## PRACTICE PROBLEMS

1. Derive the equations for combined values for resistors, capacitors and inductors in series and parallel.

2. Find the output voltage as a function of input voltage.

3. Write the differential equation for the following circuit.

4. Consider the following circuit.

a) Develop a differential equation for the circuit.

b) Put the equation in state variable matrix form.

5. Consider the following circuit. Develop a differential equation for the circuit.

6. Find the input-output equation for the circuit below, and then find the natural frequency and damping factor.

7. a) Find the differential equation for the circuit below where the input is Vi, and the output is Vo.

b) Convert the equation to an input-output equation.

c) Solve the differential equation found in part b) using the numerical values given below. Assume at time t=0, the circuit has the voltage Vo and the first derivative shown below.

8.

a) Write the differential equations for the system pictured below.

b) Put the equations in input-output form.

9. Given the circuit below, find the ratio of the output over the input (this is also known as a transfer function). Simplify the results.

10. Examine the following circuit and then derive the differential equation.

11. Examine the following circuit and then derive the differential equation.

12.

a) Find the differential equation for the circuit below.

b) Put the differential equation in state variable form and a numerical method to produce a detailed sketch of the output voltage Vo. Assume the system starts at rest, and the input is Vi=5V.

13.

a) Write the differential equations for the system pictured below.

b) Put the equations in state variable form.

c) Use numerical methods to find the ratio between input and output voltages for a range of frequencies. The general method is put in a voltage such as Vi=1sin(___t), and see what the magnitude of the output is. Divide the magnitude of the output sine wave by the input magnitude. Note: This should act as a high pass or low pass filter.

d) Plot a graph of gain against the frequency of the input.

14. Find the transfer function for the system below.

15. Develop the differential equation(s) for the system below, and use them to find the response to the following inputs. Assume that the circuit is off initially.

16. Find the transfer functions for the system below where Vi is the input and Vo is the output.