Exercises WS2010

8/6/2019 Exercises WS2010

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Prof. Dr.-Ing. Elmar Engels FEE Exercise

FB Elektrotechnik u. InformationstechnikFH Aachen

Exercise 10:

Known: U = 12 V; R1 = 20 ΩΩΩΩ, R2 = 30 ΩΩΩΩ, R3 = 10 ΩΩΩΩ. L = 3 mH, C = 5 nF .

Problem: a) Simplify the mentioned circuit if the voltage U is a DC voltage.b) How many unkown currents and voltages do we have in this circuit?c) So how many equations do we need to solve these variables?d) How many equations are contributed by Ohm’s law?e) How many (independent) equations can be derives using KCL?

f) How many (independent) equations can be derives using KVL?g) Write down the equations derived with d), e), f) and enumerate them.h) Solve the equation system and determine all currents and voltages.

R3

R1

C

R2

L

U





Exercise 11:

Known: U = 12 V; R1 = 20 ΩΩΩΩ, R2 = 30 ΩΩΩΩ, R3 = 10 ΩΩΩΩ, R4 = 50 ΩΩΩΩ.

Problem: a) Simplify the given circuit if the voltage U = V0 is a DC voltage after all dynamiceffects are stable.b) You just want to determine the current I1 that flows through the source voltage U.What is the easiest way?c) Determine current I1.





Exercise 12: Voltage Divider

The given circuit has an input voltage U. Between the terminals A and B we want to

generate1. a voltage UL with an open loop as load,2. a short-cut current IK with a closed loop.

Known: U = 230 V; UL = 57,5 V, IK = 11,5 A .

Problem: a) Determine the resistor values R1 and R2.b) Determine the power supplied by the voltage source U in the case of the external

short-cut and the open loop.

R2

U

R = RAB

R1A

B

U2





Exercise 13: Voltage Divider

The given circuit has an input voltage U.

Known: U = 230 V; R1 = 20 ΩΩΩΩ, R2 = 5 ΩΩΩΩ.

Problem: a) There is a load connected between the terminals A and B. The dissipated power

has to be P=100W. Determine the needed resistance RAB.

b) The maximum possible load shall be drained by the external load. Determine theresistance RAB in this case.c) Determine the power supplied by the voltage source in case b).d) Draw the curve of the power P as function of the ratio R2 /RAB.

0

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100

150

0 1 2 3 4 5

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R1A

B

U2





Exercise 14: Voltage DividerThe following circuit is given. The input voltage to this circuit is U.

Known: U = 230 V; R1 = 20 ΩΩΩΩ, R2 = 5 ΩΩΩΩ.

Problem: a) The terminals A and B will be short-cut (RAB=0 Ohm). Determine the short-cut

current IK that flows through the short-cut wire.b) No external load is connected to the terminals A und B. Determine the voltage U0

between terminal A and B.c) Determine the inner resistance Ri of the supply circuit that will be “seen” from an

external load.d) Determine the inner resistance Ri by short cutting the voltage source U.e) Proof that the maximum power that can be supplied to an external load is reached

if the external resistance equals the inner resistance of the supply circuit. Explainthe term impedance matching.

f) Determine the maximum power Pmax the supply circuit can supply to an externalload.

g) Determine the value of the external resistor RPmax that drains the maximum

power.

R2

U

R = RAB

R1A

B

U2





Exercise 15: Voltage DividerThe given voltage divider circuit has an input voltage U. An external load resistor isconnected to the terminals A and B. The voltage between the terminal shall be p*U2

while U2 is the open circuit voltage.

Known: U = 60 V; R1 = 100 ΩΩΩΩ, R2 = 300 ΩΩΩΩ; p = 95 % .

Problem: a) The open circuit voltage U2 between the terminals A and B has to be determined.b) Now an external load is connected to the circuit and due to the load curve of the

supply circuit the voltage drops down to p*U2. Determine the resistor valueconsisting of R2 and RAB that causes this voltage drop.

c) Determine the resistor value of RAB.

R2

U

R = RAB

R1A

B

U2





Exercise 16:

We regard the following given DC circuit.

Known: U = 12 V; R1 = 20 ΩΩΩΩ, R2 = 30 ΩΩΩΩ, R3 = 10 ΩΩΩΩ.

Problem:

Solve the problem by applying Thévenin’s theorem as you are just interested in thecurrent I3.

a) Determine the voltage U0 between the terminals A and B that would be applied to

R3 if it would be an open circuit.b) Which inner resistance Ri would be seen from the load R3?c) Draw the resulting circuit and determine the current I3.

R3

R1

R2

I2

U

A

B

I1

I3





Exercise 17: Thévenin’s theoremThe following circuit is given:

Known: U = 12 V; R1 = 20 ΩΩΩΩ, R2 = 30 ΩΩΩΩ, R3 = 10 ΩΩΩΩ, R4 = 50 ΩΩΩΩ.

Problem:You are only interested in the voltage U3 between terminals A and B. Solve theproblem by applying Thévenin’s theorem.

a) Terminals A and B are short-cut. Determine the voltage drop U2K over resistor R2.b) Determine the short-cut current I3K through the resistor R3=0 Ohms.c) Determine the inner resistance Ri of the supply circuit that can be seen at the

terminals A and B.d) Draw the resulting circuit and determine the voltage U3 between terminals A and

B if R3 = 10Ohms.

R4

R1

R2

R3

U0

A B





Exercise 18: Voltage Divider (Thévenin’s theorem)The following circuit is given:

Known: U = 230 V; Rn = n••••10 ΩΩΩΩ (n = 1, 2, ..., 5).

Problem:You are only interested in the voltage U5 between terminals A and B. Solve theproblem applying Thévenin’s theorem.

a) Determine the voltage between terminals A and B if R5 is an open circuit.

b) Determine the inner resistance of the supply circuit that can be seen from theterminals A and B.

c) Draw the resulting circuit.d) Determine the voltage between terminals A and B if the resistance values are as

noted above.

R3

U

R1

R4

R5

R2A

B



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Exercise 19: A capacitor charged through a resistor

A capacitor is partially charged up to the voltage uc(t=0) and the switch is open when

starting the observation (t=0). At the time t=0 the switch will be closed and connectsthe voltage source with the series circuit consisting of the capacitor C and the resistorR.

Known: capacity C = 70 nF, resistance R = 10 kΩ,Ω,Ω,Ω, source voltage U = 200 V

Problem: a) Determine the equation that describes the capacitor voltage uc(t) for t>=0. b) Determine the equation that describes the current i(t).

c) Determine the time constant of the circuit τ. d) Determine the capacitor voltage as function of time for the two value uc1(t=0)=0V

and uc2(t=0)=20V.e) Draw the capacitor voltage curves uc(t)/U respectively.

R

uC(t=0)

C

S

U





Exercise 20: Inductor switch off

We regard the following circuit consisting of a voltage source U, the switch S, theresistor R and the inductor L. At t=0 the switch changes from position 1 to position 2(dashed).

Known: inductance L = 30 mH, resistance R = 500 Ω,Ω,Ω,Ω, voltage U = 12 V .

Problem:

For t <<<< 0(position

1)a) The value of the inductor current I1.

b) The value of inductor energy W1.

For t = 0 (immediately after moving the switch to position 2)c) Starting value of the inductor current I0 d) Starting value of the inductor voltage U0

For t >>>> 0 (position 2)e) Diagram of the time depending inductor voltage uL(t) f) Diagram of the ratio uL(t)/U

R

S

L

t = 0

1

2

U





Exercise 21: RL/RC-Parallel Circuit

The following circuit is given. It consists of an ideal voltage source that is connected

to two parallel branches. The first branch consists of a series connection of a resistorR1 and an inductor L. The second branch consists of a series connection of a resistorR2 and a capacitor C.

Initially the switch S is open and the energy inside the inductor as well as in thecapacitor is zero.

At t=0 the switch will be closed and the ideal voltage source supplies both branches.

Known: R1 = 100 ΩΩΩΩ;;;; R2 = 50 Ω;Ω;Ω;Ω; L = 9 mH; C = 1,8 µµµµF; U = 10 V .

Problem:

a) Determine the time constant of the first branch T1, the second branch T2 and

the ratio T1 /T2=τ.

b) Determine the current i1(t=+0) and i1(t ∞) through the inductor.

c) Determine the current i2(t=0) and i2(t ∞) through the capacitor.

d) Use the following diagram to draw the ratio i1(t)/i1(t ∞) and i2(t)/i2(t=0).

R2

S

C

R1

L

i2 i1

i(t)

U





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0,8

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Prof. Dr.-Ing. Elmar Engels FEE Exercise FB Elektrotechnik u. InformationstechnikFH Aachen

Exerise 22: Charging and discharging of a capacitor

A series circuit consisting of a resistor R and a capacitor C2 will be connected to

a capacitor C1 at t0=0. At t=0 the capacitor C1 is charged to the voltage U0=Q0 /C1.The voltage across the capacitor C2 at t<0 is 0V.

C1 u1(t) u2(t)

t0=0 R

C2

Discharging capacitor 1 andcharging capacitor 2

Known:

C1 = C2 = C = 50 µµµµF, U0 = 40 V.

The shape of the current i(t) through the resistor follows the curve given by the

following diagram. The current is normalized i(t)/I0. It follows an exponentialfunction and has a time constant T=RC/2.

Problem:a) Determine the time constant T.b) Determine the resistance R.c) Determine the current I0.d) Add the curve of the normalized voltage u2(t)/U0 in the diagram.




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Exercise 23: RC-series supplied with rectangular voltage sourceA series connection consisting of a resistor R and a capacitor C is connected to avoltage source that has a rectangular voltage curve. The capacitor is dischargedat t=0.

Known: Rectangular impulse:

for t < 10 ms and t > 30 ms: ue = 0

for 10 ms < t < 30 ms: Ue = 1,5 V.

R = 2 kΩ, C = 5µF.

Problem: a) Draw the rectangular impulse in the diagram.b) Determine the time constant T of this series circuit.c) Determine the voltage uc(t) across the capacitor.d) Draw the voltage uc(t) across the capacitor in the diagram.

e) Draw the normalized current ic(t) / Imax through the capacitor in the diagram(with Imax=Ue /R).

0

0,5

1

1,5

2

0 10 20 30 40

time t / ms





Exercise 24: Comparator, inverting

The operational amplifier shall be used to convert the input voltage ue(t). Theoperational amplifier itself is connected to the supply voltages UB+ and UB-.

Diagram of the input voltage ue(t)

Known: The operational amplifier can be considered ideal.

The supply voltage is UB+ = 15 V and UB-=0V.

Problem: Draw the output voltage ua(t) in the diagram above.

-15

-10

-5

05

10

15

0 0,5 1 1,5 v o l t a g e u

( t ) / V

normalized time t / T

ue ua

UB+

UB-





Exercise 25: inverting Operational Amplifier

The operational amplifier circuit shall be used to amplify the input voltage with a fixed

gain.

UaUe

R2

R1

Known: The operational amplifier can be considered as ideal.

Problem: Determine the following values of

a) R1 in order to rate the input resistance of the network to 5k.b) R2 in order to reach a closed loop gain of V=-200.





Exercise 26: Voltage source with comparator

The given circuit consists of two operational amplifiers connected in a cascade. Both

operational amplifiers can be considered as ideal. The output voltage of bothOpAmps is limited to the range -5..+5V.

Known:

V-Source: Ri = 500 kΩΩΩΩ .

for t <<<< T: u0(t) = - 2 V •••• t/T

for t >>>> T: periodic (i.e. u0(t+T) = u0(t)): „saw-shape“.OpAmp No. 1: R1 = 1 MΩΩΩΩ .

OpAmp No. 2: R2 = 10 kΩΩΩΩ , UV = 3 V .

Problem:

a) Draw the curve of u0(t) in the following diagram.

b) Draw the curve of u1(t) in the following diagram.

c) Draw the curve of u2(t) in the following diagram.

u2

u1

R1

i(t)

u(t) u0(t) R2

Ri

uv





-5

-4

-3

-2

-1

0

1

2

3

4

5

0 0,5 1 1,5 2 2,5

V o l t a g e

/ V

rel. time t/T



Prof. Dr.-Ing. Elmar Engels FEE Excercise


Exercise 27: Inverter with comparator

The given circuit consists of two operational amplifiers connected in a cascade. Both

OpAmps can be considered as ideal. The input current is labeled with ie(t). Theoutput voltage of both OpAmps is limited to ± 5 V.

Kown: OpAmp No. 1: R1 = 10 kΩΩΩΩ , R2 = 100 kΩΩΩΩ .

OpAmp No. 2: UV = 2 V .

