电路分析 ---- 加法器

1 同相加法器

分析过程

• 虚短：\(u_{+}=u_{-}=\cfrac{R_{G}}{R_{G}+R_{F}}u_{O}\)
• \(i_{1}=\cfrac{u_{I1}-u_{+}}{R_{1}}\)；\(i_{2}=\cfrac{u_{I2}-u_{+}}{R_{2}}\)；\(i_{3}=\cfrac{u_{I3}-u_{+}}{R_{3}}\)；且有\(i_{1}+i_{2}+i_{3}=0\).
• 所以得到\(\cfrac{u_{I1}}{R_{1}}+\cfrac{u_{I2}}{R_{2}}+\cfrac{u_{I3}}{R_{3}}=(\cfrac{1}{R_{1}}+\cfrac{1}{R_{2}}+\cfrac{1}{R_{3}})u_{+}\)
• 把\(u_{+}=\cfrac{R_{G}}{R_{G}+R_{F}}u_{O}\)代入上式中，得到\(\cfrac{u_{I1}R_{2}R_{3}+u_{I2}R_{1}R_{3}+u_{I3}R_{1}R_{2}}{R_{1}R_{2}R_{3}}=\cfrac{R_{2}R_{3}+R_{1}R_{3}+R_{1}R_{2}}{R_{1}R_{2}R_{3}}\cdot \cfrac{R_{G}}{R_{G}+R_{F}}\cdot u_{O}\)
• 变形得到\(u_{O}=(1+\cfrac{R_{F}}{R_{G}})\cfrac{u_{I1}R_{2}R_{3}+u_{I2}R_{1}R_{3}+u_{I3}R_{1}R_{2}}{R_{2}R_{3}+R_{1}R_{3}+R_{1}R_{2}}\)

当三个输入电阻相等，且\(R_{F}=2R_{G}\)时，此时输入输出关系为\(u_{O}=u_{I1}+u_{I2}+u_{I3}\)，这时这个加法器叫等权加法器

2 反相加法器

分析过程

• 虚短：\(u_{-}=u_{+}=0\)
• \(i_{1}=\cfrac{u_{I1}-u_{-}}{R_{1}}=\cfrac{u_{I1}}{R_{1}}\)
• \(i_{2}=\cfrac{u_{I2}-u_{-}}{R_{2}}=\cfrac{u_{I2}}{R_{2}}\)
• \(i_{3}=\cfrac{u_{I3}-u_{-}}{R_{3}}=\cfrac{u_{I3}}{R_{3}}\)
• \(i_{g}=\cfrac{u_{O}-u_{-}}{R_{G}}=\cfrac{u_{O}}{R_{G}}\)
• 根据电流关系\(i_{1}+i_{2}+i_{3}+i_{g}=0\)
• 得到输入输出关系：\(u_{O}=-(\cfrac{R_{G}}{R_{1}}u_{I1}+\cfrac{R_{G}}{R_{2}}u_{I2}+\cfrac{R_{G}}{R_{3}}u_{I3})\)