Permittivity and Permeability of Materials Obstacle Course

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Permanent Materials

- 6061 3/8" Al rod stock
- Teflon
- Glass microscope slide
- HP Signal Generator (DC-1 GHz)
- Oscilloscope (at least 1GHz bandwidth)
- Miscellaneous electrical components

Materials to Borrow When Necessary

- Milling machine
- Lathe

Activities

Reading

1. Read the Wikipedia articles on permittivity and permeability. With the help of the instructor or TA try to achieve a physical understanding of just what the permittivity and permeability mean in a bulk material.
2. Read the first three sections of this paper (pages 1-27). Pay particular attention to the permittivity ($\epsilon$) / capacitance and permeability ($\mu$) / inductance associations.

* Permittivity of a Lossless Material From a Capacitance Measurement

1. Place three samples (air, Teflon, glass) between the aligned and polished ends of two 3/8" diameter, 1/2" lengths of 6061 Al rods (as shown at right). $d$ should be on the order of 1 mm. Measure the capacitances and, from the known surface area $A$ and spacing $d$, determine the material's relative permittivity. $C=\epsilon _{r}\epsilon _{0}{\frac {A}{d}}$ (for a capacitor with no fringing fields).
2. How do your measured permittivity values compare to standard reference values?
3. Use this web applet to build a capacitor and observe the field lines . Are there fringing fields?

$\epsilon _{r}$(air): 1.000536
$\epsilon _{r}$(Teflon): 2.1
$\epsilon _{r}$(glass): 3.7-10

* A Better Permittivity-Capacitance Measurement of a Lossless Material

1. Use this web applet to build a guarded-electrode capacitor (as shown at the right) and observe the field lines . Are there fringing fields?
2. Measure the three permittivities (air, Teflon, glass) again using this guarded-electrode setup.
3. How do these results compare to your first (unguarded) measurements?
4. How do these results compare to the standard values?

The measurements above for a lossless material amount to requiring the permittivity to be real (as opposed to complex). However, for a lossy material, the permittivity is complex and we need an additional characteristic (beyond simply the capacitance) to characterize the material. This additional characteristic is the conductance $G$. The measurement below will include the conductance of the material.

* Permittivity of a Lossy Material From a Capacitance Measurement (up to 50 MHz)

1. Employ the guarded-electrode setup above and measure the lossy material's capacitance and conductance. $C=\epsilon ^{'}{\frac {A}{d}}$ and $G=\omega \epsilon ^{''}{\frac {A}{d}}$, where $\epsilon ^{'}{\text{/}}$$\epsilon ^{''}$ are, respectively, the real and imaginary parts of the complex permittivity.
   1. Simultaneously measure the voltage across the resistor $V_{R}$ and Lock-in voltage $V_{Lock-in}$ at $V_{out}$. Employ a lock-in time constant that is much longer than the period of the AC driving voltage. The lock-in voltage $V_{Lock-in}$ is given by $V_{Lock-in}={\frac {1}{2}}V_{out}\,\,cos(\theta )$ , where $V_{out}$ is the voltage across the capacitor and $\theta$ is the phase difference between the AC driving voltage and $V_{out}$
   2. The current through the capacitor is given by $I_{C}=V_{R}/R$. The voltage across the capacitor is given by $V_{out}=Z_{C}I_{C}$, where $Z_{C}$ is the capacitor's impedance. Solve for the impedance $Z_{C}$.
   3. The capacitor's admittance $Y_{C}$ is given by $Y_{C}={\frac {1}{Z_{C}}}$, where the $Re(Y_{C})=G$ (the conductance) and the $Im(Y_{C})=B$ (the susceptance). Calculate $G$.
   4. Read section 13.1 (pages 106-107) in this [paper].
   5. Calculate the relative permittivity $\epsilon _{r}$ as $\epsilon '_{r}={\frac {C}{C_{air}}}$ and $\epsilon ''_{r}={\frac {G}{\omega C_{air}}}$.
   6. Do the above procedure for at least three frequencies between 10 MHz and 50 MHz.
   7. Plot $\epsilon '_{r}$ and $\epsilon ''_{r}$ as a function of frequency.