Alternating Current Chapter 3

• Chapter 3 is about alternating current
• Learning to apply average power is a topic
• Looking at power factors in certain AC circuits will be covered
• Phasor diagrams and potential differences are also topics

Alternating Current (AC)

• AC involves electrical current that varies periodically in both direction and magnitude over time
• AC can be supplied by an AC circuit and AC generators
• Sinusoidal output is the output of an AC generator that varies with time

Equations

• The formula for alternating current (I) is I = I₀sin(ωt), where I₀ is peak current, T is period, and ω is angular frequency
• The formula for alternating voltage (V) is V = V₀sin(ωt), where V₀ is peak voltage, T is period, and ω is angular frequency

Terminology

• Frequency (f) refers to the number of complete cycles in one second, measured in hertz (Hz) or s⁻¹
• Period (T) is the time it takes for one complete cycle, measured in seconds (s); Formula: T = 1/f
• Peak current (I₀) means the magnitude of the maximum current, measured in ampere (A)
• Angular frequency (ω) is measured in radian per second (rads⁻¹); Formula: ω = 2πf

Average/Mean Value

• This is the value of AC over half a cycle.
• It is a steady current that transfers the same amount of charge to the circuit during a time interval, as transferred by AC
• Formula: Iav = (2I₀)/π = I₀/(π/2)
• For a complete cycle, the average value of AC or EMF is zero, positive and negative values cancel out
• AC is sinusoidal and has a positive current for the first half rotation, negative for the second
• Both current directions dissipate heat through a resistor
• Mean power dissipated from a resistor is P = I²rmsR or P = V²rms/R

Root Mean Square (RMS)

• RMS current Irms) refers to the effective value of AC that produces the same power as a steady DC current when passed through a resistor
• Irms = √I2ave
• Irms = I₀/√2 ≈ 0.707 I₀ : square root of the average value of the current
• Vrms refers to the steady direct voltage value. Applying it across a resistor produces the same power as alternating voltage across that resistor
• Vrms = √V2ave
• Vrms = V₀/√2 ≈ 0.707 V₀
• For sinusoidal alternating current and voltage:
  • Irms = I₀/√2
  • Vrms = V₀/√2
• The average power equals IrmsVrms
• Peak power, P₀ = I₀V₀
• Ammeters and voltmeters used for AC measure the RMS value
• Household electricity being 240 V AC is a VRMS of 240 V

Phasor Diagrams

• Phasors are defined as vectors that rotate anticlockwise around their axis at a constant angular velocity
• A phasor diagram is a diagram containing a phasor to represent a sinusoidal alternating quantity like current and voltage
• They are used to determine phase differences in AC circuits
• Pure resistor:
  • In such a circuit, current flows I = I₀sin(ωt)
  • The voltage across the resistor VR = V₀sin(ωt)
  • Phase difference is Δ𝜙 = 0
  • Voltage V is in phase with the current I. Both reach their maximums simultaneously
• Pure capacitor:
  • When alternating voltage is applied, the voltage reaches max one quarter cycle behind the current
  • Supply voltage of, VC = V = V₀sin(ωt)
  • In an AC circuit:
    • I = I₀sin(ωt + π/2)
    • 𝛥𝜙 = −π/2
  • Voltage lags behind current, or current leads voltage by radians π/2
• Pure inductor:
  • For sinusoidal voltage, voltage reaches its maximum one quarter cycle before the current
  • Formula is I = I₀sin(ωt)
  • Can get the back EMF using the formula:
    • eB = -L(dI/dt)
    • Where eB =−LI₀ωcosωt
  • Magnitude of V and eB, where V = eB = LI₀ωcosωt or V = LI₀ωsin(ωt + π/2) , where V = V₀sin(ωt + π/2) Vo = LI₀ω
  • The phase difference is 𝛥𝜙 = π/2
  • in a pure inductor, the voltage I leads the current or I legs behind by π/2 radians

Resistance

• Resistance refers to opposition to current flow in purely resistive circuits
• Symbol = R, measured in ohms (Ω)
• R = Vrms/Irms = V₀/I₀

Reactance

• Reactance means opposition to current flow from inductance and capacitance in an AC circuit, measured in ohms (Ω)
• Capacitive reactance: Xc = 1/ωC = 1/(2πfC)
• Inductive reactance: XL = ωL = 2πfL

Impedance

• Impedance (Z) is the total opposition the AC current experiences in a circuit
• Z = Vrms/Irms or V₀/I₀
• Scalar quantity, unit: ohm (Ω)
• Acts similar to the resistance in a circuit

RL Series Circuit

• Such a circuit contains a pure resistance R ohms in series with a coil of pure inductance L
• For the voltage:
• VR = IR
• VL = IXL
• V = √(VR² + VL²)
• V = I√(R² + XL²)
• The impedance is:
  • V/I or
  • √(R²+XL²)
• In such circuit it is shown that voltage leads the current by radians, where:
  • tan = VL/VR or
  • tan / = XL/R

RC Series Circuit

• This includes the pure resistance R ohms connecting in series with a capacitance C units farads
• The current I is equal because connected in series
• Key formulas:
  • VR = IR
  • VC = IXC
  • V = √(VR² + VC²)
  • V = I√(R²+Xc²)
• The current I is such a circuit leads the source voltage byradians where: tan = VC/VR or tan = XC/R

RLC series circuit

• VL = IXL
• VR = IR
• VC = IXC
• Impedance is: sqrt(R^2 + (X_L - X_C)^2)
• Voltage leads by the following in such circuits:

Resonance:

• Resonance is a phenomemon occurring when the applied voltage frequency equals that of the LRC series circuit
• A series resonance circuit is sometimes used for radio receivers for tuning
• Impedance is at minimum

Power

• In AC circuits (RC, RL, RLC), power is only dissipated by resistance, not inductance or capacitance
• Real power (Pr), that is used or gone, equals average power (Pave)
• Pave = I²rms = Pr
• Pave = IrmsVrms

Power Factor

• Power factor = Preal/Papparent
• Power factor is the cosine of the angle between RMS voltage and current Power factor = cos Φ
• Expressed as percentage or a decimal
• A typical circuit is less than 1 or less than 100%
• The 3 cases are:
• If Phi = 0, circuit is purely resistive
• If / > 90, circuit is purely inductive
• If / < 90, circuit is purely capacitive