Ch. 7: Capacitors and Inductors Flashcards
Capacitors:
Applications of Capacitors
Block DC current while passing AC
Shift Phase
Store Energy
Suppress Noise
Start Motors
Capacitors:
Properties of
Ideal Capacitors
- When no voltage changing(DC), current = 0
- No current will flow with DC
- (Current will flow for a short time upon voltage changes, which cause charging/discharging)
- Does not dissipate energy, stored energy can be retrieved later
Capacitors:
Difference between
Real and Ideal Capacitors
Ideal capacitors have exactly 0 current flow with static voltage.
Real capacitors have small leakage current.
This can be represented with a parallel leakage model, which places a very high resistance (>100 MΩ) in parallel with the capacitor.
Can typically be ignored.
Capacitors:
Charge within a Capacitor
(q)
q = Cv
where C = capacitance in Farads(F)
Capacitors:
Energy stored
in a Capacitor (w)
ωc = (1/2)Cv2
where C=capacitance in Farads
Energy ωc is measured in Joules (J)
Capacitors:
Capacitor Current-Voltage Relationship
(Differential equation)
The current through a capacitor is proportional to the change in voltage:
i = C dv/dt
or
v(t)= 1/C ∫i(τ)dτ + v(t0)
Capacitors:
Instantaneous Power
delivered to a capacitor
(Differential Equation0
Starting with
p = i*v
Substitute for the current of the capacitor, i:
p = v* Cdv/dt
Capacitors:
Unit of Capacitance
Farad - F
1 F = 1 Coulomb/Volt = 1 amp-sec/volt
Capacitors:
Capacitance of a
Capacitor
Measure how capacitive it is, based on geometry and electrical Permittivity
C = 𝜀A/d
- A - Area of parallel plates
- d - distance between plates
- 𝜀 - Electrical Permittivity of insulating material
Free Space
Electrical Permittivity
𝜀0
𝜀0
= 8.854 pF/m
Capacitors:
Equivalent Capacitance
of
Capacitors in Series
(Similar to resistors in parallel)
The inverse of the sum of inverses of individual capacitances
Ceq = 1 / ∑ 1/Ci
Capacitors:
Capacitance of
two capacitors in series
Ceq = C1C2 / (C1 + C2)
Capacitors:
Equivalent Capacitance of
Capacitors in Parallel
Simply the sum of capacitances:
Ceq = ∑Ci
or
Ceq = C1 + C2 + … + CN
Inductors:
Properties of Ideal Inductors
- If current is constant, voltage across the inductor is 0. Acts like short in a static DC circuit
- Current through the inductor cannot change instantly
- Inductor does not dissipate energy, it stores it
Inductors:
Difference between
Ideal and Real Inductors
Ideal Inductors ignore winding resistance and capacitance.
Real Inductors
Have resistance and capacitance within the windings. Both can typically be ignored.
Capacitance really only matters at high frequencies.
Resistance can be modeled by a small resistor in series with an ideal inductor
Inductors:
Voltage across an Inductor
(Differential Equation)
v = L di/dt
This voltage OPPOSES current flow, so faster changes in current encounter more impedance
L = inductance, measured in Henries (H)
Inductors:
Unit of Inductance
the Henry (H)
1 H = 1 volt-second/ampere
Inductors:
Current through an inductor
as a function of voltage
I = (1/L) ∫v(𝝉)d𝝉 + i(t0)
Inductors:
Instantaneous Power
of an Inductor
Starting with
p = iv
Substitute in for voltage
p = (L di/dt)*i
Inductors:
Energy Stored
in an Inductor
ωL = (1/2) Li2
Inductors:
Inductance of an Inductor
Inductance is a function of the geometric size of the loops, the size of the wire, number of loops and the magnetic permeability of the core.
L = µN2A/s
- N - Number of turns
- A - Cross sectional Area
- s - Axial length of coil
- µ - Permeability of core
Inductors:
Free Space
Magnetic Permeability
µ0
µ0
= 4π x 10-7 H/m
Inductors:
Equivalent of Inductors
in Parallel
Similar to Resistors in parallel
Inverse of the sum of the inverses of individual Inductors
Leq = 1 / ∑ ( 1/Li )
or
Leq = 1 / [1/L1 + 1/L2 + … + 1/Ln]
Inductors:
Equivalent Inductance of
Inductors in Series
Similar to resistors in series:
Sum of individual inductances
Leq = ∑ Li
or
Leq = L1 + L2 +…+ Ln
