Exercise 2.4Z: Characteristics Measurement - Revision history

Guenter at 13:29, 1 October 2021

2021-10-01T13:29:30Z

Guenter at 13:17, 1 October 2021

2021-10-01T13:17:24Z

Guenter at 10:10, 20 September 2021

2021-09-20T10:10:37Z

Guenter: Guenter moved page Aufgabe 2.4Z: Kennlinienvermessung to Exercise 2.4Z: Characteristics Measurement

2021-09-20T10:08:44Z

Guenter moved page Aufgabe 2.4Z: Kennlinienvermessung to Exercise 2.4Z: Characteristics Measurement

Oezer at 19:23, 15 September 2021

2021-09-15T19:23:37Z

Oezer at 18:55, 15 September 2021

2021-09-15T18:55:45Z

Oezer at 15:13, 15 September 2021

2021-09-15T15:13:49Z

Oezer at 15:04, 15 September 2021

2021-09-15T15:04:33Z

Javier: Text replacement - "Category:Exercises for Linear and Time-Invariant Systems" to "Category:Linear and Time-Invariant Systems: Exercises"

2021-03-23T12:40:55Z

Text replacement - "Category:Exercises for Linear and Time-Invariant Systems" to "Category:Linear and Time-Invariant Systems: Exercises"

Javier: Text replacement - "Category:Aufgaben zu Lineare zeitinvariante Systeme" to "Category:Exercises for Linear and Time-Invariant Systems"

2021-03-23T12:27:55Z

Text replacement - "Category:Aufgaben zu Lineare zeitinvariante Systeme" to "Category:Exercises for Linear and Time-Invariant Systems"

