4. Alternating Current

Definitions

• Capacitance is the stored charge between 2 metal plates
• Dielectric is the insulated barrier separating 2 plates in a capacitor
• Fleming’s Left Hand Rule: When a current is passed through a coil in a magnetic field, a force acts on the coil to try and make it turn
• Fleming’s Right Hand Rule: If a conductor is moved through a magnetic field an electric current will be generated
• Amplitude is the height of a waveform
• Peak value of a waveform is the maximum positive or negative value
• Peak to peak value is the difference between the maximum positive and maximum negative value. It is normally twice the peak value.
• RMS (root mean square) is the equivalent D.C. current or voltage that would produce the same heating effect as the A.C. waveform
• Out of phase waveforms have the same frequency but start at different times
• In phase waveforms have the same frequency and start at the same time
• Capacitive reactance is a measure of a capacitor’s opposition to AC
• Inductive reactance is a measure of an inductor’s opposition to AC
• Resonant frequency is the frequency at which the inductive reactance is equal and opposite to capacitive reactance in a series tuned circuit
• Magnification / Q Factor is the ratio of the voltage across the inductor (or capacitor) to the supply voltage at resonance
• Bandwidth of a selectivity curve is the frequency range where the output is at least 0.707 of the maximum value
• Filter is a circuit that will pass some frequencies and reject others

Units of measurement

• Farads is a unit of capacitance
• Joules is the unit of work done
• Degrees is the unit for phase
• Henrys is the unit for inductance

Formula

• E = ½(CV²) where E is energy stored in capacitor
• Q = CV where Q is quantity of electricity, C is capacitance, V is voltage
• T = RC where T is time constant, C is capacitance, R is resistance
• C = C₁ + C₂ + C₃ where C is total capacitance in parallel
• 1/C = 1/C₁ + 1/C₂ + 1/C₃ where C is total capacitance in series
• RMS √2 = RMS x 1.414 = V (peak) where RMS is the RMS / DC equivalent voltage, 1.414 is root 2, V (peak) is the maximum peak voltage
• ω = 2πf where ω is angular rotation
• Xc = 1/ωC = 1/(2πfC) where Xc is capacitive reactive, ω is angular rotation
• Xc = V / I where Xc is the reactive capacitance, V is the DC Voltage, I is the AC
• XL = ωL = 2πfL where XL is inductive reactance
• Z = √(R² + XL²) where Z is impedance, R is resistance, XL is inductive reactance for an RL series circuit
• Z = √(R² + Xc²) where Z is impedance, R is resistance, Xc is capacitive reactance for an RC series circuit
• For an RLC series circuit:
  • V² = (VL - Vc)² + Vr² where V is total voltage, VL is voltage across inductor, Vc is voltage across capacitor, Vr is voltage across resistor
  • V ∝ Z
  • Z² = (XL - Xc)² + R² where Z is total impedance, XL is inductive reactance, Xc is capacitive reactance, R is resistance of the resistor

• 

  or 2πf = 1/√(LC) where fo is resonant frequency, L is inductance, C is capacitance

• Z = L/CR where Z is dynamic impedance in a parallel tuned circuit at resonance, L is inductance of the inductor, C is capacitance of the capacitor, R is resistance of the resistor
• Q = fo / (f2 – f1) where Q is the ratio, fo is resonant frequency, f1 is lower bound, f2 is upper bound

Circuit diagrams

• Capacitor-resistor in series

  

• Inductor-resistor in series

  

• RLC (resistor-inductor-capacitor) circuits:

  • series circuit

    

  • parallel circuit

    

• Parallel tuned circuit

  

Graphs

• Leading and lagging

  

• Vector diagram
  • angular rotation always goes anti-clockwise starting from 3 o’clock

  

• Capacitor current-voltage phasor diagram

  

• Inductor current-voltage phasor diagram

  

• Phasor diagram comparison

  

• Capacitive reactance

  

• Impedance

  • Capacitor-resistor in series circuit

    

  • Inductor-resistor in series circuit

    

• Total voltage Vs in RLC series circuit

  

• Total impedance Z in RLC series circuit

  

• Total current Is in RLC parallel circuit

  

• Resonant frequency

  

• Tuned circuit resonant frequency

  • series circuit

    

  • parallel circuit

    

• Bandwidth in a parallel tuned circuit

  

• response graphs

  

Image credit: Electronic tutorial

Notes

Capacitance

• 2 metals are connected by battery
• Plate “B” is now negatively charged and repels so strongly that the current ceases
• The plates have a very small capacitance for the storage of electricity
• very important to have insulation (dielectric) between the plates that will stand up to the voltage that is to be used

Capacitance of a capacitor is:

• ∝ area of the plates
• 1/∝ distance between the 2 plates
• ∝ permittivity of the dielectric between the plates

Material	Permittivity
Air	1
Dry paper	2.5
Glass	5
Mica	7

Construction of capacitors

1. Paper capacitors
   • “sandwich” of strips of foil and wax impregnated paper
   • 2 foil forms the plates
   • waxed paper is the dielectric
   • Mica capacitor
     • alternate layers of thin metal sheet
     • thin layers of mica as dielectric
   • Silvered mica
     • Silver is sprayed on the sheets of mica to form the plates
     • advantages:
   • possible to make these capacitors very accurately
   • value of the capacitance changes very little with wide temperature changes - E.g. suitable for the tuned circuits in oscillators - Ceramic capacitors - small pieces of ceramic that have a coating of silver on each side - E.g. suitable for de-coupling - disadvantages
   • large capacitance variations with changes of temperature
   • should not be used for tuned circuits - Electrolytic Capacitors - two aluminum foil strips - interleaved with an absorbent paper strip and wound very tightly into a cylinder - one capacitor plate is one of the foil strips and the other plate is the electrolyte - the oxide acting as the insulating dielectric - !! very important to ensure that electrolytic capacitors are connected right way round in any circuit - Variable capacitors (tuning capacitors) - one set of fixed plates and one set of moving plates - dielectric is usually air - When the plates are rotated they overlap, and hence the capacitance, changes

Alternating Current

• Generator
  • has a coil that continues to rotate in the same direction
  • produces electricity that flows in one direction for half a revolution and reverses in the next half cycle
  • current and voltage are in sine-wave

Capacitance

• When a voltage is applied to a capacitor the initial charge current is high at a time when the voltage is small
• In a capacitive circuit:
  • current leads the voltage by 90°
  • voltage lags the current by 90°

Inductance

• In an inductor, voltage leads the current by 90°

RLC Circuit

	Series	Parallel
Voltage across each component	calculate	same
Current across each component	same	calculate
R (resistor)	resistance in phase with voltage	resistance in phase with voltage
response curve resonant frequency	at minimum resistance	at maximum resistance
resonant frequency circuit	acceptor circuit	rejector circuit

Tuned circuits are used in oscillators and radio receivers. They can be used to select one frequency when many are present.

Selectivity

• A high Q circuit has good selectivity of frequencies
• A low Q circuit has poor selectivity of frequencies
• Parallel tuned circuit would typically have a Q of 50

Filters