Aufgaben:Exercise 2.3: Sinusoidal Characteristic - Revision history

Guenter at 13:12, 29 September 2021

2021-09-29T13:12:43Z

Guenter at 13:04, 29 September 2021

2021-09-29T13:04:38Z

Guenter at 13:02, 29 September 2021

2021-09-29T13:02:14Z

Guenter: Guenter moved page Aufgaben:Aufgabe 2.3: Sinusförmige Kennlinie to Aufgaben:Exercise 2.3: Sinusoidal Characteristic

2021-09-20T10:03:19Z

Guenter moved page Aufgabe 2.3: Sinusförmige Kennlinie to Exercise 2.3: Sinusoidal Characteristic

Guenter at 10:02, 20 September 2021

2021-09-20T10:02:10Z

Oezer at 21:50, 14 September 2021

2021-09-14T21:50:50Z

Oezer at 21:21, 14 September 2021

2021-09-14T21:21:54Z

Oezer at 21:21, 14 September 2021

2021-09-14T21:21:00Z

Oezer at 21:07, 14 September 2021

2021-09-14T21:07:39Z

Oezer at 21:07, 14 September 2021

2021-09-14T21:07:12Z

@@ Line 61: / Line 61: @@
 |type="[]"}
 + $K_{\rm g5}$&nbsp; generally represents a better approximation for&nbsp; $K$&nbsp; than&nbsp; $K_{\rm g3}$.
-- $K_{\rm g3} < K_{\rm g5}$ holds for&nbsp; $A = 1.0$ &nbsp;.
+- $K_{\rm g3} < K_{\rm g5}$ holds for&nbsp; $A = 1.0$.
-+ $K_{\rm g3} \approx K_{\rm g5}$ will hold for&nbsp; $A = 0.5$ &nbsp;.
++ $K_{\rm g3} \approx K_{\rm g5}$ will hold for&nbsp; $A = 0.5$.
@@ Line 74: / Line 74: @@
-'''(2)'''&nbsp; The analytical spectrum (positive frequencies only) of the input signal is:
+'''(2)'''&nbsp; The analytical spectrum&nbsp; (positive frequencies only)&nbsp; of the input signal is:
 :$$X_+(f) = A  \cdot {\rm \delta}(f- f_0) .$$
@@ Line 88: / Line 88: @@
 :$$A_1 = A  - {A^3}\hspace{-0.1cm}/{8},  \hspace{0.5cm}A_3 =   - {A^3}\hspace{-0.1cm}/{24}.$$
-*Using $A = 0.5$ the following is obtained: $A_1 \approx 0.484$ and&nbsp; $A_3 \approx 0.005$. Thus, the distortion factor is:
+*Using&nbsp; $A = 0.5$&nbsp; the following is obtained:&nbsp; $A_1 \approx 0.484$&nbsp; and&nbsp; $A_3 \approx 0.005$.&nbsp; Thus, the distortion factor is:
 :$$K = K_3 ={|A_3|}/{A_1}= {0.005}/{0.484} \hspace{0.15cm}\underline{ =  1.08\%}.$$
 :Note that for the approximation&nbsp; $g_3(x)$&nbsp; only the cubic part&nbsp; $K_3$&nbsp; of the distortion factor is effective.
-*For&nbsp; $A = 1.0$&nbsp; and&nbsp; $A = 1.5$&nbsp; the following numerical values arise as aresult:
+*For&nbsp; $A = 1.0$&nbsp; and&nbsp; $A = 1.5$&nbsp; the following numerical values:
 :$$A = 1.0: A_1 \approx 0.875, \hspace{0.2cm} A_3 \approx
 -0.041\hspace{0.3cm} \Rightarrow \hspace{0.3cm} \hspace{0.15cm}\underline{K \approx 4.76\%}\; \; \Rightarrow \; \; K_{g3},$$
@@ Line 119: / Line 119: @@
-'''(4)'''&nbsp; <u>Approaches 1 and 3</u> are correct:
+'''(4)'''&nbsp; <u>Approaches 1 and 3</u>&nbsp; are correct:
 *The approach&nbsp; $g_5(x)$&nbsp; is a better approximation for the sine function&nbsp; $g(x)$&nbsp; than the approximation&nbsp; $g_3(x)$ in the entire domain.
 *Thus, the value&nbsp; $K_{g5}$&nbsp; computed in the subtask&nbsp; '''(3)'''&nbsp; is a better approximation for the actual distortion factor than&nbsp; $K_{g3}$. <br>Therefore, the first statement is correct.
 *The second statement is false as already shown by the computation for&nbsp; $A=1$&nbsp;: &nbsp; $K_{g3} \approx 4.76 \%$&nbsp; is greater than&nbsp; $K_{g5} \approx 4.45 \%$.
-*The reason for this is that $g_3(x)$ is below $g_5(x)$ and thus there is also a greater deviation from the linear curve.
+*The reason for this is that&nbsp; $g_3(x)$&nbsp; is below&nbsp; $g_5(x)$&nbsp; and thus there is also a greater deviation from the linear curve.
 *For&nbsp; $A=0.5$&nbsp;,&nbsp; $K_{g5} \approx K_{g3} = 1.08 \%$&nbsp; will hold.
 *The characteristic curve on the information page shows that for&nbsp; $|x| \le 0.5$&nbsp; the functions&nbsp; $g_3(x)$&nbsp; and&nbsp; $g_5(x)$&nbsp; are indistinguishable within the accuracy of drawing.
-*This also results in the same distortion factors.
+*This also results in&nbsp; (nearly)&nbsp; the same distortion factors.
 {{ML-Fuß}}

@@ Line 58: / Line 58: @@
-{Which of the following statements are true? Here,&nbsp; $K$&nbsp; denotes the distortion factor of the sine function&nbsp; $g(x)$. <br>$K_{\rm g3}$&nbsp; and&nbsp; $K_{\rm g5}$&nbsp; are based on the approximations&nbsp; $g_3(x)$&nbsp; and&nbsp; $g_5(x)$, respectively.
+{Which of the following statements are true?&nbsp; Here,&nbsp; $K$&nbsp; denotes the distortion factor of the sine function&nbsp; $g(x)$. <br>$K_{\rm g3}$&nbsp; and&nbsp; $K_{\rm g5}$&nbsp; are based on the approximations&nbsp; $g_3(x)$&nbsp; and&nbsp; $g_5(x)$, respectively.
 |type="[]"}
 + $K_{\rm g5}$&nbsp; generally represents a better approximation for&nbsp; $K$&nbsp; than&nbsp; $K_{\rm g3}$.

@@ Line 4: / Line 4: @@
 [[File:P_ID894__LZI_A_2_3.png|right|frame|Sinusoidal characteristic curve]]
-We consider a system with input&nbsp; $x(t)$&nbsp; and output&nbsp; $y(t)$. For simplicity of description, the signals are considered to be dimensionless.
+We consider a system with input&nbsp; $x(t)$&nbsp; and output&nbsp; $y(t)$.&nbsp; For simplicity of description, the signals are considered to be dimensionless.
-The relationship between the input signal&nbsp; $x(t)$&nbsp; and the output signal&nbsp; $y(t)$&nbsp; is given by the following characteristic curve in the range between&nbsp; $-\pi/2$&nbsp; and&nbsp; $+\pi/2$&nbsp;.
+The relationship between the input signal&nbsp; $x(t)$&nbsp; and the output signal&nbsp; $y(t)$&nbsp; is given by the following characteristic curve in the range between&nbsp; $-\pi/2$&nbsp; and&nbsp; $+\pi/2$:
 :$$g(x) =  \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} -
   \hspace{0.05cm}\text{...}$$
@@ Line 17: / Line 17: @@
 :$$g_5(x) = x- x^3\hspace{-0.1cm}/{6}+x^5\hspace{-0.1cm}/{120}\hspace{0.05cm}.$$
-The input signal&nbsp; $x(t) = A \cdot \cos(\omega_0 \cdot t)$&nbsp; is always assumed wherat the values&nbsp; $A = 0.5$,&nbsp; $A = 1.0$&nbsp; and&nbsp; $A = 1.5$&nbsp; are to be considered for the (dimensionless) signal amplitude.
+*The input signal&nbsp; $x(t) = A \cdot \cos(\omega_0 \cdot t)$&nbsp; is always assumed.&nbsp;
@@ Line 27: / Line 28: @@
 ''Please note:''
-*The task belongs to the chapter&nbsp;  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion|Nonlinear Distortion]].
+*The task belongs to the chapter&nbsp;  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion|Nonlinear Distortions]].
 *The resulting signal curves for&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; are shown graphically on the page&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion#Description_of_nonlinear_systems|Description of nonlinear systems]]&nbsp;.
 *All powers required here refer to the resistance&nbsp; $R = 1 \ \rm \Omega$&nbsp; and thus have the unit&nbsp; ${\rm V}^2$.
 *The following trigonometric relations are assumed to be known:

← Older revision		Revision as of 10:02, 20 September 2021
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−	[[Category:Linear and Time-Invariant Systems: Exercises\|^2.2 ~~Nichtlineare Verzerrungen~~^]]	+	[[Category:Linear and Time-Invariant Systems: Exercises\|^2.2 Nonlinear Distortions^]]

@@ Line 126: / Line 126: @@
 *For&nbsp; $A=0.5$&nbsp;,&nbsp; $K_{g5} \approx K_{g3} = 1.08 \%$&nbsp; will hold.
 *The characteristic curve on the information page shows that for&nbsp; $|x| \le 0.5$&nbsp; the functions&nbsp; $g_3(x)$&nbsp; and&nbsp; $g_5(x)$&nbsp; are indistinguishable within the accuracy of drawing.
-*This also results in equal distortion factors.
+*This also results in the same distortion factors.
 {{ML-Fuß}}