@@ Line 21: / Line 21: @@
 In contrast, a triangular pulse with amplitude&nbsp; $A_x = 3 \ {\rm V}$&nbsp; and the one-sided pulse duration&nbsp; $T_x = 2 \ {\rm ms}$&nbsp; is considered for the subtask&nbsp; '''(5)'''&nbsp;:
 :$$x(t) =  A_x \cdot ( 1 - {|t|}/{T_x})  $$
@@ Line 77: / Line 73: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; <u>Proposed solution 3</u> is the only correct one:
-*If the input pulse&nbsp;$x(t)$&nbsp; is rectangular, then&nbsp;$x^2(t)$&nbsp; is also a rectangle with height&nbsp;$A_x^2$&nbsp; between&nbsp; $0$&nbsp; and&nbsp; $T_x$; outside zero.
+*If the input pulse&nbsp;$x(t)$&nbsp; is rectangular, then&nbsp;$x^2(t)$&nbsp; is also a rectangle with height&nbsp;$A_x^2$&nbsp; between&nbsp; $0$&nbsp; and&nbsp; $T_x$,&nbsp; outside zero.
 *The overall output signal&nbsp;$y(t)$&nbsp; is thus also rectangular with the amplitude
 :$$A_y= c_1 \cdot A_x + c_2 \cdot A_x^2 .$$
@@ Line 104: / Line 100: @@
 '''(3)'''&nbsp; The specification of a distortion factor requires the use of a harmonic oscillation at the input.
-*If&nbsp; $X_+(f) = 1 \ {\rm V} \cdot \delta (f - f_0)$ holds, then the spectrum of the analytic signal at the output is:
+*If&nbsp; $X_+(f) = 1 \ {\rm V} \cdot \delta (f - f_0)$&nbsp; holds,&nbsp; then the spectrum of the analytic signal at the output is:
 :$$ Y_{+}(f)={c_2}/{2}\cdot A_x^2 \cdot \delta(f) + c_1\cdot A_x \cdot \delta(f- f_0)+ {c_2}/{2}\cdot A_x^2 \cdot \delta(f- 2 f_0). $$
@@ Line 121: / Line 117: @@
   {c_2}\cdot A_x^2 \cdot \left( 1 - {|\hspace{0.05cm}t\hspace{0.05cm}|}/{T_x}\right)^2.$$
-*The following values occur at time&nbsp;$t = 0$&nbsp; and &nbsp;$t = T_x/2$&nbsp;:
+*The following values occur at times&nbsp;$t = 0$&nbsp; and &nbsp;$t = T_x/2$:
 :$$y(t=0) = c_1\cdot A_x  + {c_2}\cdot A_x^2  \hspace{0.15cm}\underline{= 1.95\,{\rm
   V}},$$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID898__LZI_Z_2_4.png|right|frame|Given characteristic curve&nbsp; $y = g(x)$]]
+[[File:P_ID898__LZI_Z_2_4.png|right|frame|Given characteristic&nbsp; $y = g(x)$]]
 It is known that the characteristic curve can be represented as follows for a nonlinear system:
 :$$y(t) =  c_1  \cdot x(t) + c_2  \cdot x^2(t).$$
@@ Line 9: / Line 9: @@
 Since the distortions are nonlinear no frequency response&nbsp; $H(f)$&nbsp; can be given.
-To determine the dimensionless coefficient&nbsp; $c_1$&nbsp; as well as the quadratic coefficient&nbsp; $c_2$&nbsp; different rectangular pulses&nbsp; $x(t)$&nbsp; &ndash; characterized by the amplitude&nbsp; $A_x$&nbsp; and width&nbsp; $T_x$&nbsp; &ndash; are now applied to the input and the pulse amplitude&nbsp; $A_y$&nbsp; at the output is measured in each case.
+To determine the dimensionless coefficient&nbsp; $c_1$&nbsp; as well as the quadratic coefficient&nbsp; $c_2$&nbsp; different rectangular pulses&nbsp; $x(t)$&nbsp; &ndash; characterized by the amplitude&nbsp; $A_x$&nbsp; and the width&nbsp; $T_x$&nbsp; &ndash; are now applied to the input and the pulse amplitude&nbsp; $A_y$&nbsp; at the output is measured in each case.
 The first three trials generate the following values:
@@ Line 31: / Line 31: @@
 ''Please note:''
-*The task belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion|Nonlinear Distortion]].
+*The exercise belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion|Nonlinear Distortions]].
 *The following abbreviations are used in the formulation of the questions:
@@ Line 55: / Line 55: @@
-{What distortion factor&nbsp; $K$&nbsp; is measured with the test signal&nbsp; $x(t) = 1 \hspace{0.08cm} {\rm V} \cdot \cos(\omega_0 \cdot t)$&nbsp;? That is: &nbsp; $\underline{A_x = 1\hspace{0.08cm} \rm  V}$.
+{Which distortion factor&nbsp; $K$&nbsp; is measured with the test signal&nbsp; $x(t) = 1 \hspace{0.08cm} {\rm V} \cdot \cos(\omega_0 \cdot t)$&nbsp;?&nbsp; That is: &nbsp; $\underline{A_x = 1\hspace{0.08cm} \rm  V}$.
 |type="{}"}
 $K \ = \ $  { 5 3% } $\ \%$
-{What distortion factor&nbsp; $K$&nbsp; is measured with the test signal&nbsp; $x(t) = 3 \hspace{0.08cm} {\rm V} \cdot \cos(\omega_0 \cdot t)$&nbsp;? That is: &nbsp; $\underline{A_x = 3\hspace{0.08cm} \rm  V}$.
+{Which distortion factor&nbsp; $K$&nbsp; is measured with the test signal&nbsp; $x(t) = 3 \hspace{0.08cm} {\rm V} \cdot \cos(\omega_0 \cdot t)$&nbsp;?&nbsp; That is: &nbsp; $\underline{A_x = 3\hspace{0.08cm} \rm  V}$.
 |type="{}"}
 $K \ = \ $ { 15 3% } $\ \%$
-{What output pulse&nbsp; $y(t)$&nbsp; arises as a result when the input pulse is triangular? What are the signal values at&nbsp; $ t = 0$&nbsp; and&nbsp; $ t = T_x/2$?
+{Which output pulse&nbsp; $y(t)$&nbsp; arises as a result when the input pulse is triangular?&nbsp; What are the signal values at&nbsp; $ t = 0$&nbsp; and&nbsp; $ t = T_x/2$?
 |type="{}"}
 $y(t = 0) \ = \ $ { 1.95 3% } $\ \rm V$

← Older revision		Revision as of 10:10, 20 September 2021
Line 131:		Line 131:


−	[[Category:Linear and Time-Invariant Systems: Exercises\|^2.2 ~~Nichtlineare Verzerrungen~~^]]	+	[[Category:Linear and Time-Invariant Systems: Exercises\|^2.2 Nonlinear Distortions^]]

