Appendix A: Review of Basic Circuit Theory

• Basic Electrical Parameters

  • Potential Difference (V or E)

  • Current (I)

  • Conductance (g)

  • Capacitance (C)

• Rules for Circuit Analysis

  • Conductance

  • Current

  • Capacitance

  • Potential Difference

• Current in Circuits with Capacitance

  • Circuit with Capacitor

  • Circuit with Resistor and Capacitor in Series

  • Circuit with Resistor and Capacitor in Parallel

Familiarity with the basic principles of electrical circuit theory is important for understanding the equivalent circuit model of the neuron developed in Chapters 6, 7, and 9. The appendix is divided into three parts:

1. The definition of basic electrical parameters.

2. A set of rules for elementary circuit analysis.

3. A description of current in circuits with capacitance.

Potential Difference (V or E)

Electrical charges exert an electrostatic force on other charges: like charges repel, opposite charges attract. The force decreases as the distance between two charges increases. Work is done when two charges that initially are separated are brought together. Negative work is done if their polarities are opposite and positive work if they are the same. The greater the values of the charges and the greater their initial separation, the greater the work done. (Work = where f is electrostatic force and r ₁ is the initial distance between the two charges.)

Potential difference is a measure of this work: The potential difference between two points is the work that must be done to move a unit of positive charge (one coulomb) from one point to the other (ie, it is the potential energy of the charge). One volt (V) is the energy required to move one coulomb a distance of one meter against a force of one newton.

Current (I)

A potential difference exists within a system whenever positive and negative charges are separated. Charge separation may be generated by a chemical reaction (as in a battery) or by diffusion of two electrolyte solutions with different ion concentrations across a selectively permeable barrier, such as a cell membrane. If a charge separation exists within a conducting medium, charges move between the areas of potential difference: Positive charges are attracted to the region with a more negative potential, and negative charges to the region of positive potential.

Current is defined as the net movement of charge per unit time. According to convention, the direction of current is defined as the direction of flow of positive charge. In metallic conductors current is carried by negatively charged electrons, which move in the opposite direction of conventionally defined current. In nerve and muscle cells current is carried by both positive and negative ions in solution. One ampere (A) of current represents the movement of one coulomb (of charge) per second.

Conductance (g)

Any object through which electrical charges can flow is called a conductor. The unit of electrical conductance is the siemens (S). According to Ohm's law the current that flows through a conductor is directly proportional to the potential difference across it:¹

As charge carriers move through a conductor, some of their electrical potential energy is converted into thermal energy caused by their frictional interactions with the conducting medium.

Each type of material has an intrinsic property called conductivity (σ), which is determined by its molecular structure. Metallic conductors conduct electricity extremely well and thus have high conductivities. Aqueous solutions with high-ionized salt concentrations have somewhat lower conductivity, and lipids have very low conductivity—they are poor conductors of electricity and are therefore good insulators. The conductance of an object is proportional to σ times its cross-sectional area divided by its length:

Length is defined as the direction along which one measures conductance. For example, the conductance measured along the cytoplasmic core of an axon is reduced if its length is increased or its diameter decreased (Figure A–1).

Electrical resistance (R) is the reciprocal of conductance and a measure of the resistance provided by an object to current. Resistance is measured in ohms (Ω):

Capacitance (C)

A capacitor consists of two conducting plates separated by an insulating layer. Its fundamental property is its ability to separate charges of opposite sign: Positive charges are stored on one plate, negative charges on the other. In the example in Figure A–2 a net excess of positive charges on plate x and an equal excess of negative charges on plate y results in a potential difference between the two plates.

This potential difference can be measured by determining how much work is required to move a positive test charge from y to x. Initially, the test charge is attracted by the negative charges on y and repelled by the more distant positive charges on x. The result of these electrostatic interactions is a force f that opposes the movement of the charge from y to x. However, as the test charge is moved toward x, the attraction by the negative charges on y diminishes and the repulsion by the positive charges on x increases, with the result that the net electrostatic force exerted on the test charge is constant everywhere between x and y. Work (W) is force times the distance (D) over which the force is exerted:

The work done in moving the test charge from one side of the capacitor to the other is equal to the difference in electrical potential energy, or potential difference, between x and y. In Figure A–2 it is shown as the shaded region in the plots.

Capacitance is measured in farads (F). The greater the density of charges on the capacitor plates, the greater the force acting on the test charge and the greater the resulting potential difference across the capacitor (see Figure A–2B). Thus, for a given capacitor there is a linear relationship between the amount of charge (Q) stored on its plates and the potential difference across it:

where C, the capacitance, is a constant.

The capacitance of a parallel-plate capacitor is determined by two features of its geometry: the area (A) of the two plates and the distance (D) between them. Increasing the area of the plates increases capacitance because a greater amount of charge must be deposited on each side to produce the same charge density, which is what determines the force f acting on the test charge (Figure A–2A and C). Increasing the distance between the plates does not change the force acting on the test charge, but it does increase the work that must be done to move it from one side of the capacitor to the other (Figure A–2A and D). Therefore, for a given charge separation between the two plates, the potential difference between them is proportional to the distance. Put another way, the greater the distance, the smaller the amount of charge that must be deposited on the plates to produce a given potential difference, and therefore the smaller the capacitance (Equation A–1).

