Introduction

The simplest electrical circuit consists of a source of voltage, a pathway for current to flow, and resistive elements. Electrical circuits can be divided into two types: steady-state and dynamic. In the steady-state circuit, the values of voltage, current, and resistance are fixed, whereas the values of current and voltage vary with time in dynamic circuits. In addition to resistive elements, dynamic circuits contain elements of capacitance and inductance.
 
 


Steady-State Circuits

Three important quantities in the evaluation of steady-state circuits are voltage (E), current (I), and resistance (R). Thinking of these quantities in terms of a river is a useful analogy: the voltage compares to the water pressure which develops because one point in a river is higher than another; current is like the amount of water flowing in a river; and resistive elements are like obstructions that prevent the flow of water. These three concepts are related by Ohm's law:

V = IR

where V = voltage, I = current, and R = resistance. (Note: voltage is often represented by E.) This equation states that the voltage level is proportional to the current flowing through a resistance. Therefore, the higher the resistance, the greater the voltage drop.
 
 

Voltage

Voltage (potential) is the electrical pressure difference that occurs in a circuit. Voltage may be supplied from a power plant, a car battery, specialized cells in eels, etc. Physiological voltage levels are generally quite small and measured in millivolts (mV) or microvolts (µV).

Note about prefixes: milli = 10-3

micro = 10-6

nano = 10-9

pico = 10-12

Current

Current is the quantity of electrical charge flowing per unit time in a circuit. Current may be carried as a flow of electrons along a metal wire, as ions flowing through a solution, as ions flowing through cell junctions (such as the intercalated disks in heart muscle cells), etc. Because current is defined in terms of the movement of a positive charge, current always flows from a higher positive voltage to a lower voltage. Current is measured in amperes (A), milliamperes (mA), microamperes (µA), or nanoamperes (nA). The currents generated by nerve and muscle cells are in the range of 0.1-10 nanoamperes (usually "nanoamps").

Current can be direct (DC) or alternating (AC). DC current flows continuously and in only one direction around a circuit. AC current changes direction periodically in a circuit; its amplitude variations are usually in the form of a sine wave (see Dynamic Circuits below). The frequency of the sine wave is one of the distinguishing properties of an AC signal. Frequency is equivalent to the reciprocal of the period and is measured in hertz (Hz) or cycles per second. For example, the electricity supplied to North American households is in the form of sine waves with a frequency of 60 Hz and an amplitude of 110 V; in Great Britain, its frequency is 50 Hz, and its amplitude is 220 V.
 
 

Resistance

Resistance is the measure of the difficulty with which electrons flow through a medium. Resistive elements cause voltage (electrical potential) to drop from one side of the element to the other. Resistors, light bulbs, electrodes, and cell membranes all provide resistance to current flow, causing a voltage drop. Resistance is measured in ohms (W), kilohms (kW), or megohms (MW).

Conductance (G) is the reciprocal of resistance: G=1/R The unit of conductance is the sieman (S), although it is sometimes still called the mho, because it is the reciprocal ohm. Conductances of single channels in cell membranes are usually 10-100 picosiemens (pS, which is 10-12 S).
 
 

Ground

The amplitude of a voltage, like all forms of potential energy, can be assigned only in relation to a reference point. "Ground" is the zero or reference point for voltage in a circuit. True ground is the potential of the earth. A convenient, but not always the best, path to ground is provided by the third prong of an electrical plug when it is connected to an electrical outlet.
 
 
 
 

Circuit Analysis of Steady-State Circuits The first thing we need to know about a circuit is the total voltage and the total current that flows in it. In the simplest possible circuit (Figure 1), the voltage is simply the amount supplied by the battery (V), and the current is equal to that voltage divided by the resistance of the resistor (R). In the generally accepted slang, all of the voltage supplied is "dropped" across the resistor (i.e., all of the drop in electrical potential occurs across the resistor); therefore, if the voltage were measured across the resistor, it would be equal to the total voltage of the battery.



Figure 1. Simple circuit. The voltage is supplied by a battery, with its positive and negative terminals indicated, and a single resistor.

In addition to knowing the total voltage and the total current in a system, it is often necessary to know the voltage at a particular place in a circuit and the amount of current that flows across a given resistor. For steady-state circuits, this depends upon the total potential and resistance, but it also depends upon the distribution of the resistive elements (i.e., the resistors). No matter how complex the circuit, there are essentially only two ways that resistors can be arranged: in series or in parallel.
 
  Series resistance Figure 2 is a circuit having three resistors in series. To find the amount of current that flows in this circuit, the voltage supplied by the battery must be divided by the sum of all the resistors: I = V / (R1 + R2 + R3)






Figure 2. A circuit with three resistances in series.




In other words, the entire voltage supplied across the battery is dropped across the three resistors. The portion of the voltage dropped across one of the resistors depends on the size of that resistor relative to the other two. For example, the voltage that would be measured across R3 is:














Parallel resistance

The circuit shown in Figure 3 is an example of resistances in parallel. The voltage drop across each resistor (i.e., across each leg of the circuit) is the same and is equal to the voltage supplied by the battery. The total resistance of this type of circuit is calculated as follows:






or


 
 
 
 



Figure 3. A circuit with three resistances in parallel.



 
 

The current flow across each resistor is determined by the size of that resistor. For example, the current flow in R2 is: I2 = V / R2
 
 
 
 

Dynamic Circuits

Like steady-state circuits, dynamic circuits are evaluated for voltage and current levels. However, not only do dynamic circuits contain resistive elements, they also have elements of capacitance and inductance, elements which vary with changes in voltage. If the input to such a circuit is constant (i.e., it is a DC signal), the value of the capacitance and inductance is zero. If, however, the input signal varies with time, that signal is distorted by capacitance and inductance. Figure 4 is an example of one cycle of a time-varying input current (AC current). This is a sine wave which repeats every 8 seconds, i.e., it has a frequency of 1/8 cycles per second, or 0.125 Hz. The amplitude of the signal is 0.4 mA. If this current were presented to the steady-state circuits discussed above, the current would vary in the same way with time, and the value of the voltage could be calculated using Ohm's law. In a dynamic circuit, however, capacitors and inductors influence the current flow according to how fast the current is changing, so that Ohm's law no longer applies.
 
 




Figure 4. Alternating current as a function of time.





Capacitance

A capacitive element consists of two conducting surfaces that are separated by a nonconducting (dielectric) material. When current flows into a capacitor, a positive charge builds up on the first surface and a negative charge builds up on the opposite surface. Capacitance is the amount of charge that can be stored per volt on the capacitor's surface. Its units are farads (f), although most capacitors are small fractions of a farad; therefore, most real capacitors have values measured in microfarads (µf) or picofarads (pf) are more commonly seen. The lipid bilayer in biological membranes is an excellent dielectric, so that the space between channels across cell membranes forms a very effective capacitor, thereby changing the shapes of electrical events generated by the opening and closing of channels. Also, the glass walls of a glass microelectrode serve as capacitors which distort the signals being measured by the electrode. In order to measure accurately the electrical signals generated by neurons and muscle cells, it is necessary to eliminate this source of capacitance.

Inductance

An inductor is a low-resistance wire wound into a coil. Inductance is similar to capacitance as it is also dependent on the rate of change of the input signal. However, an inductor has the opposite properties: it has its lowest resistance with low current flow and a higher resistance at higher currents. Therefore, the inductor behaves as a low resistance in steady-state. Inductive elements are seldom found in electrophysiology and therefore will not be discussed further.

Circuit Analysis of Dynamic Circuits

If a battery were connected to a capacitor, it would initially present very little resistance to electron flow, and the current would be high. However, as charges accumulate on the surfaces of the capacitor--positive on one surface and negative on the other--saturates with charge, and the resistance to current flow increases until the current stops flowing. At this point the circuit has reached steady-state and the capacitor behaves as an open circuit (i.e., its resistance is infinite--unless the voltage is so high that it breaks down the dielectric of the capacitor). The length of time that current flows depends on the size of the capacitor: a capacitor with a larger surface (i.e., a larger capacitance) will take longer to charge than will a capacitor with a smaller surface. If the direction of the applied voltage is reversed, current can flow in the opposite direction, initially to remove the accumulated charge and then to accumulate charge in the opposite direction. If the current continues to alternate, the capacitor will pass current as long as the cycle time of the AC signal is shorter than the time it takes for the capacitor to saturate with charge.

For practical purposes, circuits are never made purely of capacitors but also have resistors in them. Most circuits can be understood by considering two cases: when a capacitor is in series with a resistor and when it is in parallel with a resistor.

Capacitor in series with a resistor

Consider the current that flows in the circuit shown when a square-wave voltage is applied by the voltage, as well as the voltage that appears across the resistor:
 
 

Figure 5. A circuit with a resistor (R) and capacitor (C) in series with a square-wave generator (S).




At the positive-going part of the square wave, charge accumulates on one plate of the capacitor, and positive current flows. As charge accumulates, less current flows until no more can flow. In fact, no more current will flow as long as the voltage stays constant. The time that it takes for the circuit to reach 63% of its final state (i.e., when the voltage goes to 37% of its maximum) is called the time constant of the circuit. The time constant for this circuit is determined by the circuit capacitance (C) in µF and the circuit resistance (R) in MW: T = RC
  At the negative-going part of the square wave, all the charge that has accumulated on the capacitor now flows in the opposite direction until the capacitor is completely discharged. The size and time-course of the discharge is equal and opposite to the original charge. If the voltage across the resistor is measured, its shape is the same as the shape of the current because at every instant, this voltage is equal to the current (which is sometimes changing) times the resistance (which is constant), as defined by Ohm's law.
 
 
 
  Capacitor in parallel with a resistor A very different behavior is seen when a resistor an capacitor are in parallel:
 
 

Figure 6. The same circuit as in Fig. 5, with a resistor in parallel with the capacitor.




In this case, a large amount of current flows at the start of the square wave: some of it goes through R2 and some of it serves to charge the capacitor, which has a very low resistance at this point. (All the current, of course, flows through R1). When the capacitor is fully charged, no more current flows through it but the current continues to flow through the resistor. Note that the voltage across R2 increases with the same time-course as the decay of current across the capacitor.

In some cases, capacitors in parallel with a resistor can be a nuisance. For instance, the glass in the micropipettes used to record bioelectrical signals serves as a dielectric between the fluids inside and outside the electrode; i.e., it is a capacitor in parallel with the resistance of the microelectrode. This arrangement tends to "round-off" sharp signals--like action potentials--in the same way as it rounds off the square wave in Fig. 6. To overcome this rounding-off, the amplifiers used to record from microelectrodes have sophisticated "capacitance compensation" circuits. In other cases, however, parallel conductances are useful. For instance, amplifiers usually have a "hi-cutoff filter" to eliminate high-frequency signals that are part of the recording circuitry. Such filters are usually composed of circuits like Fig. 6.
 
 

Impedance Impedance (Z) is a measure of the opposition to current flow created by a circuit component. The impedance to current flow caused by a resistor is equal to its resistance. The impedance caused by a capacitor is known as capacitive reactance. Reactance, like resistance, is described in units of ohms. Since the degree of current limitation in a capacitive circuit depends on the frequency of the signal, capacitive reactance is given by this formula:

Xc = -1 / (2 p f C),

where: Xc is capacitive reactance in ohms

f is the frequency in Hz

C is the capacity in farads.

The total impedance of a circuit is calculated by combining the impedances (resistances and reactances) of each element. If the elements are connected in series, the impedances are added together. If the elements are connected in parallel, the formula used to derive the total resistance of a parallel resistive circuit is used, substituting impedances for resistances.

Ohm's law can be applied to impedance by substituting Z for R:
 

V = I x Z
  It is somewhat sloppy terminology, but "impedance" is often used for "resistance" when discussing the properties of an amplifier or of an electrode. Therefore, statements are often made like "This amplifier has a high input impedance" and "The electrodes all have low impedance" when what is being discussed is actually the resistance of these objects.

modified 8-27-06