{1}{2} \int_{t}^{t+T} d\left( \sin2\omega t \right) \right]} \\[6pt] &= U_{\text{p}} \sqrt{\frac{1}{2} - \frac{1}{4T} \left[ \sin\left( 2\omega t + 2\omega T \right) - \sin2\omega t \right]} \\[6pt] &= U_{\text{p}} \sqrt{\frac{1}{2} - \frac{1}{4T} \left( \sin2\omega t - \sin2\omega t \right)} \\[6pt] &= U_{\text{p}} \sqrt{\frac{1}{2}} \\[6pt] &= \frac{U_{\text{p}}}{\sqrt{2}} \end{align*} Urms=T1∫tt+T(Upsinωt)2dt =TUp2∫tt+Tsin2ωtdt =Up2T1∫tt+T(1−cos2ωt)dt =Up2T1[∫tt+Tdt−21∫tt+Td(sin2ωt)] =Up21−4T1[sin(2ωt+2ωT)−sin2ωt] =Up21−4T1(sin2ωt−sin2ωt) =Up21 =2 Up
即一个周期内正弦波的有效值为
Urms=22UP U_{rms} = \frac{\sqrt{2}}{2} U_PUrms=22 UP
正弦波取绝对值
平均值
Uave=1T∫0TUpsinωtdt 0≤ωt<π U_{\rm{ave}} = \frac{1}{T}\int_0^T U_p \sin \omega t \mathrm{d}t \;\; 0 \le \omega t < \piUave=T1∫0TUpsinωtdt0≤ωt<π
Uave=UpωT∫0Tsinωtdωt=UpωT−UpωtcosωT=Upπ−−Upπ=2πUp U_{\rm{ave}} = \frac{U_p}{\omega T} \int_0^T \sin \omega t \mathrm{d} \omega t = \frac{U_p}{\omega T} - \frac{U_p}{\omega t} \cos \omega T = \frac{U_p}{\pi} - \frac{-U_p}{\pi} = \frac{2}{\pi}U_pUave=ωTUp∫0Tsinωtdωt=ωTUp−ωtUpcosωT=πUp−π−Up=π2Up
有效值
有效值与正弦波的有效值相等
Urms=22Up U_{\rm{rms}} = \frac{\sqrt{2}}{2}U_pUrms=22 Up
正弦波的一半
i(t)={Ipsinωt0≤ωt<π0π≤ωt<2π i(t) = \begin{cases} I_{\text{p}} \sin \omega t & 0 \le \omega t < \pi \\[6pt] 0 & \pi \le \omega t < 2\pi \end{cases}i(t)=⎩ ⎨ ⎧Ipsinωt00≤ωt<ππ≤ωt<2π
平均值
Iave=1T∫0T/2Ipsinωt dt=IpωT∫0T/2sinωt d(ωt)=IpωT−IpωTcosωT2=Ip2π−Ip2πcosπ=Ipπ \begin{align*} I_{\text{ave}} &= \frac{1}{T} \int_{0}^{T/2} I_{\text{p}} \sin \omega t \, dt \\[6pt] &= \frac{I_{\text{p}}}{\omega T} \int_{0}^{T/2} \sin \omega t \, d(\omega t) \\[6pt] &= \frac{I_{\text{p}}}{\omega T} - \frac{I_{\text{p}}}{\omega T} \cos \frac{\omega T}{2} = \frac{I_{\text{p}}}{2\pi} - \frac{I_{\text{p}}}{2\pi} \cos \pi \\[6pt] &= \frac{I_{\text{p}}}{\pi} \end{align*} Iave=T1∫0T/2Ipsinωtdt=ωTIp∫0T/2sinωtd(ωt)=ωTIp−ωTIpcos2ωT=2πIp−2πIpcosπ=πIp
有效值
Irms=1T∫0T/2(Ipsinωt)2dt=Ip2 I_{\rm{rms}} = \sqrt{\frac{1}{T} \int_0^{T/2}(I_p \sin \omega t)^2 \mathrm{d}t} = \frac{I_p}{2} Irms=T1∫0T/2(Ipsinωt)2dt =2Ip