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Fixed typo
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Suba Thomas
Suba Thomas
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First off, the equations you have are not correct. For example, if equations 1 and 5 are added it results in $V_1=V_5$$V_1+V_5=0$. The l.h.s of the first four equations should be $V_1-V_5$, $V_2-V_6$, $V_3-V_7$, and $V_4-V_8$. Then you are have four equations in the four unknowns $X_1$, $X_2$, $X_3$, and $X_4$ after taking Laplace transforms with zero initial conditions.

The general method is to solve for these four unknowns and express them in terms of the $V_i$ and then you get the transfer functions for each input-output pair. The derivatives can then be obtained by multiplying with powers of the Laplace variable $s$.

Since you need just 4 transfer functions, I think it can be done a bit more easily.

For the first transfer function, you are interested only in $V_1$. Set all the other $V_i$ to zero and solve for $\frac{X_1}{V_1}$. Since you want the derivative just multiply the result by $s$.

For the second transfer function, set all the $V_i$ except $V_3$ to zero and solve for $\frac{X_2}{V_3}$. Multiply the result by $s^2$ to obtain the desired transfer function.

I hope you get the idea.

First off, the equations you have are not correct. For example, if equations 1 and 5 are added it results in $V_1=V_5$. The l.h.s of the first four equations should be $V_1-V_5$, $V_2-V_6$, $V_3-V_7$, and $V_4-V_8$. Then you are have four equations in the four unknowns $X_1$, $X_2$, $X_3$, and $X_4$ after taking Laplace transforms with zero initial conditions.

The general method is to solve for these four unknowns and express them in terms of the $V_i$ and then you get the transfer functions for each input-output pair. The derivatives can then be obtained by multiplying with powers of the Laplace variable $s$.

Since you need just 4 transfer functions, I think it can be done a bit more easily.

For the first transfer function, you are interested only in $V_1$. Set all the other $V_i$ to zero and solve for $\frac{X_1}{V_1}$. Since you want the derivative just multiply the result by $s$.

For the second transfer function, set all the $V_i$ except $V_3$ to zero and solve for $\frac{X_2}{V_3}$. Multiply the result by $s^2$ to obtain the desired transfer function.

I hope you get the idea.

First off, the equations you have are not correct. For example, if equations 1 and 5 are added it results in $V_1+V_5=0$. The l.h.s of the first four equations should be $V_1-V_5$, $V_2-V_6$, $V_3-V_7$, and $V_4-V_8$. Then you are have four equations in the four unknowns $X_1$, $X_2$, $X_3$, and $X_4$ after taking Laplace transforms with zero initial conditions.

The general method is to solve for these four unknowns and express them in terms of the $V_i$ and then you get the transfer functions for each input-output pair. The derivatives can then be obtained by multiplying with powers of the Laplace variable $s$.

Since you need just 4 transfer functions, I think it can be done a bit more easily.

For the first transfer function, you are interested only in $V_1$. Set all the other $V_i$ to zero and solve for $\frac{X_1}{V_1}$. Since you want the derivative just multiply the result by $s$.

For the second transfer function, set all the $V_i$ except $V_3$ to zero and solve for $\frac{X_2}{V_3}$. Multiply the result by $s^2$ to obtain the desired transfer function.

I hope you get the idea.

Source Link
Suba Thomas
Suba Thomas
  • 2k
  • 10
  • 14

First off, the equations you have are not correct. For example, if equations 1 and 5 are added it results in $V_1=V_5$. The l.h.s of the first four equations should be $V_1-V_5$, $V_2-V_6$, $V_3-V_7$, and $V_4-V_8$. Then you are have four equations in the four unknowns $X_1$, $X_2$, $X_3$, and $X_4$ after taking Laplace transforms with zero initial conditions.

The general method is to solve for these four unknowns and express them in terms of the $V_i$ and then you get the transfer functions for each input-output pair. The derivatives can then be obtained by multiplying with powers of the Laplace variable $s$.

Since you need just 4 transfer functions, I think it can be done a bit more easily.

For the first transfer function, you are interested only in $V_1$. Set all the other $V_i$ to zero and solve for $\frac{X_1}{V_1}$. Since you want the derivative just multiply the result by $s$.

For the second transfer function, set all the $V_i$ except $V_3$ to zero and solve for $\frac{X_2}{V_3}$. Multiply the result by $s^2$ to obtain the desired transfer function.

I hope you get the idea.