VGU - EIT - Exam: Transients and Transforms of Electric Circuits Instructor: Prof. Dr. W. Kuehn Date of Exam: Nov. 19, 2010

students family name: registration #: given name:

points achieved: grade:

points 60 66 72 78 84 90 96 102 108 114

grade 4 3,7 3,3 3,0 2,7 2,3 2,0 1,7 1,3 1,0

- no supporting material permitted except for correspondence tables -

1-2

Problem #1

A periodic square wave voltage of cycle time T = 20 ms and an amplitude U = 100 V is

connected to a series RC-circuit with R = 1 k and C = 1 F. 1. Compute the rms-values of the fundamental sinusoidal current and of the third

harmonic current using the following formula for the voltage: u(n) = 4 U/( n) 2. What is the rms-value of the total current neglecting all harmonics above n = 3? 3. Determine the real power consumption of the resistor.

Problem #2

A high pass filter comprising a resistor (R = 1 M) and a capacitor (C = 1 F) is connected to a DC voltage source (U = 100 V) at t = 0 s. The capacitor carries a charge of

0.5 x 10-4

C at that time.

1. Write down the DEQ containing the capacitor voltage and its derivative and specify the initial condition for the given circuit.

2. Determine the Laplace-Transform. 3. Compute the inverse Laplace-Transform of the capacitor voltage and determine

also the voltage over time across the resistor.

4. Plot quantitatively the resistor voltage.

Problem #3

3.1 Given is a series RLC-resonance circuit with C = 1 nF and natural resonance

frequency fn = 1 MHz.

1. Determine the inductivity L of the coil (LC

n

1 ).

2. Determine the size of the resistor for a damping ratio = 0.7 = /n where

= R/2L

3. Determine the poles p1,2 = - + 22

n and mark them in the s-plane.

3.2 The circuit of 3.1 will be connected to a DC voltage U = 100 V at t = 0 s.

The initial conditions at t = 0 s are: the capacitor carries zero charge, the current is

also zero.

4. Determine the DEQ for the capacitor voltage. 5. Determine the Laplace-Transform UC(p) of the capacitor voltage. 6. Determine the inverse Laplace-Transform L-1[(UC(p)] by using Heaviside. 7. Insert the above calculated values for p1 and p2 in the last equation and determine

the capacitor voltage uC(t).

5

1

2

8

5

1

3

8

2

1

20

2

2

6

10

4

4

10

6

24

2-2

Problem #4

1. For a short circuited series RLC circuit determine the First Order DEQ-system by using the graphical approach as demonstrated in the lecture.

The data are: C = 1 nF, L = 25 H, R = 222 Ohm. The capacitor carries a charge corresponding to 100 V at the occurrence of the short circuit.

2. Determine the Laplace-Transform of the DEQ-system (use the variable s instead of p).

3. Determine the Eigen-frequencies and compare them with the poles as determined in problem 3.1.

Problem #5

Apply the input voltage of 100 V [1(t) 1(t 1 s)] to a low pass filter comprising a

resistor (1 M) and a capacitor (1 F). 1. Determine the Fourier-Transform of the input voltage. 2. Now determine the Fourier-Transform of the output voltage. 3. Determine the time function of the output voltage using partial fraction expansion.

Problem # 6

Before the switch closes the current in the coil has reached steady state.

1. Determine the DEQ for the calculation of the current after closing the switch.

Determine the Laplace-Transform of the current.

2. Form the inverse Laplace-Transform of the current. 3. Draw the time function of the current.

Total points

10

4

6

20

4

6

10

20

5

6

6

1

18

120