Aufgaben:Exercise 2.3: Sinusoidal Characteristic - Revision history

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	''Please note:''		''Please note:''
	*The task belongs to the chapter  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion\|Nonlinear Distortions]].		*The task belongs to the chapter  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion\|Nonlinear Distortions]].
	*The resulting signal curves for  $x(t)$  and  $y(t)$  are shown graphically on the page  [[Linear_and_Time_Invariant_Systems/~~Nonlinear_Distortion~~#Description_of_nonlinear_systems\|Description of nonlinear systems]] .		*The resulting signal curves for  $x(t)$  and  $y(t)$  are shown graphically on the page  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions#Description_of_nonlinear_systems\|Description of nonlinear systems]] .
	*All powers required here refer to the resistance  $R = 1 \ \rm \Omega$  and thus have the unit  ${\rm V}^2$.		*All powers required here refer to the resistance  $R = 1 \ \rm \Omega$  and thus have the unit  ${\rm V}^2$.
	*The following trigonometric relations are assumed to be known:		*The following trigonometric relations are assumed to be known:
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	'''(3)'''  Similarly as in subtask  '''(2)''',		'''(3)'''  Similarly as in subtask  '''(2)''',
	:$$y(t) = A_1 \cdot {\rm cos}(\omega_0 t ) + A_3 \cdot {\~~rmcos~~}(3\omega_0 t )+ A_5 \cdot {\rm cos}(5\omega_0 t )$$		:$$y(t) = A_1 \cdot {\rm cos}(\omega_0 t ) + A_3 \cdot {\rm cos}(3\omega_0 t )+ A_5 \cdot {\rm cos}(5\omega_0 t )$$

	:holds with the following coefficients:		:holds with the following coefficients:
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	[[Category:Linear and Time-Invariant Systems: Exercises\|^2.2 Nonlinear Distortions^]]		[[Category:Linear and Time-Invariant Systems: Exercises\|^2.2 Nonlinear Distortions^]]
	[[de:Aufgaben:2.~~3_Sinusförmige_Kennlinie~~]]		[[de:Aufgaben:Aufgabe 2.3: Sinusförmige Kennlinie]]

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	The relationship between the input signal  $x(t)$  and the output signal  $y(t)$  is given by the following characteristic curve in the range between  $-\pi/2$  and  $+\pi/2$:		The relationship between the input signal  $x(t)$  and the output signal  $y(t)$  is given by the following characteristic curve in the range between  $-\pi/2$  and  $+\pi/2$:
	:$$g(x) = \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} -		:$$g(x) = \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} -\hspace{0.05cm}\text{...}$$
	\hspace{0.05cm}\text{...}$$

	The second part of this equation describes the series expansion of the sine function.		The second part of this equation describes the series expansion of the sine function.
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	*All powers required here refer to the resistance  $R = 1 \ \rm \Omega$  and thus have the unit  ${\rm V}^2$.		*All powers required here refer to the resistance  $R = 1 \ \rm \Omega$  and thus have the unit  ${\rm V}^2$.
	*The following trigonometric relations are assumed to be known:		*The following trigonometric relations are assumed to be known:
	:$$\cos^3(\alpha) = {3}/{4} \cdot \cos(\alpha) + {1}/{4} \cdot \cos(3\alpha)		:$$\cos^3(\alpha) = {3}/{4} \cdot \cos(\alpha) + {1}/{4} \cdot \cos(3\alpha)\hspace{0.05cm}, $$
	\hspace{0.05cm}, $$		:$$ \cos^5(\alpha) = {10}/{16} \cdot \cos(\alpha) + {5}/{16} \cdot \cos(3\alpha)+ {1}/{16} \cdot \cos(5\alpha)\hspace{0.05cm}.$$
	:$$ \cos^5(\alpha) = {10}/{16} \cdot \cos(\alpha) + {5}/{16} \cdot \cos(3\alpha)
	+ {1}/{16} \cdot \cos(5\alpha)\hspace{0.05cm}.$$


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	*Then, the following signal is applied to the output of the nonlinear characteristic curve  $g_3(x)$ :		*Then, the following signal is applied to the output of the nonlinear characteristic curve  $g_3(x)$ :
	:$$y(t) = A \cdot {\rm cos}(\omega_0 t ) - \frac{A^3}{6} \cdot		:$$y(t) = A \cdot {\rm cos}(\omega_0 t ) - \frac{A^3}{6} \cdot{\rm cos}^3(\omega_0 t )=A \cdot {\rm cos}(\omega_0 t ) - \frac{3}{4} \cdot \frac{A^3}{6} \cdot{\rm cos}(\omega_0 t )- \frac{1}{4} \cdot \frac{A^3}{6} \cdot{\rm cos}(3\omega_0 t ) = A_1 \cdot {\rm cos}(\omega_0 t )+ A_3 \cdot {\rm cos}(3\omega_0 t ).$$
	{\rm cos}^3(\omega_0 t )=
	A \cdot {\rm cos}(\omega_0 t ) - \frac{3}{4} \cdot \frac{A^3}{6} \cdot
	{\rm cos}(\omega_0 t )- \frac{1}{4} \cdot \frac{A^3}{6} \cdot
	{\rm cos}(3\omega_0 t ) = A_1 \cdot {\rm cos}(\omega_0 t )
	+ A_3 \cdot {\rm cos}(3\omega_0 t ).$$

	*For the coefficients  $A_1$  and $A_3$ the following is obtained by comparison of coefficients:		*For the coefficients  $A_1$  and $A_3$ the following is obtained by comparison of coefficients:
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	*For  $A = 1.0$  and  $A = 1.5$  the following numerical values:		*For  $A = 1.0$  and  $A = 1.5$  the following numerical values:
	:$$A = 1.0: A_1 \approx 0.875, \hspace{0.2cm} A_3 \approx		:$$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},$$
	-0.041\hspace{0.3cm} \Rightarrow \hspace{0.3cm} \hspace{0.15cm}\underline{K \approx 4.76\%}\; \; \Rightarrow \; \; K_{g3},$$		:$$A = 1.5: A_1 \approx 1.078, \hspace{0.2cm} A_3 \approx-0.140\hspace{0.3cm} \Rightarrow \hspace{0.3cm} \hspace{0.15cm}{K \approx 13 \%}.$$
	:$$A = 1.5: A_1 \approx 1.078, \hspace{0.2cm} A_3 \approx
	-0.140\hspace{0.3cm} \Rightarrow \hspace{0.3cm} \hspace{0.15cm}{K \approx 13 \%}.$$



	'''(3)'''  Similarly as in subtask  '''(2)''',		'''(3)'''  Similarly as in subtask  '''(2)''',
	:$$y(t) = A_1 \cdot {\rm cos}(\omega_0 t ) + A_3 \cdot {\rm		:$$y(t) = A_1 \cdot {\rm cos}(\omega_0 t ) + A_3 \cdot {\rmcos}(3\omega_0 t )+ A_5 \cdot {\rm cos}(5\omega_0 t )$$
	~~cos~~}(3\omega_0 t )+ A_5 \cdot {\rm cos}(5\omega_0 t )$$

	:holds 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_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}A_5 = {A^5}\hspace{-0.1cm}/{1920}.$$
	A_3 = - {A^3}\hspace{-0.1cm}/{24} + {A^5}\hspace{-0.1cm}/{384},\hspace{0.3cm}
	A_5 = {A^5}\hspace{-0.1cm}/{1920}.$$

	*From this, the following numerical values arise a result with  $A=1$ :		*From this, the following numerical values arise a result with  $A=1$ :
	:$$A_1 \approx 1 -0.125 +0.005 = 0.880,\hspace{0.3cm}		:$$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}A_5 \approx 0.0005$$
	A_3 \approx -0.042 +0.003 = -0.039,\hspace{0.3cm}		:$$\Rightarrow \hspace{0.3cm}K_3 = {\|A_3\|}/{A_1}= 0.0443,\hspace{0.3cm}K_5 ={\|A_5\|}/{A_1}= 0.0006 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} K = \sqrt{K_3^2 + K_5^2} \hspace{0.15cm}\underline{\approx 4.45\%}\; \; \Rightarrow \; \; K_{g5}.$$
	A_5 \approx 0.0005$$
	:$$\Rightarrow \hspace{0.3cm}K_3 = {\|A_3\|}/{A_1}= 0.0443,\hspace{0.3cm}K_5 =
	{\|A_5\|}/{A_1}= 0.0006 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} K = \sqrt{K_3^2 + K_5^2} \hspace{0.15cm}\underline{\approx 4.45\%}
	\; \; \Rightarrow \; \; K_{g5}.$$

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	[[Category:Linear and Time-Invariant Systems: Exercises\|^2.2 Nonlinear Distortions^]]		[[Category:Linear and Time-Invariant Systems: Exercises\|^2.2 Nonlinear Distortions^]]
			[[de:Aufgaben:2.3_Sinusförmige_Kennlinie]]

Guenter at 13:12, 29 September 2021

2021-09-29T13:12:43Z

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	\|type="[]"}		\|type="[]"}
	+ $K_{\rm g5}$  generally represents a better approximation for  $K$  than  $K_{\rm g3}$.		+ $K_{\rm g5}$  generally represents a better approximation for  $K$  than  $K_{\rm g3}$.
	- $K_{\rm g3} < K_{\rm g5}$ holds for  $A = 1.0$ ~~ ~~.		- $K_{\rm g3} < K_{\rm g5}$ holds for  $A = 1.0$.
	+ $K_{\rm g3} \approx K_{\rm g5}$ will hold for  $A = 0.5$ ~~ ~~.		+ $K_{\rm g3} \approx K_{\rm g5}$ will hold for  $A = 0.5$.