Problem:a) Draw the characteristic curve u1 = u1(ie) in the following diagram.

b) Draw the characteristic curve ua = ua(ie) in the diagram.

ua u1

R2

ie

Uv ue

R1

-7,5

-5

-2,5

0

2,5

5

7,5

-80 -60 -40 -20 0 20 40 60 80

V o l t a g e

u ( i n

V )

Input current ie (in µµµµA)



Prof. Dr.-Ing. Elmar Engels FEE Excercise


Exercise 28: Integrator (inverting)

The operational amplifier is supplied by the voltage UB. The input voltage ue(t) is a

step function that changes at t = 0 from 0 V to +1 V.

ue ua

C

R

Kown: The OpAmp can be considered as ideal.

UB = ±±±± 10 V, R = 10 kΩ,Ω,Ω,Ω, C = 40 nF .

Problem: a) Determine the input current i(t).

b) Determine the capacitor voltage uC(t).

c) Determine the output voltage ua(t).

d) Draw the step response ua(t) in the right scale.

-10

-7,5

-5

-2,5

02,5

5

7,5

10

0 2 4 6 8 10

V o l a t a g e u

a ( i n

V )

Time t (in ms)





Exercise 29: Non-inverting Amplifier

The following circuit shall be used in order amplify a voltage. The operational

amplifier is supplied with a voltage UB. The voltage that shall be amplified is ue(t).

R2

R1

UaUe

Non-inverting amplifier

Known: The OpAmp can be considered as ideal.

UB = ±±±± 16 V, ue(t) = 120 µµµµV••••sin(2ππππ•••• 5 kHz •••• t).

R1 = 2,9 kΩ,Ω,Ω,Ω, R2 = 320 kΩ.Ω.Ω.Ω.

Problem: a) Determine the closed loog gain V,

b) Determine the output voltage ua(t) .





Exercise 30: OpAmp as comparator and trigger

Initially an operational amplifier is has no external components (refer to figure 1).It acts as comparator. The output voltage is either uA,min = 0 V or uA,max = UB .

Then, the two external components R1 and R2 will be connected to theoperational amplifier. Now it acts as trigger (refer to figure 2).

Figure 1: comparator Figure 2: trigger

Known:ideal OpAmp;

UB = 10 V; R1 = 10 kΩΩΩΩ, R2 = 100 kΩΩΩΩ; RV = 5 kΩΩΩΩ, R = 15 kΩΩΩΩ.

Problem:Part 1 = comparator operation:a) Determine the threshold voltage UV ?

b) Determine the equation uD = uD (u0, UV).c) Sketch the comparator function uA (u0) in the diagram.

Part 2 = trigger operation:e) Determine the equation for the working curve uA (uE, u0).

f) Sketch the four working curves in the uA (u0) in the diagram if

uE/V sequentially is 1,5 / 1,75 / 2,75 / 3,0.

g) Determine the two trigger thresholds Udown = uE,min and Uup =

uE,max.

h) Sketch the trigger function uA (uE) in the diagram.



0

5

10

0 1 2 3 4 5

u0/V and uE/V

uA(u0) / V and uA(uE) / V



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Exercise 37b: Active, reactive and complex power

A load with impedance Z is connected to a voltage source U and drains the current I.

Known: U = 230 V, I = 4,5 A and the current follows the voltage with a phase shiftof 30°.

Problem:a) Draw the pointer diagram for U and I ,

b) Determine the power factor cos φφφφ of that load.c) Determine the modulus of the active power, the reactive power and the apparentpower.d) What is the simplest way to realize the impedance Z?



Prof. Dr.-Ing. Elmar Engels Exercise FEE


Exercise 37c: Average value, rectified value and RMS value

A periodic current i(t) switches between the peak values Iu und Id. Theperiodicity is T and the corresponding frequency f=1/T.

Periodic current i(t)

Known: Iu = + 4 mA, Id = - 6 mA, T = 20 ms .

Problem: Determine

a) the average value Ia (= arithmetic average of i(t)),

b) the rectified value Ir (= arithmetic average of i(t)),

c) the RMS value IRMS

-8

-6

-4

-2

0

2

4

6

0 0,5 1 1,5

i ( t ) / m

A

t / T





Exercise 37d: RC combination A parallel RC connection is supplied with a sinoid AC voltage of the frequency f.

