## Circuit Elements 31 Resistors

Under everyday conditions, current meets with some resistance to flow, much as friction opposes the movement of an object over a surface. Some energy or force is expended in overcoming this resistance. The voltage (or potential) difference across a given resistance is known as the voltage "drop," and the relationship between these parameters and the resultant current is given by Ohm's law:

The unit of resistance is the ohm (Q), which is defined as the resistance (R) that will dissipate 1 J of energy when a current of 1 A flows through it for a period of 1 s. Practically speaking, resistors are made from materials that do not easily allow the free movement of electrons, such as carbon. Very high resistance materials that are the most restrictive toward the movement of electrons, such as air, rubber, or glass, make the best insulators. The greater the distance that current must traverse through a resistive material, the more resistance to flow there will be. It is, thus, useful to alter the length of a resistive material to vary the current flow. As given by Ohm's law, resistance and current vary inversely with one another (R = V/I). Therefore, a reduction in the length of a resistive medium by half will lead to a doubling of the current. The potentiometer (voltmeter) uses this principle by providing a way to vary the length of a resistor (and thereby vary the current flow) to advantage.

Resistance in the acquisition of a biological test, such as an EEG, does not only derive from the material of the wiring in use. Resistance derives from any material through which current must pass. For example, resistive elements in the EEG include not just the electrode wiring but also the scalp-electrode interface and the internal circuitry of the machine. Resistance is provided by anything that lies between the positively charged terminal of a circuit (the cathode) and the negatively charged terminal (the anode). If the resistance is infinitely large, then the current becomes infinitely small (or ceases). This produces a circuit that is "open." Circuit breakers act in this way to ensure the safety of an electrical system. If resistance is reduced to a miniscule value, this permits a relatively large current, and is deemed a "short circuit." Any resistance between the anode and cathode that allows current to flow but is neither infinitely large nor extremely small is a "closed circuit."

Fig. 1. This figure contrasts the organization of a series circuit (A) and a parallel circuit (B). In a series circuit, equal current must flow through each resistor in turn. Therefore, the resistors function as a voltage divider. The resistance, Rcomb, is given by R1 + R2 + R3. The parallel circuit functions as a current divider, with equal voltage across each resistor. The combined resistance is given by 1/Rcomb = 1/R1 + 1/R2 + 1/R3.

Fig. 1. This figure contrasts the organization of a series circuit (A) and a parallel circuit (B). In a series circuit, equal current must flow through each resistor in turn. Therefore, the resistors function as a voltage divider. The resistance, Rcomb, is given by R1 + R2 + R3. The parallel circuit functions as a current divider, with equal voltage across each resistor. The combined resistance is given by 1/Rcomb = 1/R1 + 1/R2 + 1/R3.

Each element in a circuit contributes its own resistance. If multiple resistive elements exist in a succession along a circuit, they are said to be in series (Fig. 1A). If they are configured to allow current to travel in multiple alternate paths, they are said to be in parallel (Fig. 1B). Because the series configuration fractionates the total voltage across each of the resistive elements, it is also known as a voltage divider. Addition of these resistive elements creates a resistor of greater length that is equivalent to the sum of all the component resistances. Therefore, the equivalent resistance (R) for a series circuit may be obtained by summing the individual resistances in the circuit as such:

By contrast, a parallel circuit will allow current to fractionate and travel any of a number of paths, and, therefore, is known as a current divider. The several routes that the current may travel effectively reduces the total resistance to flow to less than that of any of the component resistances in the circuit. This is represented by the following relationship:

In considering a complete circuit, there are two other applicable laws. Kirchoff's current law states that the sum of current flowing into and out of any circuit node must be zero. Kirchoff's voltage law states that the sum of all voltage steps (voltage sources and drops) around a complete circuit must be zero.

### 3.2. Capacitors

A capacitor is a device that permits the storage of charge. It consists of two parallel conducting plates closely apposed to one another but separated by a small distance and an interposed insulating material, the dielectric. The gap between the plates provides a large resistance to the flow of current from plate to plate. As such, when a potential is applied across a circuit containing a capacitor, positive charge will accumulate on the positive plate, attracting negative charge to the opposite plate. Current flows between the plates via the circuit without charge actually crossing the dielectric gap between the plates. The accumulation of separated charge creates a potential difference across the plates that eventually balances the potential applied across the circuit, and current flow then ceases. Several factors affect the magnitude of charge, or capacitance, that may be stored by a capacitor. This is proportional to the size of the plates of the capacitor, inversely proportional to the distance between those plates, and is affected by the dielectric material between the plates. The MKS unit for capacitance is the farad (F). A farad will store 1 C of charge on the plates of a capacitor with an applied potential difference of 1 V. This is mathematically expressed as:

where C is the capacitance in farads, Q is the charge in coulombs, and Vis the voltage in volts across the plates. In practice, most circuits use capacitance on the order of microfarads or picofarads.

If you differentiate both sides of the above capacitance equation with respect to time and rearrange the result, you obtain the following relation:

I = C x dV/dt or current is equal to capacitance multiplied by the change in voltage with respect to time. Thus, if the voltage is unchanging (dV/dt = 0), then current flow becomes zero. This is the case with a direct current (DC) circuit, wherein current flows directly between the anode and cathode with an invariant voltage. Once the potential difference between the plates of the capacitor has equaled that applied constant voltage, current flow ceases. Conversely, a continually varying potential will be able to maintain current flow across a circuit that includes such a capacitive element. This is the effect produced by alternating current (AC) that, as the name implies, is constantly oscillating between two alternating poles. (AC will be described in more detail later.) Thus, a capacitor will pass AC flow but will block DC flow. This impeding effect of the capacitor is known as capacitive reactance and is defined as follows:

where XC is in ohms, / is the frequency of the current in hertz, and C is in farads. One can see that as the frequency of the current approaches zero (as in DC), the capacitive reactance (resistance to flow) becomes infinitely large.

Capacitance is crucial to any system that can maintain separated charge and, thereby, store potential energy for use in doing work. The lipid bilayer membrane of nerve tissue is a superb capacitor, which both permits and restricts the flow of ionic currents. It is these intermittent fluctuations in biological currents that ultimately produce the potentials of interest in clinical neurophysiology, such as in EEG. However, other sources of biological capacitance can also interfere with these signals, such as the capacitive resistance in the cerebrospinal fluid, skull, and scalp. As the equation for XC predicts, these will affect differing neuronal frequencies to different degrees. For example, 3 Hz activities through 2 ^F of capacitance will have an XC = 1/[2 ■ 3.14 ■ 3 Hz ■ (2 x 10-6) F] = 26.5 kQ, which is much larger than the 4.4 kQ reactance seen by 18-Hz beta frequencies. This illustrates how much more capacitive reactance there is to low frequencies vs higher frequencies with scalp recordings.

Multiple capacitors in a circuit interact in a manner that is opposite to the behavior of resistors. When arranged in parallel, there is an additive effect as such:

and when arranged in series, the equivalent capacitance is less than any of the individual values, as such:

Fig. 2. This figure illustrates a transformer, which is based on the principle of induction. An alternating current (AC) with voltage, V1, is applied to an inducer, represented by the coil with turns. Another coil with N2 turns shares the same rod. AC flowing through the coil induces a magnetic field that then induces a reciprocal electrical field (voltage) in the second coil. The ratio of coil loops determines the change in voltage in the second circuit; fewer turns leads to a proportionately reduced voltage in the second circuit.

Fig. 2. This figure illustrates a transformer, which is based on the principle of induction. An alternating current (AC) with voltage, V1, is applied to an inducer, represented by the coil with turns. Another coil with N2 turns shares the same rod. AC flowing through the coil induces a magnetic field that then induces a reciprocal electrical field (voltage) in the second coil. The ratio of coil loops determines the change in voltage in the second circuit; fewer turns leads to a proportionately reduced voltage in the second circuit.

### 3.3. Inductors

An inductor consists of a continuous coil of wire called a solenoid. Current flowing in this coil generates a magnetic field whose axis passes through the coil (with directionality dictated by the right hand rule). Because of the equivalence of electricity and magnetism (i.e., Maxwell's equations), this magnetic field can induce an electromotive force (emf, e) in a nearby conductor, if the magnetic field is variable over time. The magnetic field can vary if the current flow in the coil varies. The relationship of this emf to the current is:

e = - L x dl/dt, where L is a constant called the inductance of the device. The negative sign in the equation indicates that the changing current (dl) induces an emf that opposes that change. The unit of inductance (L) is the henry (H). The inductance of a solenoid is proportional to the number of turns in the coil.

A changing current (i.e., AC) passing through a coil will generate a changing magnetic field that passes through its core. If a second coil of wire is wrapped around a nearby section of this core, the changing magnetic field will generate a reciprocal emf and current in the second coil. One can tap this feature to step voltage from one value to another, as in a transformer (Fig. 2). Because inductance (L) depends on the number of turns (N) in the coils, if the number of turns in the first coil (N1) is greater than in the second coil (N2), then the inductance will decrease in the second coil. From the above equation, if L decreases, then dl/dt will increase proportionately. The induced emf (or voltage) in circuit two will decrease in proportion to the drop in inductance. Therefore, voltage varies directly with L and current varies inversely with L, whereas the total energy (power) in the system is conserved. As current steps up, voltage steps down. These vary according to the ratio L1/L2, which is directly related to N1/N2.

Inductance is similar to resistance in that it poses an impediment to the motion of charge generated by another source. For example, an AC source, with its associated emf, providing a current through a circuit with an inductor, will be opposed by the emf generated by that inductor. The inductor's emf is, in effect, subtracted from that of the circuit to determine the net potential. This property is known as the inductive reactance (XL), which is:

where XL is in ohms and the frequency (/) is in Hz.