Block 12 - Capacitors and Capacitance

Well, again

• read through the present chapter and write down anything you did not understand.
• Also here, there are some clips for more clarification under 'Embedded resources' (check the text above/below, sometimes only part of the clip is interesting).

For checking your understanding please do the following exercises:

• 5.5.1
• 5.5.3
• 5.5.4

1. Warm-up (8 min):
   1. Quick recall quiz: $C=\dfrac{Q}{U}$, $\vec{D}=\varepsilon\vec{E}$, units of $E$ (${\rm V/m}$), $D$ (${\rm C/m^2}$).
   2. Estimate: how $C$ changes when $A$ doubles or $d$ halves (plate model).
2. Core concepts & derivations (60 min):
   1. From fields to $C$ (plate capacitor): $U=\int\vec{E}\cdot{\rm d}\vec{s}=E\,d$, $Q=\oint\vec{D}\cdot{\rm d}\vec{A}=D\,A$, $\Rightarrow C=\varepsilon_0\varepsilon_{\rm r}\dfrac{A}{d}$. Dimensional check: $[\varepsilon_0\varepsilon_{\rm r}A/d]={\rm F}$.
   2. Other geometries: coaxial and spherical capacitor formulas; where fields are highest (edge intuition kept qualitative).
3. Practice (20 min):
   1. Mini-calcs: (i) $C$ of given $A,d,\varepsilon_{\rm r}$; (ii) coaxial $C$ per length; (iii) allowable $U$ from $E_0$ and $d$; (iv) energy at given $U$.
   2. Discuss the provided “glass plate in capacitor” task.
4. Wrap-up (2 min): Summary box + pitfalls checklist; connect to next block (capacitor circuits).

• A capacitor can „store“ charges. The total charge of a two-plate capacitor is in general 0.
• From the mechanical point of view a capacitor has two electrodes (= conductive areas), which are separated by a dielectric (= non-conductor).
• In a capacitor an electric field can be established without charge carriers moving internally.
• The characteristic of the capacitor is the capacitance $C$.
• In addition to the capacitance, every capacitor also has resistance and inductance. However, both of these are usually very small and are neglected for an ideal capacitor.
• Examples of capacitor are
  • the electrical component „capacitor“,
  • an open switch,
  • a wire related to ground,
  • a human being

$\rightarrow$ Thus, for any arrangement of two conductors separated by an insulating material, a capacitance can be specified.

The capacitance $C$ can be derived as follows:

1. A plate capacitor has a nearly homogenious field.
   Therefore, it is given for the voltage:
   $$U = \int \vec{E} {\rm d} \vec{s} = E \cdot l$$
   and hence
   $$E= {{U}\over{l}}$$ or with $D = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot E $ $$D= \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot {{U}\over{l}}$$
2. Furthermore, the charge $Q$ can be given as $${Q= \rlap{\Large \rlap{\int_A} \int} \, \LARGE \circ} \; \vec{D} \cdot {\rm d} \vec{A} $$ The idealizion for the plate capacitor leads to: $$Q=D \cdot A$$.
3. Thus, the charge $Q$ is given by: \begin{align*} Q = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot {{U}\over{l}} \cdot A \end{align*}
4. This means that $Q \sim U$, given the geometry (i.e., $A$ and $d$) and the dielectric ($\varepsilon_{ \rm r} $).
5. So it is reasonable to determine a proportionality factor ${{Q}\over{U}}$.

The capacitance $C$ of an idealized plate capacitor is defined as

\begin{align*} \boxed{C = \varepsilon_0 \cdot \varepsilon_{ \rm r} \cdot {{A}\over{l}} = {{Q}\over{U}}} \end{align*}

Some of the main results here are:

• The capacity can be increased by increasing the dielectric constant $\varepsilon_{ \rm r} $, given the the same geometry.
• As near together the plates are as higher the capacity will be.
• As larger the area as higher the capacity will be.

This relationship can be examined in more detail in the following simulation:

Capacitor lab

If the simulation is not displayed optimally, this link can be used.

The Abbildung 1 shows the topology of the electric field inside a plate capacitor.

• The symbol of a general capacitor is given be two parallel lines nearby each other.
  
• Since electrolytic capacitors can only withstand voltage in one direction, the polarisation is often shown by a curved electrode (US) or a unfilled one (EU).
  Be aware that electrolytic capacitors can explode, once used in the wrong direction.

To calculate the capacitance of different designs, the definition equations of $\vec{D}$ and $\vec{E}$ are used. This can be viewed in detail, e.g., in this video.
Based on the geometry, different equations result (see also Abbildung 2).

In Abbildung 3 different designs of capacitors can be seen:

1. rotary variable capacitor (also variable capacitor or trim capacitor).
   1. A variable capacitor consists of two sets of plates: a fixed set and a movable set (stator and rotor). These represent the two electrodes.
   2. The movable set can be rotated radially into the fixed set. This covers a certain area of $A$.
   3. The size of the area is increased by the number of plates. Nevertheless, only small capacities are possible because of the necessary distance.
   4. Air is usually used as the dielectric; occasionally, small plastic or ceramic plates are used to increase the dielectric constant.
2. multilayer capacitor
   1. In the multilayer capacitor, there are again two electrodes. Here, too, the area $A$ (and thus the capacitance $C$) is multiplied by the finger-shaped interlocking.
   2. Ceramic is used here as the dielectric.
   3. The multilayer ceramic capacitor is also referred to as KerKo or MLCC.
   4. The variant shown in (2) is an SMD variant (surface mound device).
3. Disk capacitor
   1. A ceramic is also used as a dielectric for the disk capacitor. This is positioned as a round disc between two electrodes.
   2. Disc capacitors are designed for higher voltages, but have a low capacitance (in the microfarad range).
4. Electrolytic capacitor, in German also referred to as Elko for Elektrolytkondensator
   1. In electrolytic capacitors, the dielectric is an oxide layer formed on the metallic electrode. The second electrode is the liquid or solid electrolyte.
   2. Different metals can be used as the oxidized electrode, e.g., aluminum, tantalum, or niobium.
   3. Because the oxide layer is very thin, a very high capacitance results (depending on the size: up to a few millifarads).
   4. Important for the application is that it is a polarized capacitor. I.e., it may only be operated in one direction with DC voltage. Otherwise, a current can flow through the capacitor, which destroys it and is usually accompanied by an explosive expansion of the electrolyte. To avoid reverse polarity, the negative pole is marked with a dash.
   5. The electrolytic capacitor is built up wrapped, and often has a cross-shaped predetermined breaking point at the top for gas leakage.
5. film capacitor, in German also referred to as Folko, for Folienkondensator.
   1. A material similar to a „chip bag“ is used as an insulator: a plastic film with a thin, metalized layer.
   2. The construction shows a high pulse load capacitance and low internal ohmic losses.
   3. In the event of an electrical breakdown, the foil enables „self-healing“: the metal coating evaporates locally around the breakdown. Thus the short-circuit is canceled again.
   4. With some manufacturers, this type is referred to as MKS (Mmetallized foilccapacitor, Polyester).
6. Supercapacitor (engl. Super-Caps)
   1. As a dielectric is - similar to the electrolytic capacitor - very thin. In the actual sense, there is no dielectric at all.
   2. The charges are not only stored in the electrode, but - similar to a battery - the charges are transferred into the electrolyte. Due to the polarization of the charges, they surround themselves with a thin (atomic) electrolyte layer. The charges then accumulate at the other electrode.
   3. Supercapacitors can achieve very large capacitance values (up to the Kilofarad range), but only have a low maximum voltage

In Abbildung 2 are shown different capacitors:

1. The above two SMD capacitors
   1. On the left a $100~{ \rm µF}$ electrolytic capacitor
   2. On the right a $100~{ \rm nF}$ MLCC in the commonly used Surface-mount_technology 0603 ($1.6~{ \rm mm}$ x $0.8~{ \rm mm}$)
2. below different THT capacitors (Through Hole Technology)
   1. A big electrolytic capacitor with $10~{ \rm mF}$ in blue, the positive terminal is marked with $+$
   2. In the second row is a Kerko with $33~{ \rm pF}$ and two Folkos with $1,5~{ \rm µF}$ each
   3. In the bottom row, you can see a trim capacitor with about $30~{ \rm pF}$ and a tantalum electrolytic capacitor and another electrolytic capacitor

Various conventions have been established for designating the capacitance value of a capacitor various conventions.

• Mixing up geometry symbols. Use $d$ (or $l$) strictly as plate spacing and $A$ as active area in $C=\varepsilon_0\varepsilon_{\rm r}\dfrac{A}{d}$. Check units to catch mistakes.
• Forgetting the field relations. $U=\int \vec{E}\cdot{\rm d}\vec{s}$ and $Q=\oint \vec{D}\cdot{\rm d}\vec{A}$; without them, layered-dielectric problems are guessed instead of solved.
• Assuming conduction through the dielectric. The apparent “current through a capacitor” is displacement-related; no charge carriers traverse the ideal dielectric.
• Real-part issues. Ignoring polarity of electrolytics and tolerance spreads ($\pm 10~\%$ and more) causes design errors; pick suitable component types.