1. Application of the Laplace Transform
Part 3

2. 6.5. Circuit Applications Steps in applying the Laplace Transform;
Transform the circuit from the time-domain to the s-domain.
Solve the circuit using nodal analysis, mesh analysis, source transformation, superposition or any other analysis technique with which we are familiar
Take the inverse Laplace transform of the solution and thus obtain the solution in the time domain
How we may transform the circuit to the s-domain?
For resistor;
Laplace transform

3. For inductor; For capacitor; Laplace transform veya Laplace transform

4. Assuming initial conditions are zero;
Example 1: Find 𝑣 0 (t) in the circuit assuming that the initial conditions are zero.

6. Example 2: In the circuit, 𝑣 0 0 =5𝑉 find 𝑣 0 (t).

7. Apply KCL;
Multiply with 10;
Partial fraction expansion;

8. Example 3: In the circuit, find i(t) for t>0.
The initial current in the inductor
Mesh analysis;

9. Partial fraction expansion;
The last value;

10. 6.6. Transfer Function Eğer; veya
The response in that condition is referred to as unit impulse response;

11. Example 4: Circuit’s ouput;
Input;
Find the circuit’s transfer function and unit impulse response.
Solution: First find the Laplace transform of x(t) and y(t).

12. Example 5: Find the transfer function H(s)= 𝑉 0 (𝑠)/ 𝐼 0 𝑠 of the circuit.
1. YOL: Current division;

13. 2. YOL: Using ladder method assuming that 𝑉 0 =1V. Then 𝐼 2 ;
The voltage in 2+1/2s impedance;
𝑉 1 voltage is the same as the voltage in s+4 impedance.

14. Example 6: 𝑥 𝑡 = 𝑒 −3𝑡 𝑢 𝑡 →𝑋 𝑠 = 1 𝑠+3
𝑌 𝑠 =𝑋 𝑠 𝐻 𝑠 = 2𝑠 (𝑠+6)(𝑠+3) =− 2 𝑠 𝑠+6
𝑦 𝑡 =−2 𝑒 −2𝑡 +4 𝑒 −6𝑡
𝐻 𝑠 = 2𝑠 𝑠+6 =𝐴+ 𝐵 𝑠+6 =2+ −12 𝑠+6 →ℎ 𝑡 =2δ 𝑡 −12 𝑒 −6𝑡

15. Example 7: