A review of basic principles of electricity & physics lec (3)

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 of
current 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.

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.

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

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:

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.

Ohms law tells us that the magnitude of this proportionality is the resistance, which we
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.

A review of basic principles of electricity & physics lec (2)

Kirchhoff’s Laws

The path by which charge may flow between the positive and negative ends of a source
of Voltage or charge is known as a circuit. Sources of current, Voltage, and resistance to
current flow are presented in a set of standard symbols that are connected together in
circuit diagrams. In such diagrams, lines represent perfect, zero resistance, conductors.
Figure A1.01 shows these standard pictures. By convention, current is said to move from
positive to negative potentials and would imply the motion of positive charges (in
actuality, most currents in man made devices result from the motion of negative
charge). Circuit diagrams frequently use arrows to indicate the direction of current flow.
In the figure, note that the current out from the top, or positive end, of the Voltage
source is identical to the current in to the bottom. The same is true of the resistor in the
circuit and indeed in any element of a circuit: The current flow in is identical to the
current flow out. This general behavior of nodes in a circuit is known as Kirchhoff’s
current law (KCL).

Kirchhoff’s current law has an analog (referred to as a “dual”) in describing Voltage
differences in circuits. Kirchhoff’s Voltage law (KVL) states that the sum of the Voltage
differences around any closed loop in a circuit must be equal to zero. In applying KVL
you must pay attention to the direction you travel in the loop, as discussed below.
Series and Parallel Connections
When electrical elements are attached end-to-end they are connected “in series”, when
the current inputs and outputs of multiple elements are held in common, the devices are
connected in parallel. Figure A1.02 shows resistors connected to a Voltage source in
series and in parallel. In the series circuit, KCL tells us that the current through all three
devices is identical, while KVL informs us that the Voltages V1 and V2 have a sum equal
to Vs. Applying KCL to the parallel circuit we see that at the two nodes, i1 + i2 = i. KVL
tells us that traveling starting at the top and going clockwise around the loop that
includes R1 and R2, V2 – V1 = 0 and that V2 = V1. Likewise, in the loop made up of Vs
and V2, V2 – Vs = 0, implying that V2 = Vs. The two Kirchhoff’s laws, in combination, are
extremely powerful organizing principles. Understanding them makes it possible to
model the behavior of the overwhelming majority of electrical circuits and devices.

Clearly, when the resistors are in series the same current flows through each. By Ohm’s
law, the voltage across R1 is equal to iR1, and the Voltage that appears across R2 is iR2.
The Voltage that appears across the Voltage source is therefore the sum of the Voltages
across each resistor, or i(R1 + R2) (Note that this is the same as the expression of KVL
for this circuit: Vs = V1 + V2. The total resistance experienced by the Voltage source in
the series circuit is thus the sum of the individual resistors: resistance in series is the
sum of the individual resistances.
In the parallel circuit, there are two paths for current to go through the resistors. Not
surprisingly more current flows through the lower resistance path, and we can compute
the individual currents by Ohm’s law. Because the Voltage (V1 and V2) across these
resistors is identical (they are connected by perfect conductors), the current, i1 in R1 is
V/R1 and i2 is V/R2. By KCL, the total current that flows from the Voltage source is
equal to the sum of the currents through the two resistors:.

which is always less than either of the resistors alone.