RC parallel circuit

Known: resistance R = 8 kΩΩΩΩ, capacity C = 10 nF,

voltage u(t) = U0 •••• cos(2ππππf••••t) with U0 = 24 V and f = 1 kHz.

Problem: Determine…

a) the currents iR(t), iC(t) and the total current i(t) by calculation in the timedomain.

b) the current IR(ω) and IC(ω) and I(ω) by using complex pointers. c) the pointer diagram including U and I. d) the phase between voltage U and current I.

RC

~

i iRiC

u





Exercise 37e: Series circuit => parallel circuit

For a fixed frequency f the circuits are identical.

R1C1

C2

R2

Known: frequency f = 50 Hz, resistance R1 = 2 kΩ,Ω,Ω,Ω, capacity C1 = 1 µµµµF

Problem: Determine

a) the impedance Z1 ,

b) the admittance Y1 ,

c) the admittance Y2 ,

d) the resistance R2,,,, capacity C2.





Exercise 37g: Impedance matching

A real harmonic voltage source with inner impedance Zi = Ri + jXi supplies a RLCcircuit Za = Ra + jXa.

Known: U = 50 V, f = 800 Hz.

Ri = 300 Ω, Xi = 200 Ω, L = 10 mH, R = Ra = 300 Ω, andC = ?

Problem:

a) Determine the total impedance connected to the ideal voltage source Ztotal(C) ?

b) Determine the modulus of the current I = U / Ztotal. c) Determine the capacity so that the modulus of the current is at maximum.

d) Determine the active power that is dissipated by the resistor Ra.





Exercise 37h: Compensation of reactive power

A harmonic voltage source (voltage U, frequency f) supplies a RLC circuit.

Known: U = 230 V, f = 50 Hz; R = 2 Ω, L = 20 mH, C = 460 µF.

Problem:

a) Determine the ratio v of the voltages UL and UR.b) Draw the pointer diagram containing U, UL and UR (scale: 1 cm ≅ 23 V)

c) Determine the modulus of UR from this diagram.

d) Determine the current IR. e) Draw the pointer diagram containing I, IR and IC (scale: 1 cm ≅ 10 A) f) Determine the component Iq of the total current I that has not the same

direction as the total voltage Ug) Determine the phase shift between the total voltage U and the total current I

L

C

R

~

UC

UL

UR

I IIC

U





Exercise 37i: RL/RC parallel circuit

We regard a black box that has two terminals. Between these terminals we canmeasure the voltage U and the current I through this box.

In this black box there are two parallel branches of a circuit as shown in the figure.

Known: U and I , R2 = R1 = R , L and C .

Probelm: Determine generally

a) the complex impedances Z1 and Z2 ,

b) the complex admittances Y1 and Y2 ,c) the complex total admittance Y ,d) the complex impedance Z .

e) Now the resistances R1 and R2 are equal and also the time constants T1 = L/R1

and T2 = R2••••C are equal. Determine the Z for this case?

Plausibility check:

d) generally for R1 / R2 = r ≠≠≠≠ 1 and T1 / T2 = ττττ ≠≠≠≠ 1 the impedance is

Z = R1••••(1 + ττττ + jωωωωT1 ••••( 1 – ττττ / (ωωωω2T12))) / (1 + r + jωωωωT1 ••••( r – ττττ / (ωωωω2T1

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I

R2 L

U



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Exercise 44: Transformer

In order to determine the values of the single-phase equivalent circuit of atransformer (U1N=400V, U2N=200V, S1N=340VA, cos ϕN =1) some measurements areperformed.

1. Passive measurement

R1 = 37,8 Ω, R2 = 15,5 Ω

2. Short-cut measurement on the secondary side:U1KS = 39,6 V, I1KS = 0,488 A

3. Open-loop measurement with measurement at the primary side

U10 = 228,7 V, I10 = 0,019 A, U20 = 140,5 V

4. Load at the secondary side:

ZV=170exp(j15°) ; U1 = U1N=400V

Problem:

a) Determine the nominal current IN b) Determine the transformation ratio N=((1+σ1)w1 /w2) and the inductance L1

from the measurement of the equivalent circuit (open-loop measurement).c) Determine the leakage coefficient based on the short-cut measurement.

(assume: jωL1 >> R2+jωL2σ)d) Determine the power factor and efficiency of the transformer for the load case

in item 4.



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Exercises WS2010