Aufgaben:Exercise 2.3Z: Asymmetrical Characteristic Operation - Revision history

Guenter at 13:50, 29 September 2021

2021-09-29T13:50:13Z

Guenter at 13:31, 29 September 2021

2021-09-29T13:31:28Z

Guenter at 13:30, 29 September 2021

2021-09-29T13:30:43Z

Guenter at 13:29, 29 September 2021

2021-09-29T13:29:55Z

Guenter at 10:09, 20 September 2021

2021-09-20T10:09:40Z

Guenter: Guenter moved page Aufgaben:Aufgabe 2.3Z: Kennlinienbetrieb asymmetrisch to Aufgaben:Exercise 2.3Z: Asymmetrical Characteristic Operation

2021-09-20T10:05:13Z

Guenter moved page Aufgabe 2.3Z: Kennlinienbetrieb asymmetrisch to Exercise 2.3Z: Asymmetrical Characteristic Operation

Oezer at 22:26, 14 September 2021

2021-09-14T22:26:55Z

Oezer at 22:24, 14 September 2021

2021-09-14T22:24:22Z

Oezer at 21:52, 14 September 2021

2021-09-14T21:52:21Z

Oezer at 21:49, 14 September 2021

2021-09-14T21:49:22Z

@@ Line 90: / Line 90: @@
   t) + A^3 \cdot \cos^3(\omega_0  t)\big].$$
-*The signal&nbsp; $y_{\rm C}(t)$&nbsp; contains a direct (DC) component&nbsp; $C - C^3/6$ which is no longer included in the signal&nbsp; $y(t)$&nbsp; due to the high-pass filter:
+*The signal&nbsp; $y_{\rm C}(t)$&nbsp; contains a direct (DC) component&nbsp; $C - C^3/6$&nbsp; which is no longer included in the signal&nbsp; $y(t)$&nbsp; due to the high-pass filter:
 :$$\underline{ A_0 = 0}.$$
-'''(2)'''&nbsp; Applying the given trigonometric relations the following coefficients with&nbsp; $A= C = 0.5$ are obtained:
+'''(2)'''&nbsp; Applying the given trigonometric relations the following coefficients with&nbsp; $A= C = 0.5$&nbsp; are obtained:
 :$$A_1 = A - {1}/{6}\cdot 3 \cdot C^2 \cdot A  - {1}/{6} \cdot {3}/{4}\cdot
   A^3 = {1}/{2} - {1}/{16} - {1}/{64} = {27}/{64}
@@ Line 103: / Line 103: @@
    A^3 = - {1}/{192}  \hspace{0.15cm}\underline{\approx -0.005}.$$
-*Higher order terms do not occur. Thus, &nbsp; $\underline{A_4  = 0}$ holds.
+*Higher order terms do not occur.&nbsp; Thus, &nbsp; $\underline{A_4  = 0}$&nbsp; holds.
-'''(3)'''&nbsp; In this task, the higher order distortion factors are&nbsp; $K_2   = 2/27 \approx 7.41\%$&nbsp; und&nbsp; $K_3   = 1/81 \approx 1.23\%$.
+'''(3)'''&nbsp; In this task,&nbsp; the higher order distortion factors are&nbsp; $K_2   = 2/27 \approx 7.41\%$&nbsp; and&nbsp; $K_3   = 1/81 \approx 1.23\%$.
 *Thereby, the following is obtained for the overall distortion factor:
 :$$K = \sqrt{K_2^2 + K_3^2} \hspace{0.15cm}\underline{\approx7.51 \%}.$$
@@ Line 121: / Line 121: @@
   -0.448}.$$
-*The signal&nbsp; $y(t)$&nbsp; is shifted downward by $0.448$&nbsp; compared to the signal&nbsp; drawn in the sketch on the information page.
+*The signal&nbsp; $y(t)$&nbsp; is shifted downward by&nbsp; $0.448$&nbsp; compared to the signal&nbsp; drawn in the sketch on the information page.
 *This signal value is obtained from the following equation considering&nbsp; $A = C = 1/2$:
 :$$C - \frac{C \cdot A^2}{4}- \frac{C^3}{6} =  {1}/{2} - {1}/{32}-  {1}/{48}   = 0.448.$$

@@ Line 39: / Line 39: @@
 ''Please note:''
-*The exercise 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 trigonometric relations are assumed to be known:

@@ Line 19: / Line 19: @@
 The sinusoidal characteristic curve&nbsp; $g(x)$&nbsp; is to be approximated by the cubic approximation&nbsp;$g_3(x)$&nbsp; throughout the whole problem according to the above equation.
-This would result in exactly the same constellation as in&nbsp; [[Aufgaben:2.3_Sinusförmige_Kennlinie|Exercise 2.3]]&nbsp; for&nbsp; $C = 0$&nbsp; in whose subtask&nbsp; '''(2)'''&nbsp; the distortion factor was calculated:
+This would result in exactly the same constellation as in&nbsp; [[Aufgaben:Exercise_2.3:_Sinusoidal_Characteristic|Exercise 2.3]]&nbsp; for&nbsp; $C = 0$&nbsp; in whose subtask&nbsp; '''(2)'''&nbsp; the distortion factor was calculated:
 *$K = K_{g3} \approx 1.08 \%$ &nbsp;für&nbsp; $A = 0.5$,
 *$K = K_{g3} \approx 4.76 \%$ &nbsp;für&nbsp; $A = 1.0$.