← Older revision	Revision as of 10:03, 20 September 2021
(No difference)

@@ Line 110: / Line 110: @@
 A_5 =  {A^5}\hspace{-0.1cm}/{1920}.$$
-*Daraus ergeben sich mit&nbsp; $A=1$&nbsp; die Zahlenwerte:
+*From this, the following numerical values arise a result with&nbsp; $A=1$&nbsp;:
 :$$A_1 \approx 1 -0.125 +0.005 = 0.880,\hspace{0.3cm}
 A_3 \approx  -0.042 +0.003 = -0.039,\hspace{0.3cm}
@@ Line 119: / Line 119: @@
-'''(4)'''&nbsp; Richtig sind die <u>Lösungsvorschläge 1 und 3</u>:
+'''(4)'''&nbsp; <u>Approaches 1 and 3</u> are correct:
-*Der Ansatz&nbsp; $g_5(x)$&nbsp; ist im gesamten Bereich eine bessere Näherung für die Sinusfunktion&nbsp; $g(x)$&nbsp; als die Näherung&nbsp; $g_3(x)$.
+*The approach&nbsp; $g_5(x)$&nbsp; is a better approximation for the sine function&nbsp; $g(x)$&nbsp; than the approximation&nbsp; $g_3(x)$ in the entire domain.
-*Deshalb ist der in der Teilaufgabe&nbsp; '''(3)'''&nbsp; berechnete Wert&nbsp; $K_{g5}$&nbsp; eine bessere Näherung für den tatsächlichen Klirrfaktor als&nbsp; $K_{g3}$. <br>die erste Aussage ist somit richtig.
+*Thus, the value&nbsp; $K_{g5}$&nbsp; computed in the subtask&nbsp; '''(3)'''&nbsp; is a better approximation for the actual distortion factor than&nbsp; $K_{g3}$. <br>Therefore, the first statement is correct.
-*Die zweite Aussage ist falsch, wie schon die Berechnung für&nbsp; $A=1$&nbsp; gezeigt hat: &nbsp; $K_{g3} \approx 4.76 \%$&nbsp; ist größer als&nbsp; $K_{g5} \approx 4.45 \%$.
+*The second statement is false as already shown by the computation for&nbsp; $A=1$&nbsp;: &nbsp; $K_{g3} \approx 4.76 \%$&nbsp; is greater than&nbsp; $K_{g5} \approx 4.45 \%$.
-*Der Grund hierfür ist, dass $g_3(x)$ unterhalb von $g_5(x)$ liegt und damit auch eine größere Abweichung vom linearen Verlauf vorliegt.
+*The reason for this is that $g_3(x)$ is below $g_5(x)$ and thus there is also a greater deviation from the linear curve.
-*Für&nbsp; $A=0.5$&nbsp; wird&nbsp; $K_{g5} \approx K_{g3} = 1.08 \%$&nbsp; gelten.
+*For&nbsp; $A=0.5$&nbsp;,&nbsp; $K_{g5} \approx K_{g3} = 1.08 \%$&nbsp; will hold.
-*Die Kennlinie auf der Angabenseite zeigt, dass   für&nbsp; $|x| \le 0.5$&nbsp; die Funktionen&nbsp; $g_3(x)$&nbsp; und&nbsp; $g_5(x)$&nbsp;  innerhalb der Zeichengenauigkeit nicht zu unterscheiden sind.
+*The characteristic curve on the information page shows that for&nbsp; $|x| \le 0.5$&nbsp; the functions&nbsp; $g_3(x)$&nbsp; and&nbsp; $g_5(x)$&nbsp; are indistinguishable within the accuracy of drawing.
-*Damit ergeben sich auch gleiche Klirrfaktoren.
+*This also results in equal distortion factors.
 {{ML-Fuß}}

@@ Line 103: / Line 103: @@
 '''(3)'''&nbsp; Similarly as in subtask&nbsp; '''(2)''',&nbsp;
 :$$y(t) = A_1 \cdot {\rm cos}(\omega_0  t ) + A_3 \cdot {\rm
-cos}(3\omega_0  t )+ A_5 \cdot {\rm cos}(5\omega_0  t )$$ holds
+cos}(3\omega_0  t )+ A_5 \cdot {\rm cos}(5\omega_0  t )$$
-:with the following coefficients:
+:holds with the following coefficients:
 :$$A_1 = A  - {A^3}\hspace{-0.1cm}/{8} + {A^5}\hspace{-0.1cm}/{192},\hspace{0.3cm}
 A_3 =  - {A^3}\hspace{-0.1cm}/{24} +  {A^5}\hspace{-0.1cm}/{384},\hspace{0.3cm}