LLC谐振电路增益公式推导
图
由图可知
G=VoutVin=sLm//Rac1sCr+sLr+sLm//RacG=\frac{V_{out}}{V_{in}}=\frac{sLm//Rac}{\frac{1}{sCr}+sLr+sLm//Rac} G=VinVout=sCr1+sLr+sLm//RacsLm//Rac
其中
sLm//Rac=sLm∗RacsLm+RacsLm//Rac=\frac{sLm*Rac}{sLm+Rac} sLm//Rac=sLm+RacsLm∗Rac
可化简为
G=sLm∗Rac(1sCr+sLr)∗(sLm+Rac)+sLm∗RacG=\frac{sLm*Rac}{(\frac{1}{sCr}+sLr)*(sLm+Rac)+sLm*Rac} G=(sCr1+sLr)∗(sLm+Rac)+sLm∗RacsLm∗Rac
其中
s=jws=jw s=jw
代入将分子分母按虚部和实部分开,得:
G=jw∗Lm∗Rac(LmCr−w2∗Lm∗Lr)+j(w∗Lr+w∗Lm−1w∗Cr)∗RacG=\frac{jw*Lm*Rac}{(\frac{Lm}{Cr}-w^2*Lm*Lr)+j(w*Lr+w*Lm-\frac{1}{w*Cr})*Rac} G=(CrLm−w2∗Lm∗Lr)+j(w∗Lr+w∗Lm−w∗Cr1)∗Racjw∗Lm∗Rac
取模长平方:
G2=w2∗Lm2∗Rac2(LmCr−w2∗Lm∗Lr)2+(w∗Lr+w∗Lm−1w∗Cr)2∗Rac2G^2=\frac{w^2*Lm^2*Rac^2}{(\frac{Lm}{Cr}-w^2*Lm*Lr)^2+(w*Lr+w*Lm-\frac{1}{w*Cr})^2*Rac^2} G2=(CrLm−w2∗Lm∗Lr)2+(w∗Lr+w∗Lm−w∗Cr1)2∗Rac2w2∗Lm2∗Rac2
令
令斜率K=LmLr,Lm=KLr代入令斜率K=\frac{Lm}{Lr},Lm=KLr代入 令斜率K=LrLm,Lm=KLr代入
则
G2=w2∗K2∗Lr2∗Rac2(K∗LrCr−w2∗K∗Lr2)2+(w∗Lr+w∗K∗Lr−1w∗Cr)2∗Rac2G^2=\frac{w^2*K^2*Lr^2*Rac^2}{(\frac{K*Lr}{Cr}-w^2*K*Lr^2)^2+(w*Lr+w*K*Lr-\frac{1}{w*Cr})^2*Rac^2} G2=(CrK∗Lr−w2∗K∗Lr2)2+(w∗Lr+w∗K∗Lr−w∗Cr1)2∗Rac2w2∗K2∗Lr2∗Rac2
分子分母同乘(wCr)的平方,则
G2=w4∗K2∗Lr2∗Cr2∗Rac2(w∗K∗Lr−w3∗K∗Lr2∗Cr)2+(w2∗Lr∗Cr+w2∗K∗Lr∗Cr−1)2∗Rac2G^2=\frac{w^4*K^2*Lr^2*Cr^2*Rac^2}{(w*K*Lr-w^3*K*Lr^2*Cr)^2+(w^2*Lr*Cr+w^2*K*Lr*Cr-1)^2*Rac^2} G2=(w∗K∗Lr−w3∗K∗Lr2∗Cr)2+(w2∗Lr∗Cr+w2∗K∗Lr∗Cr−1)2∗Rac2w4∗K2∗Lr2∗Cr2∗Rac2
分子分母同除Rac平方,则
G2=w4∗K2∗Lr2∗Cr2(w∗K∗Lr−w3∗K∗Lr2∗Cr)2Rac2+(w2∗Lr∗Cr+w2∗K∗Lr∗Cr−1)2G^2=\frac{w^4*K^2*Lr^2*Cr^2}{\frac{(w*K*Lr-w^3*K*Lr^2*Cr)^2}{Rac^2}+(w^2*Lr*Cr+w^2*K*Lr*Cr-1)^2} G2=Rac2(w∗K∗Lr−w3∗K∗Lr2∗Cr)2+(w2∗Lr∗Cr+w2∗K∗Lr∗Cr−1)2w4∗K2∗Lr2∗Cr2
