Circuit Theory/Energy Storage Elements

Energy Storage Elements

Resistors are not the only available circuit element. Far from it: There are many different types of elements that can be found in circuits. Among passive elements, there are 2 more types besides resistors: capacitors and inductors. Both capacitors and inductors can store energy, to be released back into the circuit under certain conditions. Capacitors store energy in an electric field, while inductors store energy in a magnetic field.

Capacitors

Capacitors are passive circuit elements that can be used to store energy in the form of an electric field. In the simplest case, a capacitor is a set of parallel metal plates separated by a dielectric substance.

electric charges build up on the opposite plates as a voltage is put across the capacitor. Capacitors can transfer voltage and current across the dielectric, until the electric field inside the capacitor reaches its maximum capacity. At which point, the field is saturated, and no more charges can travel from one place to the other. With a constant charge applied to the capacitor therefore, the capacitor eventually becomes an open circuit.

With a constant voltage across the capacitor, the steady-state current becomes zero. Stored energy can be discharged from a capacitor by removing an external forcing voltage, and by shorting the capacitor with a load resistance.

Capacitance

The ability of a capacitor to store energy is called Capacitance, and is measured in units called "Farads", abbreviated with an "F" (capital F). The variable most commonly associated with capacitance is "C" (upper-case C). The capacitance associated with a capacitor can be found as a ratio of the amount of charge that the element can hold, divided by the voltage across the capacitor:

$C=\frac{q}{v}$

The relationship between the current and the voltage of a capacitor is as follows:

$i=C\frac{dv}{dt}$

Energy Storage

The amount of energy that is storable in a capacitor is determined as follows:

$E=\frac{1}{2}C{v}^2$

Capacitors in Series

A set of N capacitors in series

The total capacitance of a series of capacitors is given by the following formula:

$C_{series}=\frac{\prod _n^{}{C}_n}{\sum _n^{}{C}_n}={\left(\sum _n^{}\frac{1}{x_n}\right)}^{-1}$

Impedance

Impedance is the characteristic of Capacitor resist current flows when a Voltage is applied on the Capacitor

Impedance of Capacitor is defined as the sum of its Resistance and Reactance

$Z_C=R_C+X_C$

Where:

R_C Resistance of the Capacitor

X_C Reactance of the Capacitor = 1 / jωC

j =ν - 1

ω = 2πf

C = Capacitance of the Capacitor

Direct Current

Capacitor acts as Open Circuit . At the load would see zero Voltage .

Alternating Current

When apply a Voltage on the Capacitor . The Voltage of the Reactance is lagging The Voltage of the Resistance one angle equals to 90^ο . Voltage of the Resistance is in the same phase with the Applied Voltage . The Load Voltage is at an angle θ with the total Voltage of Resistance and Reactance

• V_XC lags V_RC by 90°
• V_RC is the same phase with V_i

A capacitor is a frequency dependent element. There is one frequency at which the capacitor react or start to conduct current and this frequency is called Response Frequency denoted as ω_o = 1 / RC and the time that it takes to reach this frequency is t = RC

But how do you arrive at the frequency reponse? Idealy, when there is no voltage apply on capacitor there will be no current flows. Therefore, The impedance of the capacitor is equal to 0.

Z_C = R_C + 1 / jωC = 0 or

jω = 1 / CR_C

• ω = 0 $Z_C=R_C+\mathrm{\infty }$ . Open circuit V_o = 0
• ω = ω_o . Starts to react or conduct current. V_o ≈ V_i
• ω = infinity $Z_C=R_C+0$ . V_o ≈ V_i.

Capacitors in Parallel

A set of N capacitors in parallel

For capacitors in parallel:

$C_{parallel}=\sum _n^{}{C}_n$

For assistance remembering this formula remember the construction of a capacitor, that capacitance increases with the area of the plates.

Capacitors can "Pop"

Many capacitors are polarized in a particular way. If you apply voltage across the terminals of a polarized capacitor, the capacitor itself might pop. This is made more dangerous by the fact that many capacitors have chlorine gas inside, because the chlorine raises the capacitance of the capacitor. Popping a chlorine-filled capacitor will be very unpleasant (if not down-right dangerous).

Capacitors can Kill

Strong capacitors, specifically the capacitors in microwave ovens and CRT screens can remain charged when the device is turned off. Remember that a capacitor maintains its charge until a load is placed across its terminals (or the capacitor is shorted). For this reason, large capacitors with a high voltage across their terminals, can hold a dangerous charge even if the device is turned off, or has been out of use for a long time. Large capacitors can produce enough voltage to create a 4 amp current across the hands of a person who grabs it. 4 amps is a fatal amount of current. Be careful when dealing with old capacitors.

Inductors

An inductor is a coil of wire that stores energy in the form of a magnetic field. With a forcing voltage applied to the inductor, the magnetic field charges up. When the magnetic field has reached it's maximum capacity, the inductor no longer stores energy, and the inductor becomes a short-circuit.

Inductance

Inductance is the capacity of an inductor to store energy in the form of a magnetic field. Inductance is measured by units called "Henries" which is abbreviated with a capital "H". The variable associated with inductance is "L".

The relationship between inductance, current, and voltage through an inductor is given by the formula: v(t) = Ldi/dt

$v=L\frac{di}{dt}$

Energy Storage

The energy stored in an inductor is given with the formula:

$w=\frac{1}{2}L{i}^2$

Inductors are generally used in applications such as for limiting current through dc-dc converters, either for step-up operations, or step-down operations. Also, because inductors convert electrical energy into a magnetic field, they are the primary components of transformers, which we will discuss later.

When the forcing voltage is removed from an inductor, the energy from an inductor is discharged.

Impedance

Impedance is the characteristic of Inductor resist current flows when a Voltage is applied on the Inductor

Impedance of Inductor is defined as the sum of its Resistance and Reactance

$Z_L=R_L+X_L$

R_L Resistance of the Inductor

X_L Reactance of the Inductor = jωL

j = γ -1

ω = 2πf

L = Inductance of the Inductor

Direct Current

Inductor acts as Short Circuit . At the load would see the applied Voltage

Alternate Current

When apply a Voltage on the Inductor . The Voltage of the Reactance is leading The Voltage of its Resistance one angle equals to 90^ο . Voltage of the Resistance is in the same phase with the Applied Voltage

• V_XL leads V_RL by 90^ο
• V_RC is the same phase with V_i

Inductor is frequency dependent element . There is one frequency at which the Inductor react or start to conduct current and this frequency is called Response Frequency denoted as ω_o = R / L and the time that it takes to reach this frequency is t = L / R

How do you arrived the Frequency reponse ? Idealy, When there is no Voltage apply on Inductor . There will be no current flows . Therefore, The Impedance of the Inductor is equal to 0

Z_L = R_L + jωL = 0 or

jω = R_L / L

• ω = 0 Z_L = R_L + 0 . Inductor is Short circuited V_o ≈ V_i
• ω = ω_o Inductor Starts to React or conduct current . V_o ≈ V_i
• ω = infinity $Z_L=R_L+infinity$ . Inductor is Open circuited V_o ≈ 0

Inductors in Series

A set of N inductors in series

Like resistors, inductors appearing in series can be conceptually converted into a single inductor, with a total inductance, $L_{series}$ given as follows:

$L_{series}=\sum _n^{}{L}_n$

Inductors in Parallel

A set of N inductors in parallel

If multiple inductors are in parallel, we can calculate out the resultant inductance of the circuit as follows:

$L_{parallel}=\frac{\mathrm{\Pi }{L}_n}{\mathrm{\Sigma }{L}_n}$

Warnings

Inductors and capacitors have different associated dangers. For inductors, when the current flowing is interrupted a high voltage pulse resulting from the consequent collapse in the inductor's magnetic field can be dangerous. Using makeshift setups to conduct current through a large value inductance can be very dangerous especially when the circuit is disconnected. Give adequate planning to how your circuit and apparatus will dissipate the voltages created when power is removed from an inductor.

It is important to note also that the magnetic field of an inductor can cause magnetic interference with other electric devices, and can damage sensitive digital circuits.