@@ Line 42: / Line 42: @@
 <quiz display=simple>
-{Am Eingang liegt ein Rechteckimpuls&nbsp; $x(t)$&nbsp; mit Amplitude&nbsp; $A_x$&nbsp; und Dauer&nbsp; $T_x$&nbsp; an.&nbsp; <br>Welche Aussagen gelten für den Ausgangsimpuls&nbsp; $y(t)$?
+{A rectangular pulse&nbsp; $x(t)$&nbsp; with amplitude&nbsp; $A_x$&nbsp; and duration&nbsp; $T_x$&nbsp; is applied to the input.&nbsp; <br>Which statements hold for the output pulse&nbsp; $y(t)$?
 |type="[]"}
-- Der Ausgangsimpuls&nbsp; $y(t)$&nbsp; ist dreieckförmig.
+- The output pulse&nbsp; $y(t)$&nbsp; is triangular in shape.
-- Die Amplituden am Eingang und Ausgang sind gleich &nbsp; &rArr; &nbsp; $A_y = A_x$.
+- The amplitudes at the input and output are the same&nbsp; &rArr; &nbsp; $A_y = A_x$.
-+ Die Impulsdauer wird durch das System nicht verändert &nbsp; &rArr; &nbsp; $T_y = T_x$.
++ The pulse duration is not changed by the system&nbsp; &rArr; &nbsp; $T_y = T_x$.
-{Berechnen Sie die beiden ersten Koeffizienten der Taylorreihe.
+{Compute the first two coefficients of the Taylor series.
 |type="{}"}
 $c_1 \ = \ $ { 0.5 3% }
@@ Line 55: / Line 55: @@
-{Welcher Klirrfaktor&nbsp; $K$&nbsp; wird mit dem Testsignal&nbsp; $x(t) = 1 \hspace{0.08cm} {\rm V} \cdot \cos(\omega_0 \cdot t)$&nbsp; gemessen? Das heißt: &nbsp; $\underline{A_x = 1\hspace{0.08cm} \rm  V}$.
+{What distortion factor&nbsp; $K$&nbsp; is measured with the test signal&nbsp; $x(t) = 1 \hspace{0.08cm} {\rm V} \cdot \cos(\omega_0 \cdot t)$&nbsp;? That is: &nbsp; $\underline{A_x = 1\hspace{0.08cm} \rm  V}$.
 |type="{}"}
 $K \ = \ $  { 5 3% } $\ \%$
-{Welcher Klirrfaktor&nbsp; $K$&nbsp; wird mit dem Testsignal&nbsp; $x(t) = 3 \hspace{0.08cm} {\rm V} \cdot \cos(\omega_0 \cdot t)$&nbsp; gemessen? Das heißt: &nbsp; $\underline{A_x = 3\hspace{0.08cm} \rm  V}$.
+{What distortion factor&nbsp; $K$&nbsp; is measured with the test signal&nbsp; $x(t) = 3 \hspace{0.08cm} {\rm V} \cdot \cos(\omega_0 \cdot t)$&nbsp;? That is: &nbsp; $\underline{A_x = 3\hspace{0.08cm} \rm  V}$.
 |type="{}"}
 $K \ = \ $ { 15 3% } $\ \%$
-{Welcher Ausgangsimpuls&nbsp; $y(t)$&nbsp; ergibt sich bei dreieckförmigem Eingangsimpuls? Wie lauten die Signalwerte bei&nbsp; $ t = 0$&nbsp; und&nbsp; $ t = T_x/2$?
+{What output pulse&nbsp; $y(t)$&nbsp; arises as a result when the input pulse is triangular? What are the signal values at&nbsp; $ t = 0$&nbsp; and&nbsp; $ t = T_x/2$?
 |type="{}"}
 $y(t = 0) \ = \ $ { 1.95 3% } $\ \rm V$
@@ Line 76: / Line 76: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Richtig ist nur der <u>Lösungsvorschlag 3</u>:
+'''(1)'''&nbsp; <u>Proposed solution 3</u> is the only correct one:
-*Ist der Eingangsimpuls &nbsp;$x(t)$&nbsp; rechteckförmig, so ist auch  &nbsp;$x^2(t)$&nbsp; ein Rechteck mit Höhe &nbsp;$A_x^2$&nbsp; zwischen&nbsp; $0$&nbsp; und&nbsp; $T_x$; außerhalb Null.
+*If the input pulse&nbsp;$x(t)$&nbsp; is rectangular, then&nbsp;$x^2(t)$&nbsp; is also a rectangle with height&nbsp;$A_x^2$&nbsp; between&nbsp; $0$&nbsp; and&nbsp; $T_x$; outside zero.
-*Auch das gesamte Ausgangssignal &nbsp;$y(t)$&nbsp; ist somit rechteckförmig mit der Amplitude
+*The overall output signal&nbsp;$y(t)$&nbsp; is thus also rectangular with the amplitude
 :$$A_y= c_1 \cdot A_x + c_2 \cdot A_x^2 .$$
-*Für die Impulsdauer gilt: &nbsp; $T_y  = T_x$.
+*The following holds for the pulse duration: &nbsp; $T_y  = T_x$.
-'''(2)'''&nbsp; Mit den beiden ersten Parametersätzen kann folgendes lineares Gleichungssystem angegeben werden:
+'''(2)'''&nbsp; The following system of linear equations can be specified with the first two sets of parameters:
 :$$c_1  \cdot 1\,{\rm V} + c_2 \cdot (1\,{\rm V})^2  = 0.55\,{\rm
   V},$$
@@ Line 90: / Line 90: @@
   V}.\hspace{0.05cm}$$
-*Durch Multiplikation der ersten Gleichung mit&nbsp; $-2$&nbsp; und Addition der beiden Gleichungen erhält man:
+*The following is obtained by multiplying the first equation by&nbsp; $-2$&nbsp; and adding the two equations:
 :$$c_2 \cdot 2\,{\rm V}^2  = 0.1\,{\rm
   V} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} c_2   \hspace{0.15cm}\underline{= 0.05\cdot{1/\rm
   V}}.$$
-*Der Linearkoeffizient ist somit&nbsp; $c_1   \hspace{0.15cm}\underline{= 0.5}.$
+*The linear coefficient is thus&nbsp; $c_1   \hspace{0.15cm}\underline{= 0.5}.$
-*Der dritte Parametersatz kann genutzt werden, um das Ergebnis zu kontrollieren:
+*The third set of parameters can be used to verify the result:
 :$$c_1  \cdot 3\,{\rm V} + c_2 \cdot (3\,{\rm V})^2  = 0.5 \cdot 3\,{\rm
   V}+ 0.05 \  {1}/{\rm V}\cdot 9\,{\rm V}^2  = 1.95\,{\rm
@@ Line 102: / Line 102: @@
-'''(3)'''&nbsp; Die Angabe eines Klirrfaktors bedingt die Verwendung einer harmonischen Schwingung am Eingang.
+'''(3)'''&nbsp; The specification of a distortion factor requires the use of a harmonic oscillation at the input.
-*Ist&nbsp; $X_+(f) = 1 \ {\rm V} \cdot \delta (f - f_0)$, so lautet das Spektrum des analytischen Signals am Ausgang:
+*If&nbsp; $X_+(f) = 1 \ {\rm V} \cdot \delta (f - f_0)$ holds, then the spectrum of the analytic signal at the output is:
 :$$ Y_{+}(f)={c_2}/{2}\cdot A_x^2 \cdot \delta(f) + c_1\cdot A_x \cdot \delta(f- f_0)+ {c_2}/{2}\cdot A_x^2 \cdot \delta(f- 2 f_0). $$
-*Die Diracfunktion bei&nbsp; $f = 0$&nbsp;  folgt aus der trigonometrischen Umformung&nbsp; $\cos^2(\alpha) = 1/2 + 1/2 \cdot \cos(\alpha).$
+*The Dirac function at&nbsp; $f = 0$&nbsp; follows from the trigonometric transformation&nbsp; $\cos^2(\alpha) = 1/2 + 1/2 \cdot \cos(\alpha).$
-*Mit&nbsp; $A_1 = c_1 \cdot A_x =  0.5 \ {\rm V} $&nbsp; und&nbsp;  $A_2 = (c_2/2) \cdot A_x^2 =  0.025 \ {\rm V}^2 $&nbsp; ergibt sich somit für den Klirrfaktor:
+*With&nbsp; $A_1 = c_1 \cdot A_x =  0.5 \ {\rm V} $&nbsp; and&nbsp;  $A_2 = (c_2/2) \cdot A_x^2 =  0.025 \ {\rm V}^2 $&nbsp; the following is thus obtained for the distortion factor:
 :$$K= \frac{A_2}{A_1}= \frac{c_2/2 \cdot A_x}{c_1 }= \frac{0.025}{0.5}  \hspace{0.15cm}\underline{= 5 \%}.$$
-'''(4)'''&nbsp; Entsprechend der Musterlösung zur letzten Teilaufgabe ist&nbsp; $K$&nbsp; proportional zu&nbsp; $A_x$. Deshalb erhält man nun&nbsp; $K \hspace{0.15cm}\underline{= 15 \%}.$
+'''(4)'''&nbsp; According to the solution of the last subtask&nbsp; $K$&nbsp; is proportional to&nbsp; $A_x$. Therefore, one now obtains&nbsp; $K \hspace{0.15cm}\underline{= 15 \%}.$
-'''(5)'''&nbsp; Nun lautet das Ausgangssignal:
+'''(5)'''&nbsp; Now the output signal is:
 :$$y(t)=  c_1\cdot A_x \cdot \left( 1 - {|\hspace{0.05cm}t\hspace{0.05cm}|}/{T_x}\right) +\hspace{0.1cm}
   {c_2}\cdot A_x^2 \cdot \left( 1 - {|\hspace{0.05cm}t\hspace{0.05cm}|}/{T_x}\right)^2.$$
-*Zum Zeitpunkt &nbsp;$t = 0$&nbsp; bzw. &nbsp;$t = T_x/2$&nbsp; treten folgende Werte auf:
+*The following values occur at time&nbsp;$t = 0$&nbsp; and &nbsp;$t = T_x/2$&nbsp;:
 :$$y(t=0) = c_1\cdot A_x  + {c_2}\cdot A_x^2  \hspace{0.15cm}\underline{= 1.95\,{\rm
   V}},$$