令归一化频率X=fsfr=wwr令归一化频率X=\frac{f_s}{f_r}=\frac{w}{w_r} 令归一化频率X=frfs=wrw
由于fr=12πLrCr,则wr=1LrCr,LrCr=1wr2由于f_r=\frac{1}{2\pi \sqrt{LrCr}},则w_r=\frac{1}{\sqrt{LrCr}},LrCr=\frac{1}{w_r^2} 由于fr=2πLrCr1,则wr=LrCr1,LrCr=wr21
代入上式:
G2=w4wr4∗K2(1−w2wr2)2∗K2∗w2∗Lr2∗wr2Rac2∗wr2+(w2wr2(1+K)−1)2G^2=\frac{\frac{w^4}{w_r^4}*K^2}{\frac{(1-\frac{w^2}{w_r^2})^2*K^2*w^2*Lr^2*w_r^2}{Rac^2*w_r^2}+(\frac{w^2}{w_r^2}(1+K)-1)^2} G2=Rac2∗wr2(1−wr2w2)2∗K2∗w2∗Lr2∗wr2+(wr2w2(1+K)−1)2wr4w4∗K2
G2=X4∗K2(1−X2)2∗K2∗X2∗Lr2∗wr2Rac2+(X2∗(1+K)−1)2G^2=\frac{X^4*K^2}{\frac{(1-X^2)^2*K^2*X^2*Lr^2*w_r^2}{Rac^2}+(X^2*(1+K)-1)^2} G2=Rac2(1−X2)2∗K2∗X2∗Lr2∗wr2+(X2∗(1+K)−1)2X4∗K2
品质因数Q=2∗π∗fr∗LrRac=wr∗LrRac品质因数Q=\frac{2* \pi *f_r*Lr}{Rac}=\frac{w_r*Lr}{Rac} 品质因数Q=Rac2∗π∗fr∗Lr=Racwr∗Lr
则
G2=w4wr4∗K2(1−w2wr2)2∗K2∗w2∗Lr2Rac2∗wr2+(w2wr2(1+K)−1)2G^2=\frac{\frac{w^4}{w_r^4}*K^2}{\frac{(1-\frac{w^2}{w_r^2})^2*K^2*w^2*Lr^2}{Rac^2*w_r^2}+(\frac{w^2}{w_r^2}(1+K)-1)^2} G2=Rac2∗wr2(1−wr2w2)2∗K2∗w2∗Lr2+(wr2w2(1+K)−1)2wr4w4∗K2
G2=X4∗K2(1−X2)2∗K2∗X2∗Q2+(X2∗(1+K)−1)2G^2=\frac{X^4*K^2}{(1-X^2)^2*K^2*X^2*Q^2+(X^2*(1+K)-1)^2} G2=(1−X2)2∗K2∗X2∗Q2+(X2∗(1+K)−1)2X4∗K2
则
G=X2∗K(1−X2)2∗K2∗X2∗Q2+(X2∗(1+K)−1)2G=\frac{X^2*K}{\sqrt{(1-X^2)^2*K^2*X^2*Q^2+(X^2*(1+K)-1)^2}} G=(1−X2)2∗K2∗X2∗Q2+(X2∗(1+K)−1)2X2∗K
G=1(1X2−1)2∗Q2+(1K+1−1X2∗K)2G=\frac{1}{\sqrt{(\frac{1}{X^2}-1)^2*Q^2+(\frac{1}{K}+1-\frac{1}{X^2*K})^2}} G=(X21−1)2∗Q2+(K1+1−X2∗K1)21