Capacitors
A capacitor is a passive element designed to store energy in its electric field. Besides resistors, capacitors
are the most common electrical components. Capacitors are used extensively in electronics,
communications, computers, and power systems. For example, they are used in the tuning circuits of
radio receivers and as dynamic memory elements in computer systems.
A capacitor is typically constructed as depicted at right.
A capacitor consists of two conducting plates separated by an insulator (or
dielectric).
In many practical applications, the plates may be aluminum foil while the
dielectric may be air, ceramic, paper, or mica. When a voltage source v is
connected to the capacitor, as shown, the source deposits a positive charge q on
one plate and a negative charge − q on the other. The capacitor is said to store
the electric charge. The amount of charge stored, represented by q, is directly
proportional to the applied voltage v so that
q = Cv
where C, the constant of proportionality, is known as the capacitance of the capacitor. The unit of
capacitance is the farad (F), in honor of the English physicist Michael Faraday (1791–1867).
Capacitance is the ratio of the charge on one plate of a capacitor to the voltage difference
between the two plates, measured in farads (F).
Although the capacitance C of a capacitor is the ratio of the charge q per plate to the applied voltage v, it
does not depend on q or v. It depends on the physical dimensions of the capacitor. For example, for the
parallel-plate capacitor shown at top, the capacitance is given by
where A is the surface area of each plate, d is the distance between the plates, and  is the permittivity
of the dielectric material between the plates. Although the equation applies to only parallel-plate
capacitors, we may infer from it that, in general, three factors determine the value of the capacitance:
1. The surface area of the plates—the larger the area, the greater the capacitance.
2. The spacing between the plates—the smaller the spacing, the greater the capacitance.
3. The permittivity of the material—the higher the permittivity, the greater the capacitance.
Capacitors are commercially available in different values and types. Typically, capacitors have values in
the picofarad (pF) to microfarad (µF)range. They are described by the dielectric material they are made
of and by whether they are of fixed or variable type. The circuit
symbols for fixed and variable capacitors are shown at right. Note
that according to the passive sign convention, current is considered
to flow into the positive terminal of the capacitor when the capacitor is being charged, and out of the
positive terminal when the capacitor is discharging.
Electrolytic - Made of electrolyte, basically conductive salt in solvent. Aluminum electrodes
are used by using a thin oxidation membrane. Most common type, polarized capacitor.
Applications: Ripple filters, timing circuits. Cheap, readily available, good for storage of charge
(energy). Not very accurate, marginal electrical properties, leakage, drifting, not suitable for
use in HF (High Frequency) circuits, available in very small or very large values in uF.
The most important characteristic of electrolytic capacitors is that they have polarity. They
have a positive and a negative electrode. This means that it is very important which way round
they are connected. they will explode if the rated working voltage is exceeded or polarity is
reversed, so be careful.
Tantalum - Made of Tantalum Pentoxide. They are electrolytic capacitors but used with a material
called tantalum for the electrodes. Superior to electrolytic capacitors, excellent temperature and
frequency characteristics. When tantalum powder is baked in order to solidify it, a crack forms
inside. An electric charge can be stored on this crack. Like electrolytic, tantalums are polarized so
watch the '+' and '-' indicators. Tantalum capacitors are a little bit more expensive than aluminum
electrolytic capacitors. Capacitance can change with temperature as well as frequency, and these
types are very stable. Therefore, tantalum capacitors are used for circuits which demand high stability in the
capacitance values. Also, it is said to be common sense to use tantalum capacitors for analog signal systems,
because the current-spike noise that occurs with aluminum electrolytic capacitors does not appear. Aluminum
electrolytic capacitors are fine if you don't use them for circuits which need the high stability characteristics of
tantalum capacitors.
Mostly used in analog signal systems because of the lack of current-spike-noise. Small size fits anywhere, reliable,
most common values readily available. Expensive, easily damaged by spikes, large values exists but may be hard
to obtain.
Super Capacitors - The Electric Double Layer capacitor is a real miracle piece of work.
Capacitance is 0.47 Farad (470,000 uF). Despite the large capacitance value, its physical
dimensions are relatively small. It has a diameter of 21 mm (almost an inch) and a height of
11 mm (1/2 inch). Like other electrolytic the super capacitor is also polarized so exercise
caution in regards to the break-down voltage.
Care must be taken when using this capacitor. It has such large capacitance that, without
precautions, it would destroy part of a power supply such as the bridge rectifier, volt regulators, or whatever
because of the huge inrush current at charge. For a brief moment, this capacitor acts like a short circuit when the
capacitor is charged. Protection circuitry is a must for this type.
Polyester Film - No polarity, This capacitor uses a thin polyester film as a dielectric. Not as
high a tolerance as polypropylene, but cheap and handy, temperature stable, readily available,
widely used. Tolerance is approx 5% to 10%. Can be quite large depending on capacity or
rated voltage and so may not be suitable for all applications
Ceramic - Constructed with materials such as titanium acid barium for dielectric. Internally
these capacitors are not constructed as a coil, so they are well suited for use in high frequency
applications. Typically used to by-pass high frequency signals to ground. They are shaped like
a disk, available in very small capacitance values and very small sizes. Together with the
electrolytics the most widely available and used capacitor around. Comes in very small size
and value, very cheap, reliable. Subject to drifting depending on ambient temperature. NPO
types are the temperature stable types. They are identified by a black stripe on top
Multilayer Ceramic - Dielectric is made up of many layers. Small in size, very good temperature
stability, excellent frequency stable characteristics. Used in applications to filter or bypass the
high frequency to ground. They don't have a polarity. *Multilayer caps suffer from high-Q
internal (parallel) resonances - generally in the VHF range. The CK05 style 0.1uF/50V caps for
example resonate around 30MHz. The effect of this resonance is effectively no apparent
capacitance near the resonant frequency.
As with all ceramic capacitors, be careful bending the legs or spreading them apart to close to
the disc body or they may get damaged
Silver-Mica - Mica is used as a dielectric. Used in resonance circuits, frequency filters, and military
RF applications.
Highly stable, good temperature coefficient, excellent for endurance because of their frequency
characteristics, no large values, high voltage types available, can be expensive but worth the extra
dimes.
Adjustable Capacitors - Also called trimmer capacitors or variable capacitors. It uses ceramic or
plastic as a dielectric.
Most of them are color coded to easily recognize their tunable size. The ceramic type has the
value printed on them.
Colors are: yellow (5pF), blue (7pF), white (10pF), green (30pF), brown (60pf). There are a
couple more colors like red, beige, and purple which are not listed here. Anyways, you get the
idea...
Tuning or 'air-core' capacitors.
They use the surrounding air as a dielectric. I have seen these variable capacitor types of
incredible dimensions, especially the older ones. Amazing it all worked. Mostly used in radio
and radar equipment. This type usually have more (air) capacitors combined (ganged) and
so when the adjustment axel is turned, the capacitance of all of them changes
simultaneously. The one on the right has a polyester film as a dielectric constant and
combines two independent capacitors plus included is a trimmer cap, one for each side.
Capacitors are used to block dc, pass ac, shift phase, store energy, start motors, and suppress noise.
To obtain the current-voltage relationship of the capacitor, we take the derivative of both sides of q =
Cv.
Since I = dq / dt differentiating both sides gives
This is the current-voltage relationship for a capacitor, assuming the
positive sign convention. The relationship is illustrated at right for a
capacitor whose capacitance is independent of voltage. Capacitors that
satisfy the above equation are said to be linear. For a nonlinear
capacitor, the plot of the current-voltage relationship is not a straight line.
Although some capacitors are nonlinear, most are linear. We will assume
linear capacitors for this discussion.
The voltage-current relation of the capacitor can be obtained by integrating of the previous equation.
We get
where v(t0) = q(t0)/C is the voltage across the capacitor at time t 0 . The equation above shows that
capacitor voltage depends on the past history of the capacitor current. Hence, the capacitor has
memory—a property that is often exploited.
The instantaneous power delivered to the capacitor is
The energy stored in the capacitor is therefore
We note that v( −∞ ) = 0, because the capacitor was uncharged at t = −∞ . Thus,
This may be rewritten as
This represents the energy stored in the electric field that exists between the plates of the capacitor.
This energy can be retrieved, since an ideal capacitor cannot dissipate energy. In fact, the word
capacitor is derived from this element’s capacity to store energy in an electric field.
We should note the following important properties of a capacitor:
1. Note that when the voltage across a capacitor is not changing with time (i.e., dc voltage), the current
through the capacitor is zero. Thus,
A capacitor is an open circuit to dc.
However, if a battery (dc voltage) is connected across a capacitor, the capacitor charges.
2. The voltage on the capacitor must be continuous.
The voltage on a capacitor cannot change abruptly.
The capacitor resists an abrupt change in the voltage across it. According to
a discontinuous change in voltage requires an infinite current,
which is physically impossible.
For example, the voltage across a capacitor may take the form
shown in (a), whereas it is not physically possible for the capacitor
voltage to take the form shown in (b)because of the abrupt
change. Conversely, the current through a capacitor can change
instantaneously.
3. The ideal capacitor does not dissipate energy. It takes power from the circuit when storing energy in
its field and returns previously stored energy when delivering power to the
circuit.
4. A real, nonideal capacitor has a parallel-model leakage resistance, as
shown at right. The leakage resistance may be as high as 100 MOhm and
can be neglected for most practical applications. For this reason, we will
assume ideal capacitors.
Example: Obtain the energy stored in each capacitor shown below under DC conditions.
1.
Under DC conditions, replace each cap with an open circuit and redraw.
2. Find the voltage across each of the capacitors.
The current through the series combination of the2-k and 4-k resistors is obtained by current
division as
Hence, the voltages v 1 and v 2 across the capacitors are
v 1 = 2000i = 4 V
v 2 = 4000i = 8 V
3. Find the energy stored in each from ½ CV2.
Capacitor Voltage-current Relations
Overview:
In this assignment, we will measure the relationship between the voltage difference across a
capacitor and the current passing through it. We will apply several types of time-varying signals to a
series combination of a resistor and a capacitor. The voltage difference across the resistor, in
conjunction with Ohm’s law, will provide an estimate of the current through the capacitor. This
current can be related to the voltage difference across the capacitor.
The results of our voltage-current measurements will be compared to analytical expectations.
General Discussion:
We will use the circuit of Figure 1 in this lab assignment.
Both the voltage difference across the capacitor and the
resistor (vC(t) and vR(t)) will be measured. From this data,
we can readily compare the voltage across the capacitor
with the current through the capacitor. Since the voltage
across the resistor is measured, we can readily infer the current through the resistor via Ohm’s law:
The resistor and capacitor are in series, so the current through the capacitor is the same as the
current through the resistor, so:
Since we are also measuring the voltage difference across the capacitor, vC(t), we can readily
compare these parameters with our expectations based on our mathematical models of the
capacitor voltage-current relationships.
Pre-lab:
For the circuit of Figure 1, use the capacitor voltage-current relations to overlay sketches of the
capacitor voltage and the capacitor current (vC(t) and iC(t)) if the capacitor voltage is:
(a)
A sinusoidal wave, v(t), with frequency (f) and amplitude (A) as shown in Figure 2 (a)
(b)
A triangular wave, v(t), with frequency (f) and amplitude (A) as shown in Figure 2 (b).
Label your sketch to show the amplitude and period of the capacitor current for both of the above
cases. Your results may be dependent up on the parameters A, f, R, and C. Be sure that your sketches of
voltage and current share the same time axis!
Lab Procedures:
Construct the circuit of Figure 1, using R = 100 and C = 1F. Use channel 1 of your oscilloscope to
measure the resistor voltage difference, and channel 2 of your oscilloscope to measure the capacitor
voltage difference. Use channel 1 of your waveform generator (W1) to apply the voltage vin(t) in Figure 1.
Set up a math channel to calculate the current through the capacitor per equation (2) in the pre-lab1. Set
the oscilloscope measurements to provide at least the amplitude of each of the three displayed
waveforms.
1
Detailed instructions for doing this are provided in Appendix A.
1. Apply a sinusoidal input voltage with frequency = 1kHz, amplitude = 2V, and offset = 0V to
the circuit of Figure 1. Use your oscilloscope to display the data listed above (waveforms
corresponding to C1, C2, and M1; measurement window displaying amplitudes of C1, C2, and
M1). Record the image of the oscilloscope window, showing the waveforms and their measured
amplitudes.
