Aufgaben:Exercise 1.3Z: Calculating with Complex Numbers II - Revision history

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	'''(1)'''  The magnitude can be calculated according to the  [https://en.wikipedia.org/wiki/Pythagoras Pythagorean ] theorem:		'''(1)'''  The magnitude can be calculated according to the  [https://en.wikipedia.org/wiki/Pythagoras Pythagorean ] theorem:
	:$$\|z_1\| = \sqrt{x_1^2 + y_1^2}= \sqrt{4^2 + 3^2}\hspace{0.15cm}\underline{ = 5}.$$		:$$\|z_1\| = \sqrt{x_1^2 + y_1^2}= \sqrt{4^2 + 3^2}\hspace{0.15cm}\underline{ = 5}.$$
	*For the phase angle, the following applies according to the page  [[Signal_Representation/~~Calculating_With_Complex_Numbers~~#~~Representation_by_Magnitude_and_Phase~~\|Representation by Magnitude and Phase]]:		*For the phase angle, the following applies according to the page  [[Signal_Representation/Calculating_with_Complex_Numbers#Representation_by_magnitude_and_phase\|Representation by Magnitude and Phase]]:
	:$$\phi_1 = \arctan \frac{y_1}{x_1}= \arctan \frac{3}{4}\hspace{0.15cm}\underline{ = 36.9^{\circ}}.$$		:$$\phi_1 = \arctan \frac{y_1}{x_1}= \arctan \frac{3}{4}\hspace{0.15cm}\underline{ = 36.9^{\circ}}.$$

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	[[Category:Signal Representation: Exercises\|^1.3 Calculating with Complex Numbers		[[Category:Signal Representation: Exercises\|^1.3 Calculating with Complex Numbers
	^]]		^]]
	[[de:Aufgaben:1.~~3Z_Nochmals_komplexe_Zahlen~~]]		[[de:Aufgaben:Aufgabe 1.3Z: Nochmals komplexe Zahlen]]

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	'''(2)'''  Multiplying  $z_1$  by its conjugate complex  $z_1^{\star}$  yields the purely real quantity  $z_4$, as the following equations show:		'''(2)'''  Multiplying  $z_1$  by its conjugate complex  $z_1^{\star}$  yields the purely real quantity  $z_4$, as the following equations show:
	:$$z_4 = (x_1 + {\rm j} \cdot y_1)(x_1 - {\rm j} \cdot y_1)= {x_1^2 +		:$$z_4 = (x_1 + {\rm j} \cdot y_1)(x_1 - {\rm j} \cdot y_1)= {x_1^2 +y_1^2}= \|z_1\|^2 = 25,$$
	y_1^2}= \|z_1\|^2 = 25,$$		:$$z_4 = \|z_1\| \cdot {\rm e}^{{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm}\phi_1} \cdot \|z_1\| \cdot {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm} \phi_1}= \|z_1\|^2 = 25\hspace{0.3cm}\Rightarrow\hspace{0.3cm} x_4 \hspace{0.1cm}\underline{= 25}, \hspace{0.25cm}y_4 \hspace{0.15cm}\underline{= 0}.$$
	:$$z_4 = \|z_1\| \cdot {\rm e}^{{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm}\phi_1} \cdot \|z_1\| \cdot {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm} \phi_1}= \|z_1\|^2 = 25\hspace{0.3cm}
	\Rightarrow\hspace{0.3cm} x_4 \hspace{0.1cm}\underline{= 25}, \hspace{0.25cm}y_4 \hspace{0.15cm}\underline{= 0}.$$


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	'''(4)'''  If one writes  $z_2$  as magnitude and phase  ⇒   $\|z_2\| = 2, \ \phi_2 = 180^{\circ}$, one obtains for the product:		'''(4)'''  If one writes  $z_2$  as magnitude and phase  ⇒   $\|z_2\| = 2, \ \phi_2 = 180^{\circ}$, one obtains for the product:
	:$$\|z_6\| = \|z_1\| \cdot \|z_2\|= 5 \cdot 2 \hspace{0.15cm}\underline{= 10},$$		:$$\|z_6\| = \|z_1\| \cdot \|z_2\|= 5 \cdot 2 \hspace{0.15cm}\underline{= 10},$$
	:$$\phi_6 = \phi_1 + \phi_2 = 36.9^{\circ} + 180^{\circ} =		:$$\phi_6 = \phi_1 + \phi_2 = 36.9^{\circ} + 180^{\circ} =216.9^{\circ}\hspace{0.15cm}\underline{= -143.1^{\circ}}.$$
	216.9^{\circ}\hspace{0.15cm}\underline{= -143.1^{\circ}}.$$


	'''(5)'''  The phase is  $\phi_3 = 90^{\circ}$  (see graph above). This can be formally proven:		'''(5)'''  The phase is  $\phi_3 = 90^{\circ}$  (see graph above). This can be formally proven:
	:$$\phi_3 = \arctan \left( \frac{6}{0}\right) = \arctan (\infty)		:$$\phi_3 = \arctan \left( \frac{6}{0}\right) = \arctan (\infty)\hspace{0.2cm}\Rightarrow \hspace{0.2cm} \phi_3 \hspace{0.15cm}\underline{= 90^{\circ}}.$$
	\hspace{0.2cm}\Rightarrow \hspace{0.2cm} \phi_3 \hspace{0.15cm}\underline{= 90^{
	\circ}}.$$


