The characteristic equation for a 2^nd order system y(t) + a_1y(t) + a_0y(t) = b

Question

The characteristic equation for a 2^nd order system y(t) + a_1y(t) + a_0y(t) = b_2x(t) + b_1x(t) + b_0x(t) is written in either of the forms lambda^2 + a_1 lambda + a_0 = 0 lambda^2 + 2 zeta omega_c lambda + omega_c^2 = 0 This notation will be more thoroughly introduced in Exercise 1 before we start looking into the Sallen-Key filters. In the numerical calculations zeta = 1/Squareroot 2 omega_c = 2 pi f_c = 80 pi s^-1 = 251.3 s^-1 General considerations on a system with systems equation with real coefficients y(t) + a_1y(t) + a_0y(t) = b_2x(t) + b_1x(t) + b_0x(t) with real coefficients a_0 and a_1. If the input is given in units of [x] = V, [t] = s, and b_2 is dimensionless, what is the dimension of y, a_0, a_1, b_0, and b_1.

Explanation / Answer

Consider RHS:

It has 3 terms: b2*d2(x)/dt2 , b1*dx/dt, b0*x

The dimensions of each these terms should be same. Given b2 is dimensionless. Therefore the dimension of b2*d2(x)/dt2 will be V/s2.

The dimension of dx/dt = V/s. To get V/s2 for the term b1*dxt/dt, b1 should have the dimension 1/s

The dimension of x = V. To get the dimension of V/s2 to b0*x, b0 should be 1/s2.

Similarly, the dimension of each term on the LHS should be V/s2.

Therefore d2(y)/dt2 should have the unit of V/s2. Which implies y should have the unit of V since t has the unit of s.

Now a1* dy/dt should have the unit of V/s2. Since dy/dt will have the unit of V/s, a1 should have the unit of 1/s.

Similarly, a0*y should have the unit of V/s2. Since y has the unit of V, a0 should have the unit of 1/s2.

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