Energy Storage Devices
Though hundreds of different devices exist, we will consider just three other circuit
elements: batteries, capacitors and inductors.
A battery is a familiar device that is an approximation to an ideal Voltage source.
Physically, batteries produce voltage difference through the chemical separation of
charges. As current is allowed to flow from one pole of the battery to the other, charges
move back together releasing energy, generally into the circuit that connects the poles of
the battery. In the circuit of A1.003 (a), a wire connects the positive and negative pole of
a battery. If the wire were “ideal”, having zero resistance, we would expect that all of the
charge would flow instantaneously from the positive to the negative pole of the battery,
such that the current, dQ/dt, would be infinite. Of course this doesn’t actually happen.
Instead, it turns out that all real batteries show some resistance to current flow, which
limits the current. The circuit of A1.003 (b) shows a practical and realistic model of the
battery as an ideal Voltage source in series with a resistor. (This combination is known
as the Thévenin equivalent). Many of the electrical properties of neurons and other
excitable cells may be accurately modeled in this manner.
Capacitors
When two conductors are separated from one another by an insulator, and a source ofcurrent is applied (Figure A1.004), positive charge will accumulate on one side of the
insulator and negative charge on the other, resulting in a buildup of Voltage. As long as
the charge keeps flowing the Voltage will increase steadily, and without limit.
Analogously, if a Voltage source is applied across the separated conductors, charge will
build (immediately, in the case of ideal conductors) of such a magnitude that the
potential energy difference is equal to the Voltage of the source. When the sources are
removed, however, the charge difference, and therefore the Voltage difference, is
retained across the insulating boundary, thus storing potential energy. A device made of
conductors – typically in the form of thin films – separated by an insulating layer is
known as a capacitor. However, capacitance will exist between any conductors
separated by an insulating layer or material. Unlike resistors, capacitors store energy,
rather than dissipate it.
The capacitance, C, is measured as the ratio of the charge across the capacitor to the
applied Voltage:
Q = CV.
The circuit symbol for a capacitor is shown in figure A1.005, and represents two
separated conductive plates.
separated conductive plates.
The unit of capacitance is the Farad. A large capacitance, achieved by having a very
small insulating gap, means that a relatively small charge results in a large Voltage
potential difference, chiefly because the attractive forces between positive and negative
charge are very large over short distances. Taking the first derivative of equation 1, we
see that the current is proportional to the Voltage change over time.
small insulating gap, means that a relatively small charge results in a large Voltage
potential difference, chiefly because the attractive forces between positive and negative
charge are very large over short distances. Taking the first derivative of equation 1, we
see that the current is proportional to the Voltage change over time.
In a sense, this appears similar to Ohm’s law, except that now the current is
proportional to the rate of change of the Voltage, rather than the Voltage alone, as it is in
a resistor. The capacitor, in this case, takes the place of the resistor, but one whose
resistance depends on the rate of change of the Voltage. Rather then resistance, the term
proportional to the rate of change of the Voltage, rather than the Voltage alone, as it is in
a resistor. The capacitor, in this case, takes the place of the resistor, but one whose
resistance depends on the rate of change of the Voltage. Rather then resistance, the term
impedance, measured still in Ohms, is used to describe this behavior. In effect, this
means that while an insulating layer does not pass constant current, time-varying
currents may be passed. The units of Farads are defined such that a Voltage that
changes by 1 V/s, when applied across a 1 Farad capacitor, will result in the flow of 1
ampere (a very large amount) of current. Most often, capacitors used in electronic
circuits have capacitance of a few microfarads (μF).
Specifically, consider applying a sinusoidally varying voltage across the capacitor:
means that while an insulating layer does not pass constant current, time-varying
currents may be passed. The units of Farads are defined such that a Voltage that
changes by 1 V/s, when applied across a 1 Farad capacitor, will result in the flow of 1
ampere (a very large amount) of current. Most often, capacitors used in electronic
circuits have capacitance of a few microfarads (μF).
Specifically, consider applying a sinusoidally varying voltage across the capacitor:
that is, a sinusoidal current with a 90° (=π/4) phase lead with respect to the Voltage.
The term, ω, is the frequency, and this frequency-dependent resistance, measured in
Ohms, is given the term impedance. Specifically, the impedance goes down as the
frequency goes up. Capacitors pass greater current at the same Voltage when the
frequency is higher. They are like resistors whose resistance decreases with frequency.
Unlike resistors however, which dissipate energy as heat, the capacitor only stores
energy, by converting between potential energy in the form of Voltage, and kinetic
energy in the form of current.
The term, ω, is the frequency, and this frequency-dependent resistance, measured in
Ohms, is given the term impedance. Specifically, the impedance goes down as the
frequency goes up. Capacitors pass greater current at the same Voltage when the
frequency is higher. They are like resistors whose resistance decreases with frequency.
Unlike resistors however, which dissipate energy as heat, the capacitor only stores
energy, by converting between potential energy in the form of Voltage, and kinetic
energy in the form of current.
now see is proportional to s. The exponential term, est, simply drops out of the equation.
No assumptions were made about A or s, however, except that they are constant with
respect to t. This solution form to the differential equation 4 is known as the Laplace
transform, and is an important solution method in all forms of linear systems.