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Sahithyan's S1
Sahithyan's S1 — Electrical Fundamentals

Power and Power factor

  • In a purely resistive AC circuit, the energy delivered by the source will be dissipated as heat by the resistor.
  • In a purely capacitive or purely inductive circuit, all of the energy will be stored during a half cycle, and then returned to the source during the other – there will be no net conversion to heat.
  • When there is both a resistive component and a reactive component, some energy will be stored, and some will be converted to heat during each cycle.

Power equations

Purely resistive circuit

Suppose a circuit with load RR resistance is supplied a voltage of v(t)=Vmcosωtv(t)=V_m\cos{\omega t}.

Instantaneous power dissipated by the load is given by:

p(t)=Vm2Rcos2(ωt)p(t) = \frