Capacitance and Inductance Calculations

• This presentation covers capacitance and inductance calculations.

Course Administration

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• Quiz 3 is available on Friday 23/4.
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• There are reading period meetings for F and Pi after lunch today and N on Monday next week.
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Capacitor and Capacitance

• A capacitor consists of two metal bodies, "a" with a charge of +q and "b" with a charge of -q.
• An electric field E exists between the bodies, with the unit V/m.
• The voltage v equals V𝑎 − V𝑏 = ∫ab E ⋅ dl𝑟 between the bodies with the unit V.
• Capacitance is defined as the ratio between charge and voltage.
• The capacitance (C=q/v=q/(Va-Vb)) is measured in Farads and depends on the geometry of the metal bodies.

Capacitance in Circuits

• Charge q = Cv and current i
• Current is equal to the derivative of the charge over time
• The time derivative of q equals i, which equals the capacity times the time derivative of the voltage, this gives v
• v = (1/C) ∫ i dt

Stored Energy in a Capacitor

• w = ∫ p dt = ∫ vi dt = ∫ vC*(dv/ dt)* dt = C ∫ (d /dt)(v^2/2) dt = (1/2)Cv^2
• The energy is typically assumed as at v(−∞) = 0 and is stored in the electric field E.

Properties Of Capacitance

• With time-independent current and voltage (DC), dv/dt = 0 ⇒ i = 0. This is an open circuit for DC (direct current.)
• Continuous voltage (finite derivative) but current can be discontinuous.
• Alternating current v(t) = Vo sin(wt), i(t) = wCVo cos(wt) = wCVo sin(wt + π/2)
• Current and voltage are π/2 = 90° out of phase.
• With voltage in the form v(t) = Voest ⇒ i(t) = sCVoest = sCv(t) ⇒ v = (1/sC)i.
• This is the same form as Ohm's law with impedance 1/(sC).

Parallel and Series Connection

• Parallel connection: i = i1 + i2 + i3 = C1(dv/dt) + C2(dv/dt) + C3(dv/dt) = (C1 + C2 + C3)(dv/dt)
• i = i1 + i2 + i3 = C1(dv/dt) + C2(dv/dt) + C3(dv/dt) = Ceq(dv/dt)
• Series connection: Vab = v1 + v2 + v3 = (1/C1) ∫ idt + (1/C2) ∫ idt + (1/C3) ∫ idt = ((1/C1) + (1/C2) + (1/C3)) ∫ idt
• Vab = v1 + v2 + v3 = ((1/C1) + (1/C2) + (1/C3)) ∫ idt = (1/Ceq) ∫ idt

Flat Capacitor

• Consists of two flat metal surfaces with area A at a distance d from each other.
• There is a field image (E) in the cross-section with a positive charge on the lower plate and a negative one on the upper plate.
• There is a strong, approximately homogeneous field between the plates and a weak field outside.
• E-field from a large flat metal surface with charge q and surface A with E=q/(2ε0 A) directed out from the metal
• With charges ±q on two parallel metal surfaces with distance d.
• Spänningen fås från E-fältet :Va-Vb = ∫b r a E ∙ d𝑙𝑟= ∫0 d qt 𝜀0A 𝑒z ∙ 𝑒z dz = 𝜀0A / qd

Flat Capacitor Approximation

• Capacitance (large plates, small distance): C = 𝑞/Va - Vb = 𝜀0𝐴/d
• For this, there is a large area A and a small distance d for a large capacitance

Coaxial Cable Capacitance

• Coaxial cables consist of two long cores and therefore have capacitance/ length.
• The capacity per unit length needs to be determined on cylindrical Coaxial cables with internal core radius a and Radius outer core from b
• Electrical field E(r) will move in radial direction erċ
• Dividing the area between the cores to radius in the scale of an cylinder with infinitesimal Thickness in drċ
• The scales capacitance will be given from the plate-capacitor approach in Invers d(C-1)
• Capacity is series connected, thus the total capacitance given by integrale C-1.

Electricity Materials Properties

• With applied electric field, the centre of gravity will be moved for the positive and negative charge in the Atom and these atoms will be polarisated.
• Atoms has a dipole momentum 𝑝=qd
• Polarization will be define as P = 1/Volume Σ p in a volume.

Dielectric Material

• If the area is isolated, decreases total loading by each plate, Means the electric field and Voltage decreases.
• If the plates are plugged in with current sources, then there are additional loading from current sources to compensated for the connection.
• The usual material are liner, whereas you can assume: P= ε0 χ E with susceptible 𝜒 and this are wireless -> this has electrical flow tightness.
• where εr is the relative electric constant
• Previous bills for E (electric field) will translated to materials throughputs 80 -->

Electric Material

• The most solid materials are in scale of 2-5, however there are large variants

Electric Material in Flat Capacitor

• Bestäm capacity in a flat cap as there are 2 flat metal area
• And filled with dielectrica Disks and there are 2 disks with thickness
• We calculate it as capacitor from one serial connected capacitance with each share.

Joined Capacitance and Resistance

• There are always a material as perfect isolator, means =0 to find the electricity throughput the most electric material has to be counted with the electricity and relevant electricity constant.
• a typical example of with the most amount never as efficient as an capacitor with bar capacity, also needs to be paired with Pararell Resistances.

Magnet Field and Inductance

• Very touching strength (from the antique Greek observation in magnetistic phenomenas).
• Magnetic field of soil
• (Ferro) Magnets Material
• Magnet camera
• Able to guide electric loading with particles (particle accelerator).

Inductance

• Inductors in circuits: magnetic flow .Obs reference direction!
• Magnetic Flow = L*i -> V=L di/dt.

Electricity constant for inductions

• Time independent current and Voltage (DC) di/dt = 0 v=0.