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The networkThe following circuit shows the leadlag network which creates 2 poles and 2 zeroes:SolutionThe$V_{in}$ to $V_{out}$ transfer function can be written as:$$V_{out}/V_{in} = { {(R_3+1/{sC_2})R_2}/{R_3+1/{sC_2}+R_2} }/{ {(R_3+1/{sC_2})R_2}/{R_3+1/{sC_2}+R_2}+R_1/{sR_1C_1+1} } $$ This can be simplified to: $$V_{out}/V_{in} = { (sR_3C_2+1)R_2(sR_1C_1+1) }/{(sR_3C_2+1)R_2(sR_1C_1+1)+R_1(sC_2(R_2+R_3)+1)}$$ Zeroes and PolesSo we see the 2 zeros created are:$$z_1=1/{R_3C_2}$$ $$z_2=1/{R_1C_1}$$ The 2 poles can be calculated if we solve the denominator which can we written as: $$s^2R_1R_2R_3C_1C_2+s(R_1R_2C_1+R_2R_3C_2+R_1C_2(R_2+R_3))+R_1+R_2=0$$ or $$s^2R_3C_1C_2R_1R_2+s(R_1R_2(C_1+C_2)+R_3C_2)+1=0$$ If we substitute the following: $$A=R_3C_1C_2R_1R_2$$ $$B=R_1R_2(C_1+C_2)+R_3C_2$$ we can write (3) as: $$As^2+Bs+1=0$$ We can solve for s as: $$s={B±√{B^24A} }/{2A}$$ $$s=B/{2A}±B/{2A}√{1{4A}/B^2}$$ Now to simplify further we need to see the relative magnitudes to decide what terms can be neglected. Comparing A and B we can clearly see that if the circuit time constants are lower than even a milli second A ≪ B. That is because A is a product of 2 time constants while B is a time constant. Usually the network is designed to have 1 time constant much smaller than the other. So consider the case when one time constant is 100 times larger than the other. For example $R_3C_2 ≫ R_1R_2C_1$ . This means that the magnitude of $A/B^2$ = 100/10000 or $4A/B^2=4/100$ . So $B^2$ is still 25 times larger than 4A. So from this point on the assumption and approximation that we take is: One time constant in the circuit is more dominant than the other one. So now we can approximate the poles now as: $$s≈B/{2A}±B/{2A}(1{2A}/B^2)$$ $$p_1≈1/B ; p_2≈B/A$$ So substituting the values of A and B in we get the final poles of the circuit as: $$p_1≈1/{R_1R_2(C_1+C_2)+R_3C_2}$$ $$p_2≈{R_1R_2(C_1+C_2)+R_3C_2}/{R_3C_1C_2R_1R_2}$$ If $C_1≪C_2$ and $R_3≪R_1R_2$ then we can approximate the poles as:$$p_1≈1/{R_1R_2C_2}$$ $$p_2≈1/{R_3C_1}$$ See Also



Copyright 2018 Milind Gupta 