CLLC谐振变换器的基波分析

CLLC谐振变换器_基波分析

目录

CLLC谐振变换器_基波分析 目录电路图FHA分析基于FHA的电路增益特性分析

电路图

FHA分析

输入电压FHA分析

谐振输入假设为理想方波

V i ( t ) = 4 V i n π ∑ n = 1 , 3 , 5... ∞ 1 n s i n ( 2 π n f s t ) V_{i}(t)=\frac{4Vin}{\pi}\sum_{n=1,3,5...}^{\infin}\frac{1}{n}sin(2\pi nf_st) Vi​(t)=π4Vin​∑n=1,3,5...∞​n1​sin(2πnfs​t) （方波的傅里叶变换）

基波提取

V i ( t ) = 4 V i n π s i n ( 2 π f s t ) V_i(t)=\frac{4Vin}{\pi}sin(2\pi f_st) Vi​(t)=π4Vin​sin(2πfs​t)

有效值提取

V i , F H A = 2 2 V i n π V_{i,FHA}=\frac{2\sqrt{2}Vin}{\pi} Vi,FHA​=π22 ​Vin​输出电压FHA分析

谐振输出也设为理想方波

V o ( t ) = 4 V o π ∑ n − 1 , 3 , 5... ∞ 1 n s i n ( 2 π n f s t − ϕ ) V_{o}(t)=\frac{4Vo}{\pi}\sum_{n-1,3,5...}^{\infty}\frac{1}{n}sin(2\pi nf_st-\phi) Vo​(t)=π4Vo​∑n−1,3,5...∞​n1​sin(2πnfs​t−ϕ)（ ϕ 为相对输入方波的相移 \phi 为相对输入方波的相移 ϕ为相对输入方波的相移）

基波提取

V o ( t ) = 4 V o π s i n ( 2 π f s − ϕ ) V_{o}(t)=\frac{4V_o}{\pi} sin(2\pi f_s-\phi) Vo​(t)=π4Vo​​sin(2πfs​−ϕ)

有效值提取

V o , F H A = 2 2 V o π V_{o,FHA}=\frac{2\sqrt{2}V_o}{\pi} Vo,FHA​=π22 ​Vo​​输出电流FHA分析

i ( t ) = 2 I r c t , F H A s i n ( 2 π f s t − ϕ ) i(t)=\sqrt{2}I_{rct,FHA}sin(2\pi f_st-\phi) i(t)=2 ​Irct,FHA​sin(2πfs​t−ϕ)

平均输出电流

I o = 2 T s ∫ 0 T s 2 ∣ i r c t , F H A ( t ) ∣ d t = 2 2 π I r c t , F H A I_o=\frac{2}{T_s}\int_0^{\frac{T_s}{2}}|i_{rct,FHA}(t)|dt=\frac{2\sqrt{2}}{\pi}I_{rct,FHA} Io​=Ts​2​∫02Ts​​​∣irct,FHA​(t)∣dt=π22 ​​Irct,FHA​输出负载FHA分析

I r c t , F H A = π 2 2 I o I_{rct,FHA}=\frac{\pi}{2\sqrt{2}}I_o Irct,FHA​=22 ​π​Io​

V o , F H A = 2 2 π V o V_{o,FHA}=\frac{2\sqrt{2}}{\pi}V_o Vo,FHA​=π22 ​​Vo​

R o , e q = V o , F H A I r c t , F H A = 8 π 2 R o R_{o,eq}=\frac{V_{o,FHA}}{I_{rct,FHA}}=\frac{8}{\pi^2}R_o Ro,eq​=Irct,FHA​Vo,FHA​​=π28​Ro​

基于FHA的电路增益特性分析

H r ( s ) = Z o ( s ) Z i n ( s ) R e q Z 2 ( s ) + R e q H_{r}(s)=\frac{Z_o(s)}{Z_{in}(s)}\frac{R_{eq}}{Z_2(s)+R_{eq}} Hr​(s)=Zin​(s)Zo​(s)​Z2​(s)+Req​Req​​

