In [9]:
%pylab inline
from scipy.constants import epsilon_0, mu_0
# Interactive widget for the IPython notebook
from ipywidgets import widgets, interact, interactive, fixed
Let be:
The above quantities obey the Maxwell equations: $$\nabla \times \mathcal{H} = \frac{\partial \mathcal{D}}{\partial t} + \mathcal{J} $$ When the fields are AC, that is when the time variation is harmonic, the mathematical analysis can be simplified by using complex quantities. Any AC scalar quantities can be interpreted to $$ v = \sqrt{2} \Re\left( V e^{j \omega t} \right) $$ where $v$ si the instantaneous quantity and $V$ the complex quantity (or phasor). The factor $\sqrt{2}$ leads the magnitude of the complex quantity $V$ to be the effective root-mean-square (rms) value of the instantaneous quantity $v$.
The Maxwell equations in complex form are thus $$ \nabla \times E = - j \omega B $$ $$ \nabla \times H = j\omega D + J $$
Suppose we have a RF current $I$ flowing in coaxial line of diameter $D$. By applying Maxwell-Ampere law: $$ \oint H \cdot dl = I $$ where the complex quantity $I$ is the effective root-mean-square value of the instantaneous quantity $i$, following the interpretation $$ i = \sqrt{2} \Re\left( I e^{j \omega t} \right) $$ ie $$ I = I_{rms} = I_{peak} / \sqrt{2} $$
Integrating around a coaxial conductor, we thus have: $$ \pi D H_0 = I $$ where $H_0$ is the amplitude of $H$ at the surface of the conductor ($r=D/2$).
A wave starting at the surface of a good conductor and propagating inward is very quickly damped to insignificant values. The field is localized in a thin layer, a phenomenon known as skin effect. The distance in which a wave is attenuated to $1/e$ (36.8%) of its initial value is called the skin depth $\delta$, defined by: $$ \delta =\sqrt{\frac{2}{\omega \mu \sigma}} = \sqrt{\frac{2 \rho }{\omega \mu}} $$
In [10]:
def skin_depth(f=55.5e6, sigma=5e7):
'''
Skin depth calculation
f_MHz : frequency in Hz
sigma : metal conductivity in S/m
'''
delta = sqrt(2/(2*pi*f * mu_0 * sigma))
return delta
@interact(f_MHz=(20, 80, 0.1), sigma_e7=(1,6,0.1))
def delta_widget(f_MHz=55.5, sigma_e7=5):
delta = skin_depth(f_MHz*1e6, sigma_e7*1e7)
print('Skin depth delta= {} µm'.format(delta/1e-6))
The density of power flow into the conductor, which must also be that dissipated within the conductor, is given by: $$ S = E \times H^* = \eta |H_0|^2 $$ where $\eta=R + jX$ is the metal intrinsic impedance and $H_0$ the amplitude of $H$ at the surface. The time-average power dissipation per unit area (in W/m²) of surface cross section the the real part of the above power flow, or: $$ P_d = R |H_0|^2 $$ where $R$ is the intrinsic resistance or the surface impedance of the metal (in $\Omega$ per square): $$ R = \frac{1}{\sigma \delta} = \sqrt{\frac{\omega \mu}{2\sigma}} $$ see also http://www.microwaves101.com/encyclopedias/rf-sheet-resistance
Using the previous relation, the time-average power dissipation in W/m² in the conductor is thus in term of $I_{peak}$: $$ P_d = \frac{R}{2} \frac{|I_{peak}|^2}{(\pi D)^2} $$
Let be $V$ and $I$ the peak voltage and current on a coaxial transmission line under with a standing wave-ratio $S$. The maximum peak current in the line will be:
$$ I_{max} = \sqrt{\frac{2 P_{inc} S}{Z_0}} $$where $P_{inc}$ is the incident power and $Z_0$ the line characteristic impedance.
So finally, the time-average power dissipation in W/m² in the conductor is: $$ P_d = R \frac{P_{inc} S}{Z_0 (\pi D)^2} $$
In [11]:
# 9 inch 30 Ohm coxial line diameters [m]
line_Dint = 140e-3
line_Dout = 230e-3
# T-resonator DUT [m]
line_Dint = 128e-3
line_Dout = 219e-3
# conductor conductivity [S/m]
line_sigma_Cu = 4.4e7# 5.8e7 # inner conductor
line_sigma_Al = 3.5e7 # outer conductor
# source frequency [Hz]
line_freq = 60e6
# Line SWR
line_SWR = 2
# Input power [Watts]
line_Pinc = 0.5e6
In [12]:
def ohmic_losses_heat_flux(f, I_peak, D, sigma=5.8e7):
"""
Calculates the ohmic losses heat flux in [W/m^2] for a given current
on a coaxial conductor
Args:
f : frequency [Hz]
I_peak : peak current [A]
D : conductor diameter [m]
sigma: conductor conductivity [S/m]. Default: copper value = 5.8e7 S/m
Returns:
phi : ohmic losses heat flux [W/m^2]
"""
# surface resistance [Ohm]
Rs = 1/(skin_depth(f, sigma)*sigma) # sqrt(2.*pi*f*mu_0/(2.*sigma))
# RF resistive losses
phi = Rs/2 * abs(I_peak)**2 / (pi*D)**2
return phi
@interact(f_MHz=(40,70,0.1), I_peak_A=(0,3000,1), D_mm=(50,500,10), sigma_1e7=(0, 6, 0.1))
def widget_phi(f_MHz=line_freq/1e6, I_peak_A=100, D_mm=line_Dint/1e-3, sigma_1e7=line_sigma_Cu/1e7):
phi = ohmic_losses_heat_flux(f=f_MHz*1e6, I_peak=I_peak_A, D=D_mm*1e-3, sigma=sigma_1e7*1e7)
print('Heat Dissipation heat flux : phi={} W/m²'.format(phi))
In [13]:
# coaxial line characteristic impedance
line_Z0 = 1/(2*pi)*sqrt(mu_0/epsilon_0) * log(line_Dout/line_Dint)
print('Characteristic Impedance : Z0={} Ohm'.format(line_Z0))
def max_peak_current(Pin=1e6, SWR=1, Z0=30):
Imax = np.sqrt(2*Pin*SWR/Z0)
return Imax
@interact(Pin_MW=(0.1, 3, 0.1), SWR=(1,5,0.1), Z0=(10, 70, 5))
def widget_Imax(Pin_MW=line_Pinc/1e6, SWR=line_SWR, Z0=30):
Imax = max_peak_current(Pin_MW*1e6, SWR=SWR, Z0=Z0)
print('Maximum peak current in the line for SWR={} is I={} A'.format(line_SWR, Imax))
In [14]:
Imax=max_peak_current(Pin=1e6, SWR=1, Z0=30)
# Ohmic loss in the inner conductor
ohmic_losses_heat_flux(line_freq, I_peak=Imax, D=line_Dint, sigma=line_sigma_Cu)
Out[14]:
In [15]:
# Ohmic loss in the outer conductor
ohmic_losses_heat_flux(line_freq, Imax, D=line_Dout, sigma=line_sigma_Al)
Out[15]: