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Complex Representation of Signals

Most generally, we can use complex or imaginary values for s (and/or A for that matter)
in equation 9, where s is known as the “complex frequency.” In electrical engineering it
is common practice to use the convention that
j = (-1)^.5 in order to avoid confusion with
the use of i to represent current; we will use that convention for the remainder of this

section. When s is purely imaginary, with the value s = jω and A is real, we can take
advantage of Euler’s relation, that:

We can apply a sinusoidally varying voltage at the frequency ω, V= cos(ωt), across the
capacitor. Note that the V is the real part of Aejωt = Re(Aest).
Substituting into equation 8:

We are interested only the real part of the current, of course, so equation 11 simplifies to
iC = –ωCAsin(ωt), the result of equation 6 above. Looking again at equation 9, when
s = jω the ratio of the voltage to the current is 1/ jω C.
Generalization of Ohm’s law
The ratio of a complex voltage, V, to a complex current, I, is represented by the
impedance, Z. Because sinusoidally varying signals are conveniently represented by
complex numbers (and, as we will see when we discuss the Fourier transform in a
separate section, this can be a generalized representation of most any signal we
encounter), this is an exceptionally powerful formalization. For complex signals, Ohm’s
law states simply that:

The magnitude of the impedance is given in units of Ohms, like resistance, but in
general also incorporates a phase shift. For capacitors, as noted above, the impedance is
1/jωC. Applying Voltage as a cosine wave results in a current that is an inverted sine
wave – a 90° phase shift. In general, multiplication by j represents a 90° phase shift.
As we noted above, the voltage and current relationships for most electrical circuits
(specifically, linear electrical circuits), can be found through the application of Ohm’s
law coupled with KVL and KCL. The Laplace transform solution, as in equation 9, allows
this to be generalized readily to the analysis of devices like capacitors, which store
energy.
One should be a little bit careful here concerning the values of ω, which are in units of
radians/second or angular frequency. In common use we generally speak of frequencies
in cycles/second or “Hertz”. As one cycle is 2π radians, ω=2πf, where the letter f denotes
the frequency in cycles/second. When the input voltage is a sinusoid with frequency, f,
equation 9 may be written as:

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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.

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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.

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Introduction

While this book is not intended as a comprehensive course in electricity and magnetism

there are a few principles that are so ubiquitous in functional magnetic resonance that

they appear repeatedly throughout this text and therefore warrant this brief overview.

Charge

Electrical charge is considered to be a fundamental property of materials. Physicists

recognize that charge exists in only two forms, positive and negative, and that it is

quantal in nature, with the smallest amount of charge being that of a single electron or

proton, each being exactly 1 unit of negative or positive charge, respectively. A single

unit of charge is extremely small, of course, and charge is more commonly measured in

units of Coulombs, equivalent to about 6.242 X 1018 unit charges. Positive and negative

charges exhibit a strong attractive force, whose magnitude is proportional inversely to

the square root of the distance that separates them. In its most stable state, bulk matter

has a net charge of zero, meaning that it contains an identical number of positive and

negative charges.

Voltage

When charges become separated by distance, the presence of an attractive force between

implies an increase in potential energy, which is released when the charges are moved

together. This energy difference is known as Voltage and is measured, naturally, in

Volts. Because the potential energy of the Voltage is also measure of the force that would

tend to move the charge, it is known also as the potential difference, or simply the

potential, the “electromotive force” or the e.m.f. and these terms are used

interchangeably, which can at times be confusing. Batteries are familiar voltage sources

that rely on chemical means to store potential energy. For convenience, the units of

Volts are defined in terms of other fundamental physical constants and units. One

Joule of work is required to move one Coulomb of charge through a potential difference

of 1 Volt. In practice, this means that a Coulomb is actually defined to set unit values of

Volts and Joules. Voltage must always refer to the energy difference between two points.

It is never actually correct to discuss the Voltage at a point, though you will often see

such a statement. In those cases, the reference point is assumed implicitly, usually to

refer to a “ground” or common point in an electrical circuit.

At the atomic level charges may become separated. In some molecules, such as salts like

sodium chloride, the electronegativity of one atom (chloride) is so much greater than

that of the other (sodium) that in a covalent atomic bond between these elements the

electron or electrons are almost completely transferred from one atom to the other.

Such bonds are dissociated easily in aqueous solution so that the individual atoms now

become “ions” or charged particles. In water, the atoms of salts appear in ionic form, so

that atoms of sodium, potassium, chloride, magnesium and many others move relatively

freely of their oppositely charged complement. Not only atoms, but also molecules, can

exist in ionic form, and many proteins, for example, carry a net negative charge. Of

course some ions may be quite large and there may be physical impediments to their

motion that result in different bulk properties for ions and small charges, such as

electrons. These effects are significant in some circumstances, but in most of the

 discussion that follows, and throughout most of this book, we can consider the
properties of ions interchangeably with the properties of charge.
Current and Resistance
The motion of charge is known as current; specifically, the current, i, is equal to the
change in charge, Q, with time, so that:

Where V is the Voltage, i is the current, and R the resistance. Materials whose resistance

is extremely high are termed insulators and those whose resistance is low are called,

conductors. Good insulators may have resistance of gigaOhms (109 Ohms) or more,

whereas good conductors, such as copper wire, will have resistance of microOhms. More

accurately, we refer to resistivity, which is the measured resistance normalized by the

area and length of a conductor, so that it is a material property. Most biological

materials fall in a more intermediate range with resistances of thousands to millions of

Ohms. In a perfect conductor, where the resistance is zero, the voltage at all points along

the conductor is identical. In general, moving charge from a source of higher potential

energy to lower (current flowing from positive to negative ends of a source) must result

in energy dissipation. Resistors dissipate this energy as heat.