其中 { Z i n ( s ) = Z 1 ( s ) + Z o ( s ) Z o ( s ) = Z m ( s ) ( Z 2 ( s ) + R e q ) Z m ( s ) + Z 2 ( s ) + R e q Z 1 ( s ) = s L 1 + 1 s C 1 Z 2 ( s ) = s L 2 + 1 s C 2 Z m ( s ) = s L m \begin{cases} Z_{in}(s)=Z_1(s)+Z_o(s)\\ Z_o(s)=\frac{Z_m(s)(Z_2(s)+R_{eq})}{Z_m(s)+Z_2(s)+R_{eq}}\\ Z_1(s)=sL_1+\frac{1}{sC_1}\\ Z_2(s)=sL_2+\frac{1}{sC_2}\\ Z_m(s)=sL_m \end{cases} ⎩ ⎨ ⎧​Zin​(s)=Z1​(s)+Zo​(s)Zo​(s)=Zm​(s)+Z2​(s)+Req​Zm​(s)(Z2​(s)+Req​)​Z1​(s)=sL1​+sC1​1​Z2​(s)=sL2​+sC2​1​Zm​(s)=sLm​​

得

H r ( s ) = C 1 C 2 L m R s 3 C 1 C 2 L 1 L 2 s 4 + C 1 C 2 L 1 L m s 4 + C 1 C 2 L 1 R s 3 + C 1 C 2 L 2 L m s 4 + C 1 C 2 L m R s 3 + C 1 L 1 s 2 + C 1 L m s 2 + C 2 L 2 s 2 + C 2 L m s 2 + C 2 R s + 1 H_r(s)=\frac{{C_1}{C_2}{L_m} R s^3}{{C_1} {C_2} {L_1} {L_2}s^4+{C_1} {C_2} {L_1} {L_m} s^4+{C_1} {C_2} {L_1} Rs^3+{C_1} {C_2} {L_2} {L_m} s^4+{C_1} {C_2} {L_m} Rs^3+{C_1} {L_1} s^2+{C_1} {L_m} s^2+{C_2} {L_2}s^2+{C_2} {L_m} s^2+{C_2} R s+1} Hr​(s)=C1​C2​L1​L2​s4+C1​C2​L1​Lm​s4+C1​C2​L1​Rs3+C1​C2​L2​Lm​s4+C1​C2​Lm​Rs3+C1​L1​s2+C1​Lm​s2+C2​L2​s2+C2​Lm​s2+C2​Rs+1C1​C2​Lm​Rs3​

H r ( j w ) = C 1 C 2 j 3 L m R w 3 C 2 j w ( j w ( C 1 j 2 w 2 ( L m ( L 1 + L 2 ) + L 1 L 2 ) + L 2 + L m ) + C 1 j 2 R w 2 ( L 1 + L m ) + R ) + C 1 j 2 w 2 ( L 1 + L m ) + 1 H_r(jw)=\frac{{C_1} {C_2} j^3 {L_m} R w^3}{{C_2} j w \left(j w\left({C_1} j^2 w^2 ({L_m} ({L_1}+{L_2})+{L_1}{L_2})+{L_2}+{L_m}\right)+{C_1} j^2 R w^2({L_1}+{L_m})+R\right)+{C_1} j^2 w^2 ({L_1}+{L_m})+1} Hr​(jw)=C2​jw(jw(C1​j2w2(Lm​(L1​+L2​)+L1​L2​)+L2​+Lm​)+C1​j2Rw2(L1​+Lm​)+R)+C1​j2w2(L1​+Lm​)+1C1​C2​j3Lm​Rw3​

H r ( j w ) = − L m R w j ( C 1 C 2 w 4 ( L m ( L 1 + L 2 ) + L 1 L 2 ) − C 2 w 2 ( L 2 + L m ) C 1 C 2 w 2 − C 1 w 2 ( L 1 + L m ) − 1 C 1 C 2 w 2 ) + R ( C 1 w 2 ( L 1 + L m ) − 1 ) C 1 w H_r(jw)=-\frac{{L_m} R w}{j \left(\frac{{C_1} {C_2} w^4 ({L_m} ({L_1}+{L_2})+{L_1} {L_2})-{C_2} w^2 ({L_2}+{L_m})}{{C_1}{C_2} w^2}-\frac{{C_1} w^2 ({L_1}+{L_m})-1}{{C_1} {C_2} w^2}\right)+\frac{R \left({C_1} w^2({L_1}+{L_m})-1\right)}{{C_1} w}} Hr​(jw)=−j(C1​C2​w2C1​C2​w4(Lm​(L1​+L2​)+L1​L2​)−C2​w2(L2​+Lm​)​−C1​C2​w2C1​w2(L1​+Lm​)−1​)+C1​wR(C1​w2(L1​+Lm​)−1)​Lm​Rw​

H r ( j w ) = k w C 1 R j ( 1 C 1 L 1 w 2 − k − 1 g + C 1 L 1 w 2 ( h k + h + k ) − h − k ) + R ( C 1 ( k + 1 ) w − 1 L 1 w ) H_r(jw)=\frac{kwC_1R}{j \left(\frac{\frac{1}{{C_1} {L_1} w^2}-k-1}{g}+{C_1} {L_1} w^2 (h k+h+k)-h-k\right)+R \left({C_1} (k+1)w-\frac{1}{{L_1} w}\right)} Hr​(jw)=j(gC1​L1​w21​−k−1​+C1​L1​w2(hk+h+k)−h−k)+R(C1​(k+1)w−L1​w1​)kwC1​R​

H r ( j w ) = k w C 1 R R ( C 1 ( k + 1 ) w − 1 L 1 w ) + j ( − k + w 1 2 w 2 − 1 g + w 2 ( h k + h + k ) w 1 2 − h − k ) H_r(jw)=\frac{kwC_1R}{R \left({C_1} (k+1) w-\frac{1}{{L_1} w}\right)+j \left(\frac{-k+\frac{{w_1}^2}{w^2}-1}{g}+\frac{w^2 (hk+h+k)}{{w_1}^2}-h-k\right)} Hr​(jw)=R(C1​(k+1)w−L1​w1​)+j(g−k+w2w1​2​−1​+w1​2w2(hk+h+k)​−h−k)kwC1​R​

上下同除以 k w C 1 R 并且代入 R = Z 1 Q , Z 1 = L 1 C 1 , L 1 C 1 = w 1 , w 1 w s = w n , 得 上下同除以kwC1R并且代入R=\frac{Z_1}{Q},Z_1=\sqrt{\frac{L_1}{C_1}},\sqrt{L_1C_1}=w_1,\frac{w_1}{w_s}=w_n,得 上下同除以kwC1R并且代入R=QZ1​​,Z1​=C1​L1​​ ​,L1​C1​ ​=w1​,ws​w1​​=wn​,得

H r ( j w ) = − 1 j Q ( g h + g k + k + 1 ) C 1 g k w s L 1 C 1 + j Q w 1 2 C 1 g k w s 3 L 1 C 1 + j Q w s ( h k + h + k ) C 1 k w 1 2 L 1 C 1 − 1 C 1 k L 1 w s 2 + k + 1 k H_r(jw)=-\frac{1}{\frac{j Q (g h+g k+k+1)}{{C_1} g k {w_s}\sqrt{\frac{{L_1}}{{C_1}}}}+\frac{j Q {w_1}^2}{{C_1} g k{w_s}^3 \sqrt{\frac{{L_1}}{{C_1}}}}+\frac{j Q {w_s} (hk+h+k)}{{C_1} k {w_1}^2\sqrt{\frac{{L_1}}{{C_1}}}}-\frac{1}{{C_1} k {L_1}{w_s}^2}+\frac{k+1}{k}} Hr​(jw)=−C1​gkws​C1​L1​​ ​jQ(gh+gk+k+1)​+C1​gkws​3C1​L1​​ ​jQw1​2​+C1​kw1​2C1​L1​​ ​jQws​(hk+h+k)​−C1​kL1​ws​21​+kk+1​1​