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	'''(2)'''  The analytical spectrum (positive frequencies only) of the input signal is:		'''(2)'''  The analytical spectrum  (positive frequencies only)  of the input signal is:
	:$$X_+(f) = A \cdot {\rm \delta}(f- f_0) .$$		:$$X_+(f) = A \cdot {\rm \delta}(f- f_0) .$$

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	:$$A_1 = A - {A^3}\hspace{-0.1cm}/{8}, \hspace{0.5cm}A_3 = - {A^3}\hspace{-0.1cm}/{24}.$$		:$$A_1 = A - {A^3}\hspace{-0.1cm}/{8}, \hspace{0.5cm}A_3 = - {A^3}\hspace{-0.1cm}/{24}.$$

	*Using $A = 0.5$ the following is obtained: $A_1 \approx 0.484$ and  $A_3 \approx 0.005$. Thus, the distortion factor is:		*Using  $A = 0.5$  the following is obtained:  $A_1 \approx 0.484$  and  $A_3 \approx 0.005$.  Thus, the distortion factor is:
	:$$K = K_3 ={\|A_3\|}/{A_1}= {0.005}/{0.484} \hspace{0.15cm}\underline{ = 1.08\%}.$$		:$$K = K_3 ={\|A_3\|}/{A_1}= {0.005}/{0.484} \hspace{0.15cm}\underline{ = 1.08\%}.$$

	:Note that for the approximation  $g_3(x)$  only the cubic part  $K_3$  of the distortion factor is effective.		:Note that for the approximation  $g_3(x)$  only the cubic part  $K_3$  of the distortion factor is effective.

	*For  $A = 1.0$  and  $A = 1.5$  the following numerical values ~~arise as aresult~~:		*For  $A = 1.0$  and  $A = 1.5$  the following numerical values:
	:$$A = 1.0: A_1 \approx 0.875, \hspace{0.2cm} A_3 \approx		:$$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},$$		-0.041\hspace{0.3cm} \Rightarrow \hspace{0.3cm} \hspace{0.15cm}\underline{K \approx 4.76\%}\; \; \Rightarrow \; \; K_{g3},$$
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	'''(4)'''  <u>Approaches 1 and 3</u> are correct:		'''(4)'''  <u>Approaches 1 and 3</u>  are correct:
	*The approach  $g_5(x)$  is a better approximation for the sine function  $g(x)$  than the approximation  $g_3(x)$ in the entire domain.		*The approach  $g_5(x)$  is a better approximation for the sine function  $g(x)$  than the approximation  $g_3(x)$ in the entire domain.
	*Thus, the value  $K_{g5}$  computed in the subtask  '''(3)'''  is a better approximation for the actual distortion factor than  $K_{g3}$. <br>Therefore, the first statement is correct.		*Thus, the value  $K_{g5}$  computed in the subtask  '''(3)'''  is a better approximation for the actual distortion factor than  $K_{g3}$. <br>Therefore, the first statement is correct.
	*The second statement is false as already shown by the computation for  $A=1$ :   $K_{g3} \approx 4.76 \%$  is greater than  $K_{g5} \approx 4.45 \%$.		*The second statement is false as already shown by the computation for  $A=1$ :   $K_{g3} \approx 4.76 \%$  is greater than  $K_{g5} \approx 4.45 \%$.
	*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  $g_3(x)$  is below  $g_5(x)$  and thus there is also a greater deviation from the linear curve.
	*For  $A=0.5$ ,  $K_{g5} \approx K_{g3} = 1.08 \%$  will hold.		*For  $A=0.5$ ,  $K_{g5} \approx K_{g3} = 1.08 \%$  will hold.
	*The characteristic curve on the information page shows that for  $\|x\| \le 0.5$  the functions  $g_3(x)$  and  $g_5(x)$  are indistinguishable within the accuracy of drawing.		*The characteristic curve on the information page shows that for  $\|x\| \le 0.5$  the functions  $g_3(x)$  and  $g_5(x)$  are indistinguishable within the accuracy of drawing.
	*This also results in the same distortion factors.		*This also results in  (nearly)  the same distortion factors.
	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 13:04, 29 September 2021

2021-09-29T13:04:38Z

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

Guenter at 13:02, 29 September 2021

2021-09-29T13:02:14Z

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	[[File:P_ID894__LZI_A_2_3.png\|right\|frame\|Sinusoidal characteristic curve]]		[[File:P_ID894__LZI_A_2_3.png\|right\|frame\|Sinusoidal characteristic curve]]
	We consider a system with input  $x(t)$  and output  $y(t)$. For simplicity of description, the signals are considered to be dimensionless.		We consider a system with input  $x(t)$  and output  $y(t)$.  For simplicity of description, the signals are considered to be dimensionless.

	The relationship between the input signal  $x(t)$  and the output signal  $y(t)$  is given by the following characteristic curve in the range between  $-\pi/2$  and  $+\pi/2$~~ .~~		The relationship between the input signal  $x(t)$  and the output signal  $y(t)$  is given by the following characteristic curve in the range between  $-\pi/2$  and  $+\pi/2$:
	:$$g(x) = \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} -		:$$g(x) = \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} -
	\hspace{0.05cm}\text{...}$$		\hspace{0.05cm}\text{...}$$
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	:$$g_5(x) = x- x^3\hspace{-0.1cm}/{6}+x^5\hspace{-0.1cm}/{120}\hspace{0.05cm}.$$		:$$g_5(x) = x- x^3\hspace{-0.1cm}/{6}+x^5\hspace{-0.1cm}/{120}\hspace{0.05cm}.$$

	The input signal  $x(t) = A \cdot \cos(\omega_0 \cdot t)$  is always assumed ~~wherat the~~ values  $A = 0.5$,  $A = 1.0$  and  $A = 1.5$  are to be considered for the (dimensionless) signal amplitude.		*The input signal  $x(t) = A \cdot \cos(\omega_0 \cdot t)$  is always assumed.
			*The values  $A = 0.5$,  $A = 1.0$  and  $A = 1.5$  are to be considered for the (dimensionless) signal amplitude.


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	''Please note:''		''Please note:''
	*The task belongs to the chapter  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion\|Nonlinear ~~Distortion~~]].		*The task belongs to the chapter  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion\|Nonlinear Distortions]].
	*The resulting signal curves for  $x(t)$  and  $y(t)$  are shown graphically on the page  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion#Description_of_nonlinear_systems\|Description of nonlinear systems]] .		*The resulting signal curves for  $x(t)$  and  $y(t)$  are shown graphically on the page  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortion#Description_of_nonlinear_systems\|Description of nonlinear systems]] .

	*All powers required here refer to the resistance  $R = 1 \ \rm \Omega$  and thus have the unit  ${\rm V}^2$.		*All powers required here refer to the resistance  $R = 1 \ \rm \Omega$  and thus have the unit  ${\rm V}^2$.
	*The following trigonometric relations are assumed to be known:		*The following trigonometric relations are assumed to be known:

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 Aufgaben:Aufgabe 2.3: Sinusförmige Kennlinie to Aufgaben:Exercise 2.3: Sinusoidal Characteristic

← Older revision	Revision as of 12:03, 20 September 2021
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Guenter at 10:02, 20 September 2021

2021-09-20T10:02:10Z

← Older revision		Revision as of 12:02, 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^]]

Oezer at 21:50, 14 September 2021

2021-09-14T21:50:50Z

← Older revision		Revision as of 23:50, 14 September 2021
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	*For  $A=0.5$ ,  $K_{g5} \approx K_{g3} = 1.08 \%$  will hold.		*For  $A=0.5$ ,  $K_{g5} \approx K_{g3} = 1.08 \%$  will hold.
	*The characteristic curve on the information page shows that for  $\|x\| \le 0.5$  the functions  $g_3(x)$  and  $g_5(x)$  are indistinguishable within the accuracy of drawing.		*The characteristic curve on the information page shows that for  $\|x\| \le 0.5$  the functions  $g_3(x)$  and  $g_5(x)$  are indistinguishable within the accuracy of drawing.
	*This also results in ~~equal~~ distortion factors.		*This also results in the same distortion factors.
	{{ML-Fuß}}		{{ML-Fuß}}

Oezer at 21:21, 14 September 2021

2021-09-14T21:21:54Z

← Older revision		Revision as of 23:21, 14 September 2021
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	We consider a system with input  $x(t)$  and output  $y(t)$. For simplicity of description, the signals are considered to be dimensionless.		We consider a system with input  $x(t)$  and output  $y(t)$. For simplicity of description, the signals are considered to be dimensionless.

	The relationship between the input signal  $x(t)$  and the output signal  $y(t)$  is given in the range between  $-\pi/2$  and  $+\pi/2$  ~~by the following characteristic curve~~.		The relationship between the input signal  $x(t)$  and the output signal  $y(t)$  is given by the following characteristic curve in the range between  $-\pi/2$  and  $+\pi/2$ .
	:$$g(x) = \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} -		:$$g(x) = \sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} -
	\hspace{0.05cm}\text{...}$$		\hspace{0.05cm}\text{...}$$