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Linear_and_Time_Invariant_Systems/Nonlinear_Distortion}}
-[[File:P_ID895__LZI_Z_2_3.png|right|frame|Influence of nonlinear distortions]]
+[[File:P_ID895__LZI_Z_2_3.png|right|frame|System and signal examples]]
 The cosine signal
 :$$x(t) =  A \cdot \cos(\omega_0 t)$$
-is applied to the input of a system&nbsp; $S$&nbsp; where&nbsp; $A = 0.5$&nbsp; shall always hold for the amplitude. The system&nbsp; $S$&nbsp; consists of
+is applied to the input of a system&nbsp; $S$&nbsp; where&nbsp; $A = 0.5$&nbsp; shall always hold for the amplitude.&nbsp; The system&nbsp; $S$&nbsp; consists of
 *the addition of a direct (DC) component&nbsp; $C$,
 *a nonlinearity with the characteristic curve
 :$$g(x) =  \sin(x) \hspace{0.05cm} \approx x -{x^3}\hspace{-0.1cm}/{6} = g_3(x),$$
-*as well as an ideal high-pass filter that allows all frequencies to pass unaltered except for a direct (DC) signal&nbsp; $(f = 0)$&nbsp;.
+*as well as an ideal high-pass filter that allows all frequencies to pass unaltered except for a direct (DC) signal&nbsp; $(f = 0)$.
@@ Line 19: / Line 19: @@
 The sinusoidal characteristic curve&nbsp; $g(x)$&nbsp; is to be approximated by the cubic approximation&nbsp;$g_3(x)$&nbsp; throughout the whole problem according to the above equation.
-This would result in exactly the same constellation as in&nbsp; [[Aufgaben:2.3_Sinusförmige_Kennlinie|Exercise 2.3]] for&nbsp; $C = 0$&nbsp; in whose subtask&nbsp; '''(2)'''&nbsp; the distortion factor was calculated:
+This would result in exactly the same constellation as in&nbsp; [[Aufgaben:2.3_Sinusförmige_Kennlinie|Exercise 2.3]]&nbsp; for&nbsp; $C = 0$&nbsp; in whose subtask&nbsp; '''(2)'''&nbsp; the distortion factor was calculated:
 *$K = K_{g3} \approx 1.08 \%$ &nbsp;für&nbsp; $A = 0.5$,
 *$K = K_{g3} \approx 4.76 \%$ &nbsp;für&nbsp; $A = 1.0$.
@@ Line 27: / Line 27: @@
 :$$x_{\rm C}(t) =  C + A \cdot \cos(\omega_0 t) = {1}/{2} + {1}/{2}\cdot \cos(\omega_0 t).$$
-*So, the characteristic curve is operated asymmetrically with values between&nbsp; $0$&nbsp; and&nbsp; $1$.
+*So,&nbsp; the characteristic curve is operated asymmetrically with values between&nbsp; $0$&nbsp; and&nbsp; $1$.
-*In the above graph, the signals&nbsp; $x_{\rm C}(t)$&nbsp; and&nbsp; $y_{\rm C}(t)$&nbsp; are plotted additionally directly before and after the characteristic curve&nbsp; $g(x)$&nbsp;.
+*In the above graph,&nbsp; the signals&nbsp; $x_{\rm C}(t)$&nbsp; and&nbsp; $y_{\rm C}(t)$&nbsp; are plotted additionally directly before and after the characteristic curve&nbsp; $g(x)$&nbsp;.
@@ Line 39: / Line 39: @@
 ''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 Distortion]].
 *The following trigonometric relations are assumed to be known:
@@ Line 51: / Line 51: @@
 <quiz display=simple>
-{Compute the output signal&nbsp; $y(t)$&nbsp; considering the high-pass filter. What is the direct (DC) signal component&nbsp; $A_0$?
+{Compute the output signal&nbsp; $y(t)$&nbsp; considering the high-pass filter.&nbsp; What is the direct (DC) signal component&nbsp; $A_0$?
 |type="{}"}
 $A_0 \ = \ $ { 0. }

← Older revision		Revision as of 10:09, 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^]]

← Older revision	Revision as of 10:05, 20 September 2021
(No difference)

← Older revision		Revision as of 22:26, 14 September 2021
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	A^3 = - {1}/{192} \hspace{0.15cm}\underline{\approx -0.005}.$$		A^3 = - {1}/{192} \hspace{0.15cm}\underline{\approx -0.005}.$$

−	*Higher order terms do not occur. Thus, h  $\underline{A_4 = 0}$ holds.	+	*Higher order terms do not occur. Thus,   $\underline{A_4 = 0}$ holds.

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Lineare zeitinvariante Systeme/Nichtlineare Verzerrungen
+{{quiz-Header|Buchseite=Linear_and_Time_Invariant_Systems/Nonlinear_Distortion}}
 [[File:P_ID895__LZI_Z_2_3.png|right|frame|Einfluss nichtlinearer Verzerrungen]]
@@ Line 39: / Line 38: @@
-''Hinweise:''
+''Please note:''
-*Die Aufgabe bezieht sich auf das 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]].
-*Als bekannt vorausgesetzt werden die folgenden trigonometrischen Beziehungen:
+*The following trigonometric relations are assumed to be known:
 :$$\cos^2(\alpha) =  {1}/{2}  + {1}/{2}
 \cdot \cos(2\alpha)\hspace{0.05cm}, \hspace{0.3cm}
@@ Line 49: / Line 48: @@
-===Fragebogen===
+===Questions===
 <quiz display=simple>
@@ Line 79: / Line 78: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp; Unter Berücksichtigung der kubischen Näherung &nbsp;$g_3(x)$&nbsp; erhält man vor dem Hochpass: