Chapter 9: Electronic circuits (C4466667)

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1 Circuit elements

Current is the flow of electrically charged particles, defined by $I=\dfrac{Q}{t}$, measured in the SI units amperes ($A$), or $C/s$. Note that although charge flows by means of electrons in metals (as a conductor of electricity, the most typical scenario), conventional current is defined as the flow of positive charge, due to Ben Franklin’s incorrect conjecture that electrons were positive. Current flow through a wire is analogous to fluid flow (rate) through a pipe (discussed ), in various respects, including that just like fluid which has random translational motion and uniform translational motion, current also has random motion and drift velocity respectively. Drift velocity is the net flow of charge. Batteries are represented by a repeated set of two parallel lines of unequal length. The longer side is the positive terminal, and the shorter side is the negative terminal.

Elements either oppose electric current, known as resistors; or do not oppose electric current, known as conductors. Semiconductors lie between these two extremes. Resistors are represented by a single jagged line.

Resistivity of a material is given by $R=\dfrac{\rho l}{A}$, where $\rho$ is electrical resistivity, $l$ is length, and $A$ is cross-sectional area. Resistance can be analogized with hydraulics discussed , such that the longer and thinner the pipes, the greater the resistance.

Capacitor is an element which has the ability to store an electric charge (whilst maintaining a low voltage drop) for delivery when required, or alternatively, $C=\dfrac{Q}{V}$. Capacitors are represented by two parallel lines of equal length.

The simplest capacitor is a parallel-plate capacitor, which are two parallel conductive plates separated by a distance. As a result of separating two conductive plates, the two plates take on equal but opposite charges. As a result, there is an electric field (charge in itself distorts space-time), and therefore an according voltage drop [due to difference in electrical potential energy of the two plates]. Voltage of a capacitor is $V=\dfrac{Qd}{\epsilon A}$, where $\epsilon=k\epsilon_{0}$ ($k$ is relative permittivity of the dielectric and $\epsilon_{0}$ is the permittivity of free space), $Q$ is the amount of charge on either (but not both) plate, $d$ is distance between the plates, and $A$ is the surface area of either (but not both) plate. Note therefore that voltage is directly proportional with charge of the plate, and inversely proportional with area of the plate. The material [or lack of material: air] between the plates is known as a dielectric, and is an insulator (otherwise, the capacitor would be discharged), which can be polarized when placed in an electric field, such that an internal electric field can be generated, weakening the overall field [and therefore voltage] of the dielectric. This makes sense because the strength of the dielectric ($\epsilon$) is inversely proportional to voltage, a polar dielectric will cause decreased voltage. Since $E=\dfrac{V}{d}$ from , therefore $E=\dfrac{Qd}{\epsilon A}.\dfrac{1}{d}=\dfrac{Q}{\epsilon A}$. Capacitance is $C=\dfrac{Q}{V}$, and since $V=\dfrac{Qd}{\epsilon A}$ from , in-substituting, $C=Q.\dfrac{\epsilon A}{Qd}=\dfrac{\epsilon A}{d}$. Electrostatic potential energy stored on a capacitor can be calculated by $PE=\dfrac{1}{2}QV$, and since $Q=CV$, in-substituting, $PE=\dfrac{1}{2}CV^2$, or alternatively, $PE=\dfrac{Q^2}{2C}$.

Remember from  that power is the rate at which energy is transferred. Electric power is given by $P=IV$. Since $V=IR$, in-substituting, we can have $P=I^2 R$ or alternatively, $P=\dfrac{V^2}{R}$. Note that these last two formulae are only applicable to determine power dissipation from electrical to heat energy.

 Formative learning activity Maps to RK9.A Give examples of circuit elements, and what they do

2 Circuits

Circuits are closed loop networks, consisting of an electrical source, and elements, permitting flow of current.

Series circuits require a current to go through every component of that circuit. The total resistance of resistors in series is equal to the sum of their resistances, $R_{total}=R_{1}+R_{2}+ ... +R_{n}$. Parallel circuits are connected such that current can circumvent a component of that circuit, but current is nevertheless supplied to that component. The total resistance of resistors in parallel is equal to the sum of the reciprocals of their resistances, $\dfrac{1}{R_{total}}=\dfrac{1}{R_{1}}+\dfrac{1}{R_{2}}+ ... + \dfrac{1}{R_{n}}$. Capacitors can also be totaled like resistors, but are totaled in the opposing manner: when in series, they sum reciprocals $\dfrac{1}{C_{total}}=\dfrac{1}{C_{1}}+\dfrac{1}{C_{2}}+ ... + \dfrac{1}{C_{n}}$, and when in parallel they are directly summed $C_{total}=C_{1}+C_{2}+ ... +C_{n}$. The total battery, resistance and capacitance is useful to calculate as a set of elements can be reduced to a single element. Once reduced to a single battery, resistance and capacitance, the circuit can be solved using Ohm’s law, which is $V=IR$, where $V$ is voltage, $I$ is current, and $R$ is resistance. Keep in mind that current going through each component in series is the same, and voltage going through each component in parallel is the same.

When a node is reached, current in each respective node can be determined using Kirchhoff’s laws. Node is the junction of wires coming together, or splitting away. Kirchhoff’s 1st law is that the currents flowing into a node is equal to the currents flowing out of a node. The equivalent hydraulic analogy, is that the rate of fluid flow into a pipe, is equal to the rate of fluid flow out of a pipe. Kirchoff’s 2nd law is that the sum of potential differences [or voltage drop] around any closed network is 0. The equivalent gravitational analogy, is that if a mass is moved up and down but eventually returns to its original height, the sum of the change in gravitational potential energy is 0.

 Formative learning activity Maps to RK9.B What are circuits?

3 Alternating currents

Although the  circuits assume direct current (DC), which is where electricity flows only in one direction, this is not always the case. Alternating current (AC) is where electrical charge periodically reverses direction. The usual waveform of AC is a sine wave. Note that if current is sinusoidal, that the average current is 0. Because this doesn’t provide sufficient information on current, root mean square (RMS) current is used instead. There is also a RMS voltage. Root mean square refers to the square root of the mean of the squares of the values. Note that because the mean is found of the squares of the values (rather than the mean of the values per se), the RMS is slightly higher than the average of the positive of the values. For a sinusoidal current, $I_{rms}=\dfrac{I_{peak}}{\sqrt{2}}$; and for a sinusoidal voltage, $V_{rms}=\dfrac{V_{peak}}{\sqrt{2}}$. For example, the typical voltage is 230V, meaning the related peak voltage is $V_{peak}=230\sqrt{2}\approx 325$.

 Formative learning activity Maps to RK9.C What is alternating current?

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