H r ( j w n ) = − 1 − j Q ( g h + g k + k + 1 ) g k w n + j Q g k w n 3 + j Q w n ( h k + h + k ) k − 1 k w n 2 + k + 1 k H_r(jw_n)=-\frac{1}{-\frac{j Q (g h+g k+k+1)}{g k {w_n}}+\frac{j Q}{g k {w_n}^3}+\frac{j Q{w_n} (h k+h+k)}{k}-\frac{1}{k {w_n}^2}+\frac{k+1}{k}} Hr​(jwn​)=−−gkwn​jQ(gh+gk+k+1)​+gkwn​3jQ​+kjQwn​(hk+h+k)​−kwn​21​+kk+1​1​

整理得

H r ( j w n ) = 1 j Q k ( − g h + g k + k + 1 g w n + 1 g w n 3 + w n ( h k + h + k ) ) − 1 k w n 2 + k + 1 k H_r(jw_n)=\frac{1}{\frac{j Q}{k} \left(-\frac{g h+g k+k+1}{g {w_n}}+\frac{1}{g {w_n}^3}+{w_n} (hk+h+k)\right)-\frac{1}{k {w_n}^2}+\frac{k+1}{k}} Hr​(jwn​)=kjQ​(−gwn​gh+gk+k+1​+gwn​31​+wn​(hk+h+k))−kwn​21​+kk+1​1​

增益为

M ( w n ) = ∣ ∣ H r ( j w n ) ∣ ∣ = 1 ( j Q k ) 2 ( − g h + g k + k + 1 g w n + 1 g w n 3 + w n ( h k + h + k ) ) 2 + ( − 1 k w n 2 + k + 1 k ) 2 M(w_n)=||H_r(jw_n)||=\frac{1}{\sqrt{(\frac{j Q}{k})^2 \left(-\frac{g h+g k+k+1}{g {w_n}}+\frac{1}{g {w_n}^3}+{w_n} (hk+h+k)\right)^2+(-\frac{1}{k {w_n}^2}+\frac{k+1}{k})^2}} M(wn​)=∣∣Hr​(jwn​)∣∣=(kjQ​)2(−gwn​gh+gk+k+1​+gwn​31​+wn​(hk+h+k))2+(−kwn​21​+kk+1​)2 ​1​

令 { a = k + h + k h b = k + k g + h + 1 g c = 1 g \begin{cases} a=k+h+kh \\ b=k+\frac{k}{g}+h+\frac{1}{g}\\ c=\frac{1}{g} \end{cases} ⎩ ⎨ ⎧​a=k+h+khb=k+gk​+h+g1​c=g1​​

M ( w n ) = ∣ ∣ H r ( j w n ) ∣ ∣ = 1 ( j Q k ) 2 ( − b w n + c w n 3 + a w n ) 2 + ( − 1 k w n 2 + k + 1 k ) 2 M(w_n)=||H_r(jw_n)||=\frac{1}{\sqrt{(\frac{j Q}{k})^2 \left(-\frac{b}{ {w_n}}+\frac{c}{ {w_n}^3}+ a{w_n}\right)^2+(-\frac{1}{k {w_n}^2}+\frac{k+1}{k})^2}} M(wn​)=∣∣Hr​(jwn​)∣∣=(kjQ​)2(−wn​b​+wn​3c​+awn​)2+(−kwn​21​+kk+1​)2 ​1​