Engineering Question - Calculating Coefficients for 2nd Order Transient Analysis

[damianpd]

Homework Statement

Calculating K1 and K2 coefficients for second order transient analysis.

Relevant Equations

v(0), v'(0), i(0), i'(0)

Hello everyone,

I am struggling with calculating the coefficients for second order transient analysis.

For example, when analyzing a underdamped circuit, we know that the equation for voltage or current is x_t=e^-αt(K₁cos(sqrt(ω²-α²)t ) + K₂sin(sqrt(ω²-α²)t)).

Then in order to determine for example v_(t), I would first determine what v₍₀₎ is, and plugging in 0 for all values of t in the above equation, I would get the result v₍₀₎=K₁.

Then the next step in calculating K₂, I need to take the derivative of v₍₀₎ in order to determine the coefficient.

However, when following these steps, my solution isn't always correct when comparing it to the book solutions. Sometimes they utilize v₍₀₎ and v'₍₀₎, and other times they use v₍₀₎ and i₍₀₎ to determine these coefficients.

Some examples:
https://ibb.co/233Zd0q <---- In this they use i₍₀₎ and i'₍₀₎

https://ibb.co/DMyDfM4 <---- In this, they use v₍₀₎ and i₍₀₎

I would greatly appreciate some sort of explanation regarding this.

 

[damianpd]

Edit:
I have added my work for the second problem, in which I attempt to get K₁ and K₂ using v₍₀₎ and i₍₀₎, as well as i₍₀₎ and i'₍₀₎.

When doing it the first method, I get K₁=-4 and K₂=22, while the second method I get K₁=4 and K₂=-7.

Method 1: https://ibb.co/XL4CY11 Method 2: https://ibb.co/FxnYxPk

 

[Delta2]

You use the same constants $K_1$ and $K_2$ in both the equations for the current and the voltage (at the ends of the capacitor). This is not correct. If we use the constants $K_1$ and $K_2$ for voltage, then we should use the constants $K_1'$ and $K_2'$ for the current. There is a relationship between the two pairs of constants which you can find out by exploring the equation $$i(t)=C\frac{dv(t)}{dt}$$.
In your first method you calculate the pair of constants for the voltage, but in the second method you calculate the pair of constants for the current. That's why those two pairs differ.