	'''(6)'''  First, the more inconvenient solution:		'''(6)'''  First, the more inconvenient solution:
	:$$z_7 = \frac{z_3}{z_1}= \frac{6{\rm j}}{4 + 3{\rm j}} = \frac{6{\rm j}\cdot(4 - 3{\rm j})}{(4 + 3{\rm j})\cdot (4 - 3{\rm j})} =		:$$z_7 = \frac{z_3}{z_1}= \frac{6{\rm j}}{4 + 3{\rm j}} = \frac{6{\rm j}\cdot(4 - 3{\rm j})}{(4 + 3{\rm j})\cdot (4 - 3{\rm j})} =\frac{18 +24{\rm j}}{25} = 1.2 \cdot{\rm e}^{{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 53.1^{ \circ}}.$$
	\frac{18 +24{\rm j}}{25} = 1.2 \cdot{\rm e}^{{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 53.1^{ \circ}}.$$
	*An easier way of solving the problem is:		*An easier way of solving the problem is:
	:$$\|z_7\| = \frac{\|z_3\|}{\|z_1\|} = \frac{6}{5}\hspace{0.15cm}\underline{=1.2}, \hspace{0.3cm}\phi_7 = \phi_3 - \phi_1 = 90^{\circ} - 36.9^{\circ}		:$$\|z_7\| = \frac{\|z_3\|}{\|z_1\|} = \frac{6}{5}\hspace{0.15cm}\underline{=1.2}, \hspace{0.3cm}\phi_7 = \phi_3 - \phi_1 = 90^{\circ} - 36.9^{\circ}\hspace{0.15cm}\underline{=53.1^{\circ}}.$$
	\hspace{0.15cm}\underline{=53.1^{\circ}}.$$
	{{ML-Fuß}}		{{ML-Fuß}}

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	[[Category:Signal Representation: Exercises\|^1.3 Calculating with Complex Numbers		[[Category:Signal Representation: Exercises\|^1.3 Calculating with Complex Numbers
	^]]		^]]
			[[de:Aufgaben:1.3Z_Nochmals_komplexe_Zahlen]]

Guenter at 12:28, 24 May 2021

2021-05-24T12:28:39Z

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	{{quiz-Header\|Buchseite=~~Signaldarstellung~~/~~Prinzip der Nachrichtenübertragung~~}}		{{quiz-Header\|Buchseite=Signal_Representation/Calculating_With_Complex_Numbers}}

	[[File:P_ID802__Sig_Z_1_3.png\|right\|frame\|Considered numbers <br>in the complex plane]]		[[File:P_ID802__Sig_Z_1_3.png\|right\|frame\|Considered numbers <br>in the complex plane]]

Guenter at 16:18, 17 May 2021

2021-05-17T16:18:08Z

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	{{quiz-Header\|Buchseite=Signaldarstellung/Prinzip der Nachrichtenübertragung}}		{{quiz-Header\|Buchseite=Signaldarstellung/Prinzip der Nachrichtenübertragung}}

	[[File:P_ID802__Sig_Z_1_3.png\|right\|frame\|Considered numbers in the complex plane]]		[[File:P_ID802__Sig_Z_1_3.png\|right\|frame\|Considered numbers <br>in the complex plane]]
	The following three complex quantities are shown in the complex plane to the right:		The following three complex quantities are shown in the complex plane to the right:

Guenter at 11:36, 12 April 2021

2021-04-12T11:36:56Z

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	[[Category:Signal Representation: Exercises\|^1.3 Calculating ~~With~~ Complex Numbers		[[Category:Signal Representation: Exercises\|^1.3 Calculating with Complex Numbers
	^]]		^]]

Guenter at 15:37, 9 April 2021

2021-04-09T15:37:12Z

← Older revision		Revision as of 17:37, 9 April 2021
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	'''(1)'''  The magnitude can be calculated according to the  [https://en.wikipedia.org/wiki/Pythagoras Pythagorean ] theorem:		'''(1)'''  The magnitude can be calculated according to the  [https://en.wikipedia.org/wiki/Pythagoras Pythagorean ] theorem:
	:$$\|z_1\| = \sqrt{x_1^2 + y_1^2}= \sqrt{4^2 + 3^2}\hspace{0.15cm}\underline{ = 5}.$$		:$$\|z_1\| = \sqrt{x_1^2 + y_1^2}= \sqrt{4^2 + 3^2}\hspace{0.15cm}\underline{ = 5}.$$
	*For the phase angle, the following applies according to the page  [[Signal_Representation/Calculating_With_Complex_Numbers#~~Representation_by_Amplidute_and_Phase~~\|Representation by ~~Amplidute~~ and Phase]]:		*For the phase angle, the following applies according to the page  [[Signal_Representation/Calculating_With_Complex_Numbers#Representation_by_Magnitude_and_Phase\|Representation by Magnitude and Phase]]:
	:$$\phi_1 = \arctan \frac{y_1}{x_1}= \arctan \frac{3}{4}\hspace{0.15cm}\underline{ = 36.9^{\circ}}.$$		:$$\phi_1 = \arctan \frac{y_1}{x_1}= \arctan \frac{3}{4}\hspace{0.15cm}\underline{ = 36.9^{\circ}}.$$

Guenter at 15:28, 9 April 2021

2021-04-09T15:28:32Z

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	{Calculate  $z_5 = x_5 + {\rm j} \cdot y_5$ .		{Calculate  $z_5 = z_1 + 2 \cdot z_2 - {z_3}/{2} = x_5 + {\rm j} \cdot y_5$ .
	\|type="{}"}		\|type="{}"}
	$x_5\ = \ $ { 0. }		$x_5\ = \ $ { 0. }
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	'''(1)'''  The magnitude can be calculated according to the  [https://en.wikipedia.org/wiki/Pythagoras Pythagorean ] theorem:		'''(1)'''  The magnitude can be calculated according to the  [https://en.wikipedia.org/wiki/Pythagoras Pythagorean ] theorem:
	:$$\|z_1\| = \sqrt{x_1^2 + y_1^2}= \sqrt{4^2 + 3^2}\hspace{0.15cm}\underline{ = 5}.$$		:$$\|z_1\| = \sqrt{x_1^2 + y_1^2}= \sqrt{4^2 + 3^2}\hspace{0.15cm}\underline{ = 5}.$$
	*For the phase angle, the following applies according to the page  [[Signal_Representation/Calculating_With_Complex_Numbers#~~Darstellung_nach_Betrag_und_Phase~~\|~~Darstellung nach Betrag und~~ Phase]] :		*For the phase angle, the following applies according to the page  [[Signal_Representation/Calculating_With_Complex_Numbers#Representation_by_Amplidute_and_Phase\|Representation by Amplidute and Phase]]:
	:$$\phi_1 = \arctan \frac{y_1}{x_1}= \arctan \frac{3}{4}\hspace{0.15cm}\underline{ = 36.9^{\circ}}.$$		:$$\phi_1 = \arctan \frac{y_1}{x_1}= \arctan \frac{3}{4}\hspace{0.15cm}\underline{ = 36.9^{\circ}}.$$

Guenter at 15:19, 9 April 2021

2021-04-09T15:19:14Z

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	: $$ z_2 = -2 ,$$		: $$ z_2 = -2 ,$$
	: $$z_3 = 6\cdot{\rm j} .$$		: $$z_3 = 6\cdot{\rm j} .$$
	Within the framework of this task, the following quantities are to be calculated::		Within the framework of this task, the following quantities are to be calculated:
	: $$z_4 = z_1 \cdot z_1^{\star},$$		: $$z_4 = z_1 \cdot z_1^{\star},$$
	: $$z_5 = z_1 + 2 \cdot z_2 - {z_3}/{2},$$		: $$z_5 = z_1 + 2 \cdot z_2 - {z_3}/{2},$$
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	''Hints:''		''Hints:''
	*This exercise belongs to the chapter [[Signal_Representation/Calculating_With_Complex_Numbers\|Calculating ~~With~~ Complex Numbers]].		*This exercise belongs to the chapter [[Signal_Representation/Calculating_With_Complex_Numbers\|Calculating with Complex Numbers]].
	*The topic is also covered in the ~~teaching~~ video [[Rechnen_mit_komplexen_Zahlen_(Lernvideo)\|Rechnen mit komplexen Zahlen]] .		*The topic of this task is also covered in the (German language) learning video <br>      [[Rechnen_mit_komplexen_Zahlen_(Lernvideo)\|Rechnen mit komplexen Zahlen]]   ⇒   "Arithmetic operations involving complex numbers".
	*Enter the phase values in the range of  $-\hspace{-0.05cm}180^{\circ} < \phi ≤ +180^{\circ}$.		*Enter the phase values in the range of  $-\hspace{-0.05cm}180^{\circ} < \phi ≤ +180^{\circ}$.

Noah at 09:11, 26 March 2021

2021-03-26T09:11:59Z

← Older revision		Revision as of 11:11, 26 March 2021
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	{{quiz-Header\|Buchseite=Signaldarstellung/Prinzip der Nachrichtenübertragung}}		{{quiz-Header\|Buchseite=Signaldarstellung/Prinzip der Nachrichtenübertragung}}

	[[File:P_ID802__Sig_Z_1_3.png\|right\|frame\|~~Betrachtete Zahlen~~ in ~~der komplexen Ebene~~]]		[[File:P_ID802__Sig_Z_1_3.png\|right\|frame\|Considered numbers in the complex plane]]
	The following three complex quantities are shown in the complex plane to the right:		The following three complex quantities are shown in the complex plane to the right: