Now, onto my favorite part of electricity: circuits!

Circuits exploit the aforementioned physics concepts in order to harness and manipulate electricity. Circuits are composed of circuit elements, discrete components each designed to perform a specific function by manipulating electricity according to some physical law. An understanding of how circuit components work aids in the analysis of complicated circuits. The basic Marx Generator circuit by itself only requires three unique components: resistors, capacitors, and spark gaps. However, for the purpose of providing an adequate introduction to electronics, I shall introduce several other major components as well.

Resistors: Oppose current. Resistors add resistance, the electrical analog of friction, to a circuit. Electrical loads, such as lamps, add resistance, or impedance if reactive components are involved, to a circuit. Wires possess an innate, material specific quality called resistivity, and the resistance of a wire can be calculated as the product of the wire's resistivity and length divided by its cross-sectional area. The resistance of a resistor, the voltage across a resistor, and the current through a resistor are all related by Ohm's Law. Potentiometers, rheostats, and trimmers are types of variable resistors, which can be configured to form adjustable voltage divider circuits. Resistors are used to limit current and/or voltage in circuits. In this Instructable, we will be using resistors to slow the charge and discharge of capacitors.

Capacitors: Store energy in an electric field. Capacitors are often composed of two parallel conducting plates on which charge accumulates when a voltage is applied. Between the plates, these forms a uniform electric field having magnitude proportional to the surface charge density of the plates. As charge accumulates, the electric field, and thus voltage, between the plates increases in magnitude. Once the voltage across the capacitor equals the source voltage, current will cease to flow. Decreasing the surface area of the plates will increase the voltage per unit charge and decrease the maximum charge accumulation accordingly. In this way, the product of the voltage and charge of a capacitor remains constant and defines an innate quality of each capacitor called capacitance, C. The energy (in joules) stored in the electric field of a capacitor at any instant can be calculated as 1/2CV^2. As a capacitor is charged through a resistor (an RC circuit), the voltage difference between the capacitor and the supply decreases and charging slows. Using calculus, we can solve a first-order differential equation for the current through the RC circuit with a steady supply voltage as a function of time. The result indicates the current decreases exponentially towards zero, with steeper decrease resulting from smaller capacitance and resistance values. The product of resistance and capacitance in an RC circuit is known as the RC time constant. The capacitor's opposition to slow changing currents (i.e. low frequencies) is known as its reactance, X. In AC circuits, reactance compounds resistance to yield complex impedance Z, defined as the sum of orthogonal resistance and reactance vectors. In short, at very high frequencies (approaching infinity), capacitors offer no impedance and act as short circuits. At very low frequencies (approaching 0; DC), capacitors offer infinite impedance and act as open circuits. In this Instructable, we will be using capacitors as the primary energy storage element.

Inductors and Transformers: Store energy in a magnetic field. Inductors are the magnetic analog of capacitors and mirror their behavior. Inductors are simply coils of wire, and as such, wire itself can exhibit non-ideal parasitic inductance (likewise, two wires lying adjacent can exhibit parasitic capacitance). Inductors exploit the principles of electromagnetism described by Ampere's Law and Faraday's Law. From Ampere's Law, current running through a wire produces a magnetic field that encircles the wire. From Faraday's Law, a changing magnetic field (magnetic flux) through a circuit induces a current that counteracts the magnetic field. Combining the laws, we see that the magnetic fields resulting from individual loops in an inductor serve to sustain current flowing through the inductor. This characteristic behavior of inductors is measured as inductance, L. The energy stored in the magnetic field of an inductor at any time can be calculated as 1/2LI^2. As with the capacitor, we can solve a first-order differential equation for the current through the RL circuit (resistor-inductor circuit) as a function of time. We find that the current gradually approaches a value equal to V/R (supply voltage divided by resistance) according to an exponential with a steepness that increases with decreasing inductance and resistance values, the product of which is referred to as the RL time constant. When the current through an inductor changes, an emf (electromotive force; voltage) is induced across the inductor that directly opposes the current which caused produced it. The magnitude of the emf produced is proportional to both the rate of change of the current through the inductor and the inductance of the inductor. In this way, the inductor opposes fast changing currents (i.e. high frequencies), giving it a reactance, X, that mirrors that of the capacitor. In short, at very high frequencies (approaching infinity), capacitors offer no impedance and act as short circuits. At very low frequencies (approaching 0; DC), capacitors offer infinite impedance and act as open circuits. Thus, the inductor's frequency response is inverse of the capacitor's. The inductive reactance vector points in the opposite direction to the capacitive reactance vector. Thus, there exists some frequency at which inductive and capacitive reactances cancel. It is at this resonant frequency that voltage and current will oscillate in an inductive-capacitive (LC; tank) circuit as energy sloshes back and forth [indefinitely] between the inductor's magnetic field and the capacitor's electric field.

Two inductor coils can be wound on core to form a transformer. One coil becomes the primary winding of the transformer and the other becomes the secondary winding of the transformer. The two windings share mutual inductance, a magnetic linkage or coupling. When the current through the primary winding changes, the changing magnetic flux through the primary is transferred to the secondary via the ferrite core. This induces a current in the secondary that is proportional to the current in the primary. The ratio of turns in the primary winding to turns in the secondary winding determines the relative magnitudes of the voltages and currents in each winding. Secondary voltage is equal to primary voltage divided by the ratio. Secondary current is equal to primary current multiplied by the ratio. In this way, power is not created but rather transformed. If the ratio is greater than 1:1, secondary voltage will be greater and the transformer is considered a step-up transformer. The reciprocal is true for a step-down transformer. Keep in mind that the primary and secondary winding designations are arbitrary; a transformer may be reversed to obtain the inverse ratio. In this Instructable, we will be using a transformer to step-up the supply voltage.

Diodes: Permit current to flow in only one direction. Semiconductor diodes are composed of a single junction of two doped semiconducting materials. The forward bias voltage (typically ranging from 0.7-1.4V) is the potential difference required for current to flow through a diode in the forward direction. The reverse bias voltage (typically much higher than the forward bias voltage) is the potential difference at which the diode will break down and allow current to flow in the reverse direction (this is usually considered non-ideal behavior; however, in the case of Zener diodes, the breakdown resulting from reverse bias is exploited for its "avalanche" effect). The negative (cathode) terminal of a diode is indicated by a band (see image). Diodes are commonly used to rectify AC to DC using a four-diode configuration called a full-wave rectifier or diode bridge. In this Instructable, we will be using diodes to rectify AC in a Cockcroft–Walton voltage multiplier circuit.

Transistors: Switch and amplify current. During the second half of the 20th century, the proliferation of solid-state transistors in electronics made obsolete previous switching devices, such as relays and vacuum tubes, and sparked the digital electronics revolution. Although there are several different classes of transistors, most adhere to a common basic structure consisting of three pins: a base (or gate), collector (or drain), and emitter (or source). Bipolar junction transistors (BJTs) are composed of two adjacent semiconducting junctions in either an NPN or PNP configuration. For a BJT, a small signal at the base can modulate the flow of a larger current between the collector and emitter. The high-gain properties of some transistors can be exploited to form logic circuits with binary states. In this Instructable, we will be using a high-power NPN transistor to switch the transformer current.

Spark gaps: Conduct electricity only at high voltages. Spark gaps consist of two electrodes separated by air or other dielectric. Up to a certain voltage, the dielectric will act as an insulator and inhibit current. However, once the electric field magnitude between the electrodes has exceeded the specific dielectric strength, the dielectric will breakdown and conduct. For the ionization of air, the rough approximation of 1kV per mm of separation is commonly used. In this Instructable, we will be using spark gaps to trigger the firing of the Marx Generator.

Specialty Components - Actuators, Transducers, and Sensors: Convert electrical energy into energy of another form and vice versa. The term "transducer" is used to refer to anything that facilitates such conversion. Motors and solenoids are examples of actuators, which convert electrical energy into angular and linear motion. Microphones, speakers, and piezoelectric materials also fall under the definition of transducer. The term "sensor" may refer to a transducer that uses minimal or ambient energy to derive information, such as light intensity or chemical concentration, about the surrounding environment.

Integrated Circuits: Package entire circuits into tiny chips. The integration density of ICs has grown exponentially since the invention of the IC by Jack Kilby. This growth phenomenon, known as Moore's Law, has seen ICs become smaller, faster, and cheaper simultaneously. Current technology enables billions of individual transistors to be packaged into a single IC. In this Instructable, we will be using a TLC555 timer, a common hobbyist IC, to generate a square wave signal.