and Capacitance
1. Objective
The objective of Experiment #2 is to investigate the concepts of capacitors and capacitance.
The capacitance of coaxial and two-wire transmission lines will be measured and the basic properties
of a parallel plate capacitor will be investigated. The physical properties of a parallel plate capacitor
(plate separation, dielectric material) are varied to determine the effect on circuit behavior.
2. Introduction
A capacitor is a device used to store energy in an electric field. The basic geometry of the
ideal parallel-plate capacitor is shown in Figure 1. The plates are assumed to have an area A and the
separation distance between the plates is defined as d. The material between the two conducting
plates is a dielectric (insulator) characterized by permittivity
, (F/m). A voltage V is applied to the
capacitor which results in stored charge on the capacitor plates (total charges of +Q and
!Q). The
capacitance C defines the relationship between the applied voltage and the stored charge.
(1)
The vector electric field E (V/m) within the ideal capacitor is assumed to be uniform everywhere
between the plates so that the potential difference between the plates may be written as

(2)
Figure 1
. Ideal parallel plate capacitor. Figure 2.
Coaxial
transmission line.
The assumption of a uniform electric field between the plates of the capacitor is accurate for large,
closely-spaced plates. The uniform electric field within the ideal parallel plate capacitor requires that
the charge density on the capacitor plates also be uniform. The boundary condition for the electric
flux on the surface of the capacitor plate gives
(3)
Inserting Equations (2) and (3) into the capacitance definition of Equation (1) yields the ideal parallel-
plate capacitance formula:
(4)
The relationship between charge, voltage and capacitance holds true for both DC and time-varying
voltages. Since the capacitance C is constant for a given conductor geometry, this implies that a
time-varying voltage across a capacitor must yield a capacitor charge with the same time variation
[Q(t) = CV(t)].
In an actual parallel plate capacitor, the charge density is larger at the plate edges than the
interior region of the plate. This charge distribution causes fringing (or bending) of the electric field
at the edges of the capacitor. For large, closely-spaced plates, the fringing effect is minimal and the
ideal capacitor equation in (4) is an accurate approximation. As the spacing of the plates increases,
the fringing effect becomes more significant and the accuracy of Equation (4) is reduced. For the
ideal capacitor, the shape of the capacitor plates is immaterial. However, for actual capacitors,
conductors with sharp corners produce larger charge densities and more pronounced fringing.
When the atoms of the dielectric material insulating the conductors of the capacitor are
exposed to the capacitor electric field, the equivalent charges in the atoms are displaced (polarized).
The relative permittivity of the dielectric material is a measure of the degree of polarization within
the dielectric. More polarization occurs in dielectrics with higher permittivities. The additional
energy required to polarize the dielectric results in a higher stored energy within the capacitor. This
effect is seen in the parallel plate capacitance formula of Equation (4).
Capacitance exists whenever two conductors are in close proximity separated by an insulating
material. Even devices which are not specifically designed to store energy have capacitance. For
example, transmission lines designed to carry signals from one point to another via guided
electromagnetic waves contain capacitance based on their physical structure. The coaxial
transmission line shown in Figure 2 can be shown to contain the following capacitance per unit length:
(5)
Equation (5) represents the capacitance of an ideal coaxial capacitor
in which the fringing effects at the cable ends are neglected.
A two-wire transmission line formed by parallel conductors
also has capacitance. Given equal wire radii of a and a center-to-
center spacing between the wires of d, the ideal two-wire transmission
line capacitance per unit length (neglecting fringing at the conductor
ends) is given by (6)
where
, is the total permittivity of the insulating medium between the wires.
3. Equipment List
LCR Meter
Adjustable Parallel Plate Capacitors (283 mm × 283 mm, 200 mm × 200 mm)
Dielectric plates (glass, polystyrene), Coaxial cable, two insulated copper wires
Calipers, 120 k
S resistor, 100 :F capacitor
Breadboard Unit
Digital multimeter
4. Procedure
Note:
Be sure to discharge any capacitor before making a capacitance measurement.
1.
Use the LCR Meter to measure the capacitance of the segment of coaxial cable provided to
you. Measure the length of your cable segment and compute its total capacitance using
Equation (5) for the ideal coaxial capacitor (see Table 1 for the physical characteristics of
your coaxial cable). Compare your measured capacitance and your computed capacitance
with the manufacturer specification provided in Table 1.
Cable
a (mm)
b (mm)
,
r
Nominal capacitance (pF/ft)
RG 58C/U
0.451
1.47
2.3
28.5
RG 59/U
0.292
1.85
2.3
21.0
RG 213U
1.13
3.62
2.3
29.5
RG 223U
0.445
1.47
2.3
28.5
Table 1. Coaxial cable specifications.
2.
Using calipers, measure the conductor dimensions (conductor diameter, insulation diameter)
of the two straight insulated copper wires which are provided. Also measure the length of
the two-wires. Form a two-wire transmission line by placing the conductors parallel to each
other for the following spacings: [1] conductor insulation touching, [2] center-to-center wire
spacing of 0.5 inch, and [3] center-to-center spacing of 1 inch. Measure the capacitance in
each case with the LCR Meter. Discuss your results comparing them with the ideal two-wire
line given in Equation (6). Note that your two-wire line has an inhomogeneous insulating medium (air/dielectric).
3.
Use the LCR Meter to measure the capacitance of the smaller (200 mm × 200 mm) parallel
plate capacitor for plate spacings of d = 1, 2, 3, 4 and 6 mm with air as the insulating medium
between the plates. Use the insulating spacers on the corners of the capacitor plates to
achieve the necessary plate spacing. Each spacer provides either 1 mm or 3 mm of plate
separation, depending on its orientation. Align the leads from the LCR meter to the capacitor
in such a way as to minimize their capacitive contribution to the measurement. Repeat these
measurements for the larger (283 mm × 283 mm) parallel plate capacitor. Note that the plate
area of the larger capacitor is twice that of the smaller capacitor. Compare the measured
capacitances of these two air-insulated capacitors with the values obtained using the ideal
parallel plate capacitor equation. Discuss your results. How does fringing effect the
percentage errors obtained for the two capacitors?
4.
Connect the smaller and larger air-insulated parallel plate capacitors in series using plate
spacings of 1 mm for both capacitors. Measure the capacitance of the series combination.
Repeat the measurement for the capacitors in parallel. Compare your measured series and
parallel capacitances with the appropriate circuit equation using the measured capacitances
for the individual capacitors found in part 3.
5.
Measure and record the thicknesses of the glass and polystyrene plates. Using the larger
parallel plate capacitor, measure the capacitance of the glass-insulated capacitor and the
polystyrene-insulated capacitor. Compare these values with the equivalent air-insulated
capacitor and discuss the differences in capacitance with regard to polarization. Determine
the approximate permittivities of glass and polystyrene based on your measurements.
Compare these with the permittivity values of these materials found in your textbook.
6.
Place both the glass and polystyrene panels between the capacitor plates in the form of a
dielectric sandwich. Measure the capacitance of this inhomogeneous dielectric capacitor.
Compare your measured results with what you would expect to measure based on the results
for the individual plexiglass-insulated and fiberglass-insulated capacitors. Show the
equivalent circuit of this inhomogeneous dielectric capacitor in your report.
7.
Connect a parallel RC circuit with a capacitance of C = 100
:F and a resistance of R = 120
k
S. Use the LCR meter to measure the actual values of R and C for your components.
Note: You are using electrolytic capacitors so that the polarity of the voltage source
across the capacitor is important. Be careful to connect the ground connection of your
voltage source to the terminal labeled by a
! on the capacitor. Apply a DC voltage
of 12 V across the parallel circuit and measure the capacitor voltage with a multimeter.
Disconnect the voltage source (open circuit the source connection so the capacitor can
discharge only through the resistor) and measure the capacitor voltage at t = 0 s to t = 45 s in
5 s intervals.
(a.)
From circuit analysis, we may write a simple expression for the time-
dependent voltage across the capacitor V(t) in terms of the initial voltage V
o
,
the resistance R and the capacitance C. Using the same terms (V
o
, R and C)
and the definition of capacitance, determine a similar expression for the
capacitor charge with respect to time [Q(t)]. Plot the theoretical charge
expression and the measured charge (computed from the measured voltage)
verses time on the same plot.
(b.)
Based on each measurement of the capacitor ch