Consider the second order low pass filter circuit shown below.
The differential equation associated with the input voltage vi(t) and the output voltage vo(t) of this system can be written in the following form:
A⋅d2vodt2+B⋅dvodt+C⋅vo=vi
Where the coefficients A, B, and C can all be written in terms of the component values L, C and R
(a) What is the algebraic expression for coefficient A associated with the d2vodt2 term?
L*C
To find the algebraic expression for coefficient A associated with the d2vodt2 term, we need to analyze the given circuit and derive the transfer function that relates the output voltage vo(t) to the input voltage vi(t).
Step 1: Circuit Analysis
Let's analyze the given circuit to determine the transfer function. Based on the given statement, we have a second-order low-pass filter circuit.
Step 2: Derive the Transfer Function
To derive the transfer function, we can use the technique of circuit analysis based on Kirchhoff's voltage law and Kirchhoff's current law. The transfer function relates the Laplace transform of the output voltage, Vo(s), to the Laplace transform of the input voltage, Vi(s).
Step 3: Apply Laplace Transform and Solve
Assuming the initial conditions are zero, we can apply the Laplace transform to the differential equation you provided:
A⋅s^2Vo(s) + B⋅sVo(s) + C⋅Vo(s) = Vi(s)
Rearranging the equation, we can write it in terms of the Laplace domain:
Vo(s)(As^2 + Bs + C) = Vi(s)
Dividing both sides by (As^2 + Bs + C), we get:
Vo(s) = Vi(s) / (As^2 + Bs + C)
The expression within the parentheses (As^2 + Bs + C) is the characteristic equation of the transfer function.
Step 4: Find Coefficient A
The coefficient A in the differential equation is the coefficient of the d2vodt2 term. In the Laplace domain transfer function, this term corresponds to the coefficient of s^2.
Therefore, the algebraic expression for coefficient A can be written as:
A = LC
where L is the inductance and C is the capacitance in the circuit.
Hence, the algebraic expression for coefficient A associated with the d2vodt2 term is A = LC.