These geometrical determinants of capacitance can be summarized by the equation

As shown in Equation A–1, the separation of positive and negative charges on the two plates of a capacitor results in a potential difference between them. The converse of this statement is also true: The potential difference across a capacitor is determined by the net positive and negative charge on its plates. For the potential across a capacitor to change, the amount of electrical charges stored on the two conducting plates must change first.

Familiarity with a few basic rules for electric circuit analysis will help in understanding the equivalent circuits used throughout the textbook.

Conductance

The symbol for a conductor is:

A variable conductor is represented as:

A pathway with infinite conductance (zero resistance) is called a short circuit and is represented by a line:

Conductances in parallel add:

Conductances in series add reciprocally:

Resistances in series add, while resistances in parallel add reciprocally.

Current

An arrow denotes the direction of current (net movement of positive charge). Ohm's law is

When charge flows through a conductor, the end that the current enters is positive with respect to the end that it leaves:

The algebraic sum of all currents entering or leaving a junction is zero. (We arbitrarily define current approaching a junction as positive, and current leaving a junction as negative.) In the following circuit for junction x

the currents are

In the following circuit for junction y

the currents are

Current follows the path of greatest conductance (least resistance). For conductance pathways in parallel, the current through each path is proportional to its conductance value divided by the total conductance of the parallel combination:

Capacitance

The symbol for a capacitor is:

The potential difference across a capacitor is proportional to the charge stored on its plates:

Potential Difference

The symbol for a battery or electromotive force (E) is

The positive pole is always represented by the longer bar.

Batteries in series add algebraically, but attention must be paid to their polarities. If their polarities are the same, their absolute values add:

If their polarities are opposite, they subtract:

(Here the convention used for potential difference is that V _AB = V _A − V _B.)

A battery drives a current around the circuit from its positive to its negative terminal:

For purposes of calculating the total resistance of a circuit, the internal resistance of a battery is set at zero.

The potential differences across parallel branches of a circuit are equal:

As one goes around a closed loop in a circuit, the algebraic sum of all the potential differences is zero:

Thus,

Circuits that have capacitive elements are much more complex than those which have only batteries and conductors because in capacitive circuits current varies with time. The time dependence of changes in current and voltage in capacitive circuits is illustrated qualitatively in the following three examples.

Circuit with Capacitor

Current does not cross the insulating gap in a capacitor; rather, it builds up positive and negative charges on the capacitor plates. However, we can measure a current into and out of the terminal of a capacitor. Consider the circuit shown in Figure A–3A. When switch S is closed, the battery E moves a net positive charge onto plate a, and an equal amount of net positive charge is withdrawn from plate b. The result is a counterclockwise current in the circuit (Figure A–3B).

Because this current flows into or out of the terminals of a capacitor, it is called a capacitive current (I _c). Because there is no resistance in this circuit, the battery E can generate a very large amplitude of current that will charge the capacitance to a value Q = E ×C instantaneously (Figure A–3D).

Circuit with Resistor and Capacitor in Series

Now consider what happens if a resistor is added in series with the capacitor (Figure A–4A). The maximum current that can be generated when switch S is closed (Figure A–4B) is now limited by Ohm's law (I = V /R). Therefore the capacitor charges more slowly. When the potential across the capacitor has finally reached the value V _c = Q /C = E (Figure A–4C), there is no longer a difference in potential around the loop (ie, the battery voltage E is equal and opposite to the voltage across the capacitor, V _c). The two thus cancel out, and no net potential difference is left to drive a current around the loop.

The potential difference is greatest, and current is at a maximum, immediately after the switch is closed. As the capacitor begins to charge, the net potential difference (V _c + E) available to drive a current becomes smaller and current decreases. This results in an exponential change in voltage as well as current across the resistor and the capacitor. Note that in this circuit resistive current must equal capacitative current at all times (see earlier section, Rules for Circuit Analysis).

Circuit with Resistor and Capacitor in Parallel

Consider now what happens if we place a parallel resistor and capacitor in series with a constant-current generator that generates a total current I _T (Figure A–5A). When the switch (S) is closed, charge starts to flow around the loop (Figure A–5B). In the first instant of time after the current begins to flow, all the charge flows into the capacitor (ie, I _T = I _c). However, as charge builds up on the plates of the capacitor, a potential difference V _c is generated.

Because the resistor and capacitor are in parallel, the potential across them must be equal; thus part of the total current begins to flow through the resistor, such that I _R × R = V _R = V _c. As less and less charge flows into the capacitor, the rate of charging slows; this accounts for the exponential shape of the curve of voltage versus time. Eventually, the voltage reaches a plateau and no longer changes. When this occurs, all the charge flows through the resistor and V _c = V _R = I _T × R (Figure A–5C).