Resistors in DC Circuits

Resistors are the building blocks of electrical circuits, offering resistance to the flow of electric current by dissipating energy as heat. In direct current (DC) circuits, their behavior is governed by Ohm's Law, which states:

$V=IR$

This law finds its foundation in Kirchhoff's Laws, particularly Kirchhoff's Voltage Law and Kirchhoff's Current Law.

Kirchhoff’s Voltage Law states that the sum of the voltages around any closed loop in a circuit must be zero:

$V_1+V_2+V_3+V_4=0$

Kirchhoff’s Current Law states that the total current entering a junction must equal the total current leaving the junction:

$\Sigma{I_{in}}=\Sigma{I_{out}}$

Internal Resistance can be accounted for in circuits, adding complexity to calculations. It's often represented as $r$, and for a voltage source with internal resistance, the effective voltage across the terminals is given by:

$V=\epsilon-Ir$ , where $\epsilon$ is the EMF, or electromotive force that would be supplied by the power source, assuming there was no internal resistance (an ideal power source).

Capacitors in DC Circuits

Capacitors are devices that store electrical energy in an electric field between two charged plates. In DC circuits, capacitors can accumulate charge and maintain a voltage across their terminals. The behavior of capacitors is described by the equation:

$Q=CV$

The Capacitor Construction Equation is used to describe the properties of a capacitor:

$C=\frac{\epsilon_r\epsilon_0A}{d}$ , where $C$ is the capacitance, $A$ is the overlapping area of the two plates, $d$ is the distance between the plates, $\epsilon_0$ is the permittivity of free space ($8.84\times10^{-12}F m^{-1}$) and $\epsilon_r$ is the dielectric constant.

The Dielectric Constant compares the capacitance of a capacitor with a dielectric to one without. A dielectric is typically an insulating material that helps store charge. The dielectric constant of air is approximately 1, so it can be ignored in the equation if there is no material between the plates.

RC Circuits contain a capacitor and a resistor. The capacitor will store charge, building up a voltage until it is fully charged. Then, as the supply voltage is now zero, it will discharge in the opposite direction to the supply, proving a voltage to the resistor until fully discharged.

Time Constants ($\tau$) describe the way capacitors charge and discharge. One time constant is the time it takes for the capacitor to either:

• Charge to 63% of the maximum value or;

• Discharge by 63% of the maximum value.

The time constant can be calculated using:

$\tau=RC$

After $5\tau$ , we say that the capacitor is essentially fully charged or discharged (99.9%).

Energy is supplied to a capacitor by the power source, and is calculated using the equation:

$E=QV$

The capacitor will store energy in the electric field between its plates. However, only half of the supplied energy will make it to the capacitor, as the other half will be dissipated as heat by resistance (in wires, resistors, or internal resistance). Therefore, the energy stored in a capacitor is given by:

$E=\frac{1}{2}QV$ or $E=\frac{1}{2}CV²$

Capacitors in Series and Parallel

When electrical components are placed in series, the voltage used by each component will sum to the supply voltage. This also holds true for capacitors. In a series capacitor network:

• Voltages sum to the supply voltage;

• Each capacitor has that same charge;

• $\frac{1}{C_T}=\frac{1}{C_1}+\frac{1}{C_2}…$

When electrical components are placed in parallel, each branch has equal voltage to the supply. This also holds true for capacitors. In a parallel capacitor network:

• Each capacitor has the same voltage across it;

• Charges are different, based on $Q=CV$ ;

• $C_T=C_1+C_2…$

Inductors in DC Circuits

Inductors are components that store energy in a magnetic field when current passes through them. In DC circuits, inductors exhibit unique characteristics related to magnetic flux, Faraday's Law and Lenz’s Law.

Magnetic Flux is a measure of the number of magnetic field lines passing through a defined area or surface. Magnetic flux is greatest when the field lines are perpendicular (at right angles) to the surface. It can be found using:

$\Phi=BAsin\theta$

Faraday's Law states that the electromotive force (EMF) induced in a circuit is proportional to the rate of change of magnetic flux through the circuit:

$\epsilon=\frac{-\Delta\Phi}{\Delta{t}}$ ,where $\epsilon$ is the induced EMF, and $\Phi$ is the magnetic flux.

Lenz’s Law states that the induced EMF will oppose the change in magnetic flux that created it. Using the right-hand grip rule, we can find the direction of the current and magnetic field lines.

Self Induction occurs when current flows through a coil and a magnetic field is created. Using the right-hand grip rule, we can find the direction of the current and magnetic field lines. If the current is changing (e.g. in AC or if it is turned on and off) there will be a changing magnetic flux. This will induce a back EMF which will resist the growth/decay in the primary current.