2. Apply a sinusoidal input voltage with frequency = 2 kHz, amplitude = 2V, and offset = 0V to
the circuit of Figure 1. Use your oscilloscope to display the data listed above (waveforms
corresponding to C1, C2, and M1; measurement window displaying amplitudes of C1, C2, and
M1). Record the image of the oscilloscope window, showing the waveforms and their measured
amplitudes.
3. Apply a triangular input voltage with frequency = 100 Hz, amplitude = 4V, and offset = 0V to
the circuit of Figure 1. Use your oscilloscope to display the data listed above (waveforms
corresponding to C1, C2, and M1; measurement window displaying amplitudes of C1, C2, and
M1). Record the image of the oscilloscope window, showing the waveforms and their measured
amplitudes.
Post-lab Exercises:
For the three cases in the lab procedures (1kHz sinusoid, 2kHz sinusoid, 100Hz triangular wave), use
your pre-lab results to sketch the expected capacitor current waveforms corresponding to the capacitor
voltage waveforms you measured in the lab procedures. Comment briefly on the agreement between
the measured and expected capacitor currents for each of these cases. In your comments, be sure to
include a quantitative comparison (including percent difference) between the expected and measured
amplitudes of the capacitor current.
Include the following in your blog:
1. Attach the sketches of the capacitor voltage and current for both sinusoidal and triangular inputs.
2. Attach an image of the oscilloscope window, showing the capacitor voltage and current waveforms
and the measured amplitudes of the waveforms for a 1kHz sinusoidal input.
3. Attach an image of the oscilloscope window, showing the capacitor voltage and current waveforms
and the measured amplitudes of the waveforms for a 2kHz sinusoidal input.
4. Attach an image of the oscilloscope window, showing the capacitor voltage and current waveforms
and the measured amplitudes of the waveforms for a 100Hz triangular input.
Appendix A - Math channel to calculate current from resistor's
voltage
The analog discovery provides capabilities for performing mathematical operations on the displayed
waveforms and displaying the result. Essentially, there are two basic “types” of mathematical
operations which can be performed: “Simple” and “Custom”. The simple math operations consist of
addition, subtraction, or multiplication of the two channels. The custom operations are much more
wide-ranging. In order to determine the resistor current, we want to divide the resistor voltage by a
constant (the resistance value), so we will create a custom math channel. To do this, follow the
steps below:
1. Click on
Select
from the resulting drop-down
menu and choose
.
2.
A custom math function window will open, as shown below. Type the desired math function
(typically a function of the scope channels, C1 and C2) in the text box in this window or use the
buttons in the window to create the function. We are using channel 1 (C1) to measure the
resistor voltage. The current through the resistor is simply the resistor voltage divided by the
resistance value (100Q), so our function
is: C1/100, also shown in the Figure below. Click
to display the function
in the main window.
3. The properties of the math channel display can be adjusted using the channel’s control box, just
as any with any other channel displayed by the scope. A typical control box is shown below:
4.
2
The units of our math channel are amperes. It is nice to have
the displayed units agree with the actual units of the measurement. To change the units, click
on the
icon on the control box and select
from the resulting drop-down
menu. Volts, will typically be the default unit; it you want the vertical axis in amps, click the
icon and select A from the resulting menu2
Choices of units are volts (V), amps (A), and watts (W).
SERIES AND PARALLEL CAPACITORS
We know from resistive circuits that series-parallel combination is a
powerful tool for reducing circuits. This technique can be extended to
series-parallel connections of capacitors, which are sometimes
encountered. We desire to replace these capacitors by a single
equivalent capacitor C eq .
In order to obtain the equivalent capacitor C eq of N capacitors in parallel,
consider the circuit in a. The equivalent circuit is in b. Note that the
capacitors have the same voltage v across them. Applying KCL to (a),
i = i1 + i2 + i3 + · · · + iN
But i k = C k dv/dt. Hence,
Where
C eq = C 1 + C 2 + C 3 + · · · + C N
The equivalent capacitance of N parallel-connected capacitors is the sum of the individual capacitances.
We observe that capacitors in parallel combine in the same manner as resistors in series.
We now obtain C eq of N capacitors connected in series by comparing
the circuit in (a) with the equivalent circuit in (b).
Note that the same current i flows (and consequently the same
charge) through the capacitors. Applying KVL to the loop in (a),
v = v 1 + v2 + v3 + · · · + v N
Where
The initial voltage v(t 0 ) across C eq is required by KVL to be the sum of the capacitor voltages at t 0 . Or,
v(t 0) = v 1 (t 0) + v 2 (t 0) + · · · + vN (t 0)
The equivalent capacitance of series-connected capacitors is the reciprocal of the
sum of the reciprocals of the individual capacitances.
Note that capacitors in series combine in the same manner as resistors in parallel.
Example:
Find the equivalent capacitance for the collection of capacitors shown:
INDUCTORS
An inductor is a passive element designed to store energy in its magnetic field. Inductors find numerous
applications in electronic and power systems. They are used in power supplies, transformers, radios,
TVs, radars, and electric motors.
Any conductor of electric current has inductive properties and may
be regarded as an inductor. But in order to enhance the inductive
effect, a practical inductor is usually formed into a cylindrical coil
with many turns of conducting wire, as shown
An inductor consists of a coil of conducting wire.
If current is allowed to pass through an inductor, it is found that the voltage across the inductor is
directly proportional to the time rate of change of the current. Using the passive sign convention,
where L is the constant of proportionality called the inductance of the inductor. The unit of inductance is
the henry (H), named in honor of the American inventor Joseph Henry (1797–1878). 1 henry equals 1
volt-second per ampere.
Inductance is the property whereby an inductor exhibits opposition to the change
of current flowing through it, measured in henrys (H).
The inductance of an inductor depends on its physical dimension and construction. Formulas for
calculating the inductance of inductors of different shapes are derived from electromagnetic theory and
can be found in standard electrical engineering handbooks. For example, for the inductor (solenoid)
shown
where N is the number of turns, l is the length, A is the cross-sectional area, and µ is the permeability of
the core. We can see from the equation above that inductance can be increased by increasing the
number of turns of coil, using material with higher permeability as the core, increasing the crosssectional area, or reducing the length of the coil.
Like capacitors, commercially available inductors come in different
values and types. Typical practical inductors have inductance
values ranging from a few microhenrys (µH), as in communication
systems, to tens of henrys (H) as in power systems. Inductors may
be fixed or variable. The core may be made of iron, steel, plastic, or
air. The terms coil and choke are also used for inductors. The circuit symbols for inductors are shown
below following the passive sign convention.
Common inductors are shown below.
is the voltage-current relationship for an inductor. The figure at right shows
this relationship graphically for an inductor whose inductance is independent of
current. Such an inductor is known as a linear inductor. For a nonlinear
inductor, the plot will not be a straight line because its inductance varies with
current. We will assume linear inductors unless stated otherwise.
The current-voltage relationship is obtained from the above equation as
Integrating gives
Or
where i(t 0 ) is the total current for −∞ < t < t0 and i( −∞ ) = 0. The idea of making i( −∞ ) = 0 is practical
and reasonable, because there must be a time in the past when there was no current in the inductor.
The inductor is designed to store energy in its magnetic field. The power delivered to the inductor is
The energy stored is
Since i( −∞ ) = 0
We should note the following important properties of an inductor.
1. Note that the voltage across an inductor is zero when the current is constant.
An inductor acts like a short circuit to dc.
2. An important property of the inductor is its opposition to the change in current flowing through it.
The current through an inductor cannot change instantaneously.
A discontinuous change in the current through an inductor requires an infinite voltage, which is not
physically possible. Thus, an inductor opposes an abrupt change in the current through it. For example,
the current through an inductor may take the form shown in (a),
whereas the inductor current cannot take the form shown in (b) in
real-life situations due to the discontinuities.
However, the voltage across an inductor can change abruptly.
3. Like the ideal capacitor, the ideal inductor does not dissipate energy. The energy stored in it can be
retrieved at a later time. The inductor takes power from the circuit when storing energy and delivers
power to the circuit when returning previously stored energy.
4. A practical, nonideal inductor has a significant resistive component, as
shown. This is due to the fact that the inductor is made of a conducting
material such as copper, which has some resistance. This resistance is called
the winding resistance Rw , and it appears in series with the inductance of the
inductor. The presence of R w makes it both an energy storage device and an
energy dissipation device.
Since Rw is usually very small, it is ignored in most cases. The nonideal inductor also has a winding
capacitance Cw due to the capacitive coupling between the conducting coils. Cw is very small and can be
ignored in most cases, except at high frequencies. We will assume ideal inductors.
Example:
The voltage across a 2-H inductor is 20 ( 1-e-2t) V . If the initial current through the inductor is 0.3 A, find
the current and the energy stored in the inductor at t = 1s.
Inductor Voltage-current Relations
Overview:
In this assignment, we will measure the relationship between the voltage difference across a
capacitor and the current passing through it. We will apply several types of time-varying signals to a
series combination of a resistor and a capacitor. The voltage difference across the resistor, in
conjunction with Ohm’s law, will provide an estimate of the current through the capacitor. This
current can be related to the voltage difference across the capacitor.
The results of our voltage-current measurements will be compared to analytical expectations.
General Discussion:
We will use the circuit of Figure 1 in this lab assignment.
Both the voltage difference across the inductor and the
resistor (vL(t) and vR(t)) will be measured. From this data,
we can readily compare the voltage across the inductor
with the current through the inductor. Since the voltage
across the resistor is measured, we can readily infer the
current through the resistor via Ohm’s law:
The resistor and inductor are in series, so the current through the inductor is the same as the current
through the
resistor, so:
Since we are also measuring the voltage difference across the inductor, vL(t), we can readily compare
these parameters with our expectations based on our mathematical models of the capacitor voltagecurrent relationships.
Pre-lab:
In this lab, we will apply sinusoidal signals to the
inductor of Figure 1. Mathematically, the form of
the inductor current will be: iL (t) = A cos(2,ft)
where A is the amplitude of the sinusoid (in volts)
and f is the frequency (in Hz). The waveform is
shown graphically in Figure 2. For the circuit of
Figure 1, use the inductor voltage-current relations
to calculate the inductor voltage resulting from
application of the voltage of equation (3). Your
results may be dependent up on the parameters A, f, and L.
Lab Procedures:
Construct the circuit of Figure 1 with L = 1mH and R = 100. Use channel 1 of your oscilloscope to
measure the resistor voltage difference, and channel 2 of your oscilloscope to measure the inductor
voltage difference. Use channel 1 of your waveform generator (W1) to apply the voltage vin(t) in Figure 1.
Set up a math channel to calculate the current through the inductor per equation (2) in the pre-lab1. Set
the oscilloscope measurements to provide at least the amplitude of each of the three displayed
waveforms.
1. Apply a sinusoidal input voltage with frequency = 1kHz, amplitude = 2V, and offset = 0V to the
circuit of Figure 1. Use your oscilloscope to display the data listed above (waveforms
corresponding to C1, C2, and M1; measurement window displaying amplitudes of C1, C2, and
M1). Export the data in the oscilloscope time window to a .csv file for later processing.
2. Apply a sinusoidal input voltage with frequency = 2 kHz, amplitude = 2V, and offset = 0V to the
circuit of Figure 1. Use your oscilloscope to display the data listed above (waveforms
corresponding to C1, C2, and M1; measurement window displaying amplitudes of C1, C2, and
M1). Export the data in the oscilloscope time window to a .csv file for later processing.
Post-lab Exercises:
Import the data acquired in the lab procedures into your favorite numerical analysis software (e.g.
Excel, Matlab, Octave, etc.) Use the software and the results of your pre-lab analysis to calculate the
expected inductor voltage waveforms corresponding to the inductor current waveforms you measured
in the lab procedures. Use the software to overlay plots of the expected and measured inductor
voltages for each of the cases tested in the lab procedures. Comment briefly on the agreement
between the measured and expected inductor voltages for each of the cases. In your comments, be
sure to include a quantitative comparison (including percent difference) between the expected and
measured amplitudes of the inductor voltages.
Include the following in your blog:
1. Provide the mathematical expression for the inductor voltage resulting from the provided current
waveform.
2. Attach an image of the oscilloscope window, showing the inductor voltage, the resistor voltage, and
the inductor current and the measured amplitudes of the waveforms for a 1kHz sinusoidal input
3. Attach an image of the oscilloscope window, showing the inductor voltage, the resistor voltage, and
the inductor current and the measured amplitudes of the waveforms for a 2kHz sinusoidal input.
4. Attach plots of:
a. Expected (per your pre-lab analysis) and measured inductor voltages for a 1kHz sinusoidal input
b. Expected (per your pre-lab analysis) and measured inductor voltages for a 2kHz sinusoidal input