@@ Line 19: / Line 19: @@
 For the subtasks&nbsp; '''(3)'''&nbsp; and&nbsp; '''(4)'''&nbsp; let the input signal&nbsp; $x(t)$&nbsp; be a harmonic oscillation because only for such an oscillation a distortion factor can be specified.
-Dagegen wird für die Teilaufgabe&nbsp; '''(5)'''&nbsp; ein Dreieckimpuls mit Amplitude&nbsp; $A_x = 3 \ {\rm V}$&nbsp; und der einseitigen Impulsdauer&nbsp; $T_x = 2 \ {\rm ms}$&nbsp; betrachtet:
+In contrast, a triangular pulse with amplitude&nbsp; $A_x = 3 \ {\rm V}$&nbsp; and the one-sided pulse duration&nbsp; $T_x = 2 \ {\rm ms}$&nbsp; is considered for the subtask&nbsp; '''(5)'''&nbsp;:
 :$$x(t) =  A_x \cdot ( 1 - {|t|}/{T_x})  $$

← Older revision	Revision as of 10:08, 20 September 2021
(No difference)

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Lineare zeitinvariante Systeme/Nichtlineare Verzerrungen
+{{quiz-Header|Buchseite=Linear_and_Time_Invariant_Systems/Nonlinear_Distortion
 }}
@@ Line 30: / Line 30: @@
-''Hinweise:''
+''Please note:''
-*Die Aufgabe gehört zum Kapitel&nbsp;  [[Linear_and_Time_Invariant_Systems/Nichtlineare_Verzerrungen|Nichtlineare Verzerrungen]].
+*The task belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion|Nonlinear Distortion]].
-*Im Fragenkatalog werden folgende Abkürzungen benutzt:
+*The following abbreviations are used in the formulation of the questions:
 :$$y_1(t) =  c_1  \cdot x(t), \hspace{0.5cm} y_2(t) = c_2  \cdot
 x^2(t).$$
@@ Line 39: / Line 39: @@
-===Fragebogen===
+===Questions===
 <quiz display=simple>
@@ Line 74: / Line 74: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp; Richtig ist nur der <u>Lösungsvorschlag 3</u>:

← Older revision		Revision as of 12:40, 23 March 2021
Line 131:		Line 131:


−	[[Category:~~Exercises for~~ Linear and Time-Invariant Systems\|^2.2 Nichtlineare Verzerrungen^]]	+	[[Category:Linear and Time-Invariant Systems: Exercises\|^2.2 Nichtlineare Verzerrungen^]]

← Older revision		Revision as of 12:27, 23 March 2021
Line 131:		Line 131:


−	[[Category:~~Aufgaben zu Lineare zeitinvariante Systeme~~\|^2.2 Nichtlineare Verzerrungen^]]	+	[[Category:Exercises for Linear and Time-Invariant Systems\|^2.2 Nichtlineare Verzerrungen^]]