Inductors are designed to oppose changes in current in a circuit. If an ideal inductor is used, it will not add any resistance to the circuit. However, a real inductor will add some resistance, reducing the overall current in the circuit. The inductance of an inductor can be influenced by:

1. Number of turns in the coil → More turns means more induced voltage, so a higher inductance;

2. Coil area → A larger coil area means more magnetic flux, so there is a higher induced voltage and higher inductance;

3. Coil length → The shorter the coil with the same number of turns means that there is more magnetic flux, so a higher induced voltage and a higher inductance;

4. Core material → If high permeability materials (e.g. soft iron) are added the magnetic flux will increase. Therefore the induced voltage and inductance will be higher.

The induced (back) EMF can be found using:

$\epsilon=\frac{-L\Delta{I}}{\Delta{t}}$

Inductors store energy in the induced magnetic field. The energy stored in an inductor is given by:

$E=\frac{1}{2}LI²$

An RL Circuit contains an ideal inductor and a resistor. When the switch is closed, the change in current induces a back EMF in the inductor. Initially the opposing voltages result in 0V across the resistor. As the current stabilises the change in magnetic flux will reduce to zero, so the back EMF induced will also reduce to zero. When the switch is opened, the change in current will induce a back EMF in the inductor which will keep the voltage the same as it originally was. The resistor will use this EMF. Over time, this will reduce to zero. The current in the circuit and the voltage across the resistor will always be in phase. $V_S=V_L+V_R$

The Time Constant for an RL circuit is the time it takes for the voltage or current to increase/decrease by 63%. It is given by:

$\tau=\frac{L}{R}$

Mutual Inductance occurs when the magnetic field created by one coil induces an EMF in a second coil. This is because if the coils are close enough together, the magnetic field lines will pass through the second coil. If the current in the first coil is changing, it will have a changing magnetic flux. Therefore, there will be a changing magnetic flux in the second coil which will result in an induced EMF to oppose the change.

Transformers are components that use mutual inductance to change the voltage of a power supply. A basic transformer has two sets of windings on different sides of an iron core. A power supply will be attached to the primary windings, creating a current and a magnetic field. This field will create a magnetic flux in the secondary windings. By constantly changing the current in the primary windings, there will be a constantly changing magnetic flux in the secondary windings, so a voltage will be induced.

Using an iron core for the loop provides a pathway for the magnetic field to travel between the primary and secondary windings.

The output voltage of a transformer is based on the ratio of the number of turns on each set of windings:

$\frac{N_P}{N_S}=\frac{V_P}{V_S}$

In an ideal transformer, there are no power losses. This gives the relationship:

$V_PI_P=V_SI_S$

In reality, there are power losses where some is lost as heat. This impacts the current flowing out of the transformer but not the voltage. This scenario is given by the relationship:

$I_S=\frac{Efficiency\times{P_P}}{V_S}$ , where the efficiency is given as a percentage.

There are three types of transformer:

1. Step Down Transformer → There are fewer turns in the secondary windings. This results in a smaller voltage and larger current output.

2. Step Up Transformer → There are more turns in the secondary windings. This results in a larger voltage and smaller current output.

3. Isolating Transformer → There are the same number of turns in the primary and secondary windings, maintaining the current, voltage and power. This is used to protect the circuit against spikes in the current by using mutual inductance to oppose the change.

Eddy Currents are swirling currents induced in solid conductive materials when exposed to a changing magnetic field. These currents circulate within the material and generate their own magnetic fields, which oppose the original magnetic field. They are often undesirable in electrical devices as they can cause energy loss and heating, but they are also utilized beneficially in applications such as electromagnetic braking and metal sorting.

AC Circuits

In AC Circuits, the current follows a sinusoidal curve (a sine wave). As it is constantly changing, there are some unique features seen with resistors, capacitors, and inductors in these circuits. When the resistance in an AC circuit is greater than $1\Omega$ then the maximum voltage will be greater than the maximum current and the minimum voltage will be smaller than the minimum current. When time=0, voltage and current also =0.

We can find the instantaneous current or voltage in an AC circuit using the following formulae:

$V=V_{MAX}\sin\omega{t}$ and $I=I_{MAX}\sin\omega{t}$

$\omega$ is the angular speed and is measured in radians per second. This can be found using:

$\omega=2\pi{f}=\frac{2\pi}{T}$ , where T is the time period (s) and f is the frequency (Hz).

Due to the constant oscillation of voltage and current in an AC circuit, there is no fixed value to use in calculations. This is where Root Mean Square (RMS) values come in. These are averages, which are the DC equivalent of heating through a resistor, given by:

$V_{RMS}=\frac{V_{MAX}}{\sqrt2}$ and $I_{RMS}=\frac{I_{MAX}}{\sqrt2}$

Power in AC circuits will always be positive, as it is found by multiplying current and voltage (which are always in phase). The average power is given by $P_{AV}=\frac{1}{2}P_{MAX}$ .

Phasors are like vectors for electrical quantities. They are used to find out which quantities are lagging or leading others. When given the phasors for two of the supply voltage, or the voltage across the capacitor, inductor, or resistor, the other quantity and the amount it is lagging/leading by can be found by using trigonometry.

Capacitors in AC circuits will charge until $V_C=V_S$ (so there is no current). However, as both quantities are constantly changing there may be a situation where V_C>V_S, when the capacitor will begin discharging (so the current will be negative). The current will not be in phase with the supply voltage in this circuit.

Capacitive Reactance is a quantity that describes how much a capacitor impedes the current in a circuit. It is measured in $\Omega$ , but is not the same as resistance, as it is dependent on the frequency:

$X_C=\frac{1}{2\pi{fC}}=\frac{1}{\omega{C}}$

When the frequency is higher, the current is oscillating rapidly, meaning that the capacitor has less time to charge. This means that there is less impedance due to the capacitor, so the capacitive reactance is less.

Similar to Ohm’s Law, the voltage across the capacitor can be found using:

$V_C=IX_C$

In an RC Circuit in AC, the current and voltage across the resistor are in phase. As the voltage across the capacitor takes time to build up, it lags behind the voltage across the resistor. Therefore we can say that:

• $V_R$ reaches a maximum value before $V_C$.

• $V_R$ is at the maximum value when $V_C=0$. This is because $V_C$ impedes current flow and therefore reduces $V_R$.

• When $V_C$ is at the maximum value, $V_R=0$. This is because $V_C=V_S$ so there is no current and no voltage across the resistor.

• This means that $V_C$ lags $V_R$ by $\frac{1}{4}$ cycle ($90^o$ or $\frac{\pi}{2}$ rad).

Inductive Reactance is the amount that an inductor impedes the flow of current in a circuit. It can be found using:

$X_L=2\pi{f}L=\omega{L}$

When the frequency of the current is high, the inductor will experience a large change in magnetic flux. This causes a large induced voltage and more opposition to current flow, so more reactance. When the frequency is low, the reactance will be low as well. Additionally, the inductance affects the reactance. A higher inductance means that more back EMF is produced, so there is a higher reactance.

In RL Circuits in AC, the current and voltage across the resistor are in phase. As the induced voltage is dependent on the rate of change of the current, it will be greatest when $I=0$ and zero when the current is at its maximum value. Therefore, we can say that:

• $V_L$ reaches a maximum value before $V_R$.

• $V_L$ is at the maximum value when $V_R=0$. This is because the rate of change of current is greatest when $\frac{I}{V_R}=0$.

• $V_R$ is at the maximum value when $V_R=0$.

• $V_L$ leads $V_R$ by $\frac{1}{4}$ cycle ($90^o$ or $\frac{\pi}{2}$ rad).

LCR Circuits contain a capacitor, inductor and resistor. The capacitor and inductor voltage phasors are $\frac{1}{2}$ cycle out of phase, meaning that they work against each other. Because of this, the difference between them can be used to calculate the supply voltage phasor using vector addition. Using trigonometry:

$V_S=\sqrt{((V_L-V_C)²+V_R²)}$

$\theta=\sin^{-1}(\frac{V_L-V_C}{V_R})$

Impedance is the total opposition to current in a circuit when a current is applied. Impedance can be found using:

$V=IZ$

Or it can be found vectorially using:

$Z=\sqrt{(R²+(X_L-X_C)²)}$

When calculating impedance for RC or RL circuits, make the missing quantity =0.

Resonance occurs in LCR circuits as the result of oscillations as stored energy is passed from the inductor to the capacitor. Resonance is greatest when $X_L=X_C$. At resonance, $Z=R$ and current will be at its maximum value. The resonant frequency ($f_o$) is the frequency that produces the maximum current. This can be calculated using:

$f_o=\frac{1}{2\pi\sqrt{LC}}$

Useful Formulae

Resistors in DC Circuits:

• $V=IR$

• $V_T=\epsilon-Ir$

• $P=IV=I²R$

Capacitors in DC Circuits:

• $E=\frac{1}{2}QV$

• $Q=CV$

• $C=\frac{\epsilon_0\epsilon_rA}{d}$

• $\frac{1}{C_T}=\frac{1}{C_1}+\frac{1}{C_2}…$

• $C_T=C_1+C_2…$

• $\tau=RC$

Inductors in DC Circuits:

• $\Phi=BA\sin\theta$

• $\epsilon=\frac{-\Delta{\Phi}}{\Delta{t}}$

• $\epsilon=\frac{-L\Delta{I}}{\Delta{t}}$

• $E=\frac{1}{2}LI²$

• $\tau=\frac{L}{R}$

• $\frac{N_P}{N_S}=\frac{V_P}{V_S}$

AC Circuits:

• $f=\frac{1}{T}$

• $I=I_{MAX}\sin\omega{t}$

• $V=V_{MAX}\sin\omega{t}$

• $I_{RMS}=\frac{I_{MAX}}{\sqrt2}$

• $V_{RMS}=\frac{V_{RMS}}{\sqrt2}$

• $X_C=\frac{1}{\omega{C}}$

• $X_L=\omega{L}$

• $V=IZ$

• $\omega=2\pi{f}$

• $f_o=\frac{1}{2\pi\sqrt{LC}}$