Revision as of 10:29, 21 September 2014

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⌘ Maxwell's Equation in a source free environment

Source free environment and free space:

$\nabla \cdot \vec E = 0 \qquad \qquad \qquad \ \ (1)$

$\nabla \times \vec E = -\frac{\partial}{\partial t} \vec B \qquad \qquad (2)$

$\nabla \cdot \vec B = 0 \qquad \qquad \qquad \ \ (3)$

$\nabla \times \vec B = \mu_0 \epsilon_0 \frac{\partial}{\partial t} \vec E \qquad \ \ \ (4)$

where div is a scalar function
$\mbox{div}\,\vec v = {\partial v_x \over \partial x} + {\partial v_y \over \partial y} + {\partial v_z \over \partial z} = \nabla \cdot \vec v$
and curl is a vector function
$\mbox{curl}\;\vec v = \left( {\partial v_z \over \partial y} - {\partial v_y \over \partial z} \right) \mathbf{i} + \left( {\partial v_x \over \partial z} - {\partial v_z \over \partial x} \right) \mathbf{j} + \left( {\partial v_y \over \partial x} - {\partial v_x \over \partial y} \right) \mathbf{k} = \nabla \times \vec v$

[Source: Wikipedia]

⌘ Wave equation

Taking the curl of Maxwell's equation

$\nabla \times \nabla \times \vec E = -\frac{\partial } {\partial t} \nabla \times \vec B = -\mu_0 \varepsilon_0 \frac{\partial^2 \vec E } {\partial t^2}$

$\nabla \times \nabla \times \vec B = \mu_0 \varepsilon_0 \frac{\partial } {\partial t} \nabla \times \vec E = -\mu_o \varepsilon_o \frac{\partial^2 \vec B}{\partial t^2}$

yields the wave equation:
${\partial^2 \vec E \over \partial t^2} \ - \ {c_0}^2 \cdot \nabla^2 \vec E \ \ = \ \ 0$

${\partial^2 \vec B \over \partial t^2} \ - \ {c_0}^2 \cdot \nabla^2 \vec B \ \ = \ \ 0$

with $c_0 = { 1 \over \sqrt{ \mu_0 \varepsilon_0 } } = 2.99792458 \times 10^8$ m/s

[Source: Wikipedia]

⌘ Homogeneous electromagnetic wave

single frequency

$\vec E(\vec r) = E_0 e^{j(\omega t - \vec k \cdot \vec r) }$ ,

$\vec B(\vec r) = B_0 e^{j(\omega t - \vec k \cdot \vec r) }$ ,

[Source: Wikipedia]

where

$\vec r = (x, y, z)$ and $\vec k = (k_x, k_y, k_z)$ so?
$j \,$ is the imaginary unit
$\omega = 2 \pi f \,$ is the angular frequency, [rad/s]
$f \,$ is the frequency [1/s]
$e^{j \omega t} = \cos(\omega t) + j \sin(\omega t)$ is Euler's formula

with $c = { c_0 \over n } = { 1 \over \sqrt{ \mu \varepsilon } }$ and $n = \sqrt{ \mu \varepsilon \over \mu_0 \varepsilon_0$

⌘ Comments and tasks

What is the difference between a static and a dynamic field
Develop the relations for a plain wave

Assume a plane wave: $E_x, H_y$ . Show that $\frac{E_x}{H_y}=Z_0 = \sqrt{\mu_0/\varepsilon_0}$

Calculation of a plane wave, proove that Z_0 = ... = E_x/H_y

⌘ Task: Plane wave propagation

Assume a plane wave: $E_x, H_y$ . Show that $\frac{E_x}{H_y}=Z_0 = \sqrt{\mu_0/\varepsilon_0}$

What is the relation between a plane wave and an omnidirectional wave?

⌘ Free space propagation

develop propagation equation, see (http://www.antenna-theory.com/basics/friis.php)

Power received in an area in a distance R from transmitter:

area of a sphere is $A_s = 4*\pi*R^2$
power transmitted from isotropic antenna is $P_t$
antenna area of receiver is $A_r = \lamda^2/4\pi$
power received in A_r = P_r

$P_r = P_t * A_r / A_s = P_r = P_t * A_r / (4*\pi*R^2)$

thus

$P_r = P_t \ G_t\ G_r\ \left (\frac{\lambda}{4\pi r} \right )^2\cdot$

convert into dB
provide examples for f = 10 MHz, 1 GHz, 100 GHz
discuss influences on radiation pattern

How much is 0 dB_m and 10 dB_m?

Convert dBm to mW is: mW = 10^(x/10), x = number of dBm
Convert mW to dBm is: dBm = 10*log10(y), y = number of mW

So you get:

0 dBm = 10^(0/10) = 1 mW
10 dBm = 10^(10/10) = 10 mW

Free space attenuation $L = 92,4 + 20 \log(d \mathrm{[km]}) + 20 \log(f \mathrm{[/GHz]})$

⌘Comments

Free space propagation from a transmit (t) to a receive (r) station.

Calculation of free space attenuation. Note the increased free-space attenuation of approx 5 dB from 900 to 1800 Unik/MHz, and a further increase of 3 dB from 1800 (GSM 1800) to 2450 Unik/MHz (802.11b). Note also that increasing the distance by a factor of 10 will increase the power requirements by 20 dB.

Free space propagation Calculation: http://spreadsheets.google.com/pub?key=p0EyjWrbirGKJXK43uluJfg

@@ Line 1: / Line 1: @@
-==⌘ Background: Wave Propagation ==
+=⌘ Maxwell's Equation in a source free environment =
+Source free environment and free space:
+<math> \nabla \cdot \vec E = 0  \qquad \qquad \qquad \ \ (1) </math>
+<math>\nabla \times \vec E = -\frac{\partial}{\partial t} \vec B \qquad \qquad (2) </math>
+<math>  \nabla \cdot \vec B = 0 \qquad \qquad \qquad \ \ (3) </math>
+<math> \nabla \times \vec B = \mu_0 \epsilon_0 \frac{\partial}{\partial t} \vec E  \qquad \ \ \ (4) </math>
+where div is a scalar function<br />
+<math>\mbox{div}\,\vec v = {\partial v_x \over \partial x} + {\partial v_y \over \partial y} + {\partial v_z \over \partial z} = \nabla \cdot \vec v </math><br />
+and curl is a vector function <br />
+<math>\mbox{curl}\;\vec v = \left( {\partial v_z \over \partial y} - {\partial v_y \over \partial z} \right) \mathbf{i} + \left( {\partial v_x \over \partial z} - {\partial v_z \over \partial x} \right) \mathbf{j} + \left( {\partial v_y \over \partial x} - {\partial v_x \over \partial y} \right) \mathbf{k} = \nabla \times \vec v</math>
+[Source: Wikipedia]
+=⌘ Wave equation =
+Taking the curl of Maxwell's equation
+<math>\nabla \times \nabla \times \vec E = -\frac{\partial } {\partial t} \nabla \times \vec B = -\mu_0 \varepsilon_0 \frac{\partial^2 \vec E }  {\partial t^2} </math>
+<math>\nabla \times \nabla \times \vec B = \mu_0 \varepsilon_0 \frac{\partial } {\partial t} \nabla \times \vec E = -\mu_o \varepsilon_o \frac{\partial^2 \vec B}{\partial t^2} </math>
+yields the wave equation:<br />
+<math>{\partial^2 \vec E \over \partial t^2} \ - \  {c_0}^2 \cdot \nabla^2 \vec E  \ \ = \ \ 0 </math>
+<math>{\partial^2 \vec B \over \partial t^2} \ - \  {c_0}^2 \cdot \nabla^2 \vec B  \ \ = \ \ 0 </math>
+with <math> c_0 = { 1 \over \sqrt{ \mu_0 \varepsilon_0 } } = 2.99792458 \times 10^8 </math> m/s
+[Source: Wikipedia]
+=⌘ Homogeneous electromagnetic wave =
+single frequency
+<math>\vec E(\vec r) = E_0 e^{j(\omega t  - \vec k \cdot \vec r) }  </math>,
+<math>\vec B(\vec r) = B_0 e^{j(\omega t - \vec k \cdot \vec r) }  </math>,
+[Source: Wikipedia]
+where
+* <math>\vec r = (x, y, z) </math> and  <math>\vec k = (k_x, k_y, k_z) </math> <span style="color:#000B80">so?
+* <math>j \, </math> is the imaginary unit
+* <math>\omega = 2 \pi f \,  </math> is the angular frequency, [rad/s]
+* <math> f \,</math> is the frequency [1/s]
+* <math>  e^{j \omega t} = \cos(\omega t) + j \sin(\omega t)  </math> is Euler's formula
+with <math>c = { c_0 \over n } =  { 1 \over \sqrt{ \mu \varepsilon } } </math> and
+<math>n = \sqrt{ \mu \varepsilon \over  \mu_0 \varepsilon_0  </math>
+==⌘ Comments and tasks ==
+* What is the difference between a static and a dynamic field
+* Develop the relations for a plain wave
+* Assume a plane wave: <math>E_x, H_y</math>. Show that <math>\frac{E_x}{H_y}=Z_0 = \sqrt{\mu_0/\varepsilon_0}</math>
 [[File:f3-8.png|450px|right|Calculation of a plane wave, proove that Z_0 = ... = E_x/H_y]]

Difference between revisions of "B1-Free Space Propagation"

Revision as of 10:29, 21 September 2014

Contents

⌘ Maxwell's Equation in a source free environment

⌘ Wave equation

⌘ Homogeneous electromagnetic wave

⌘ Comments and tasks

⌘ Task: Plane wave propagation

⌘ Free space propagation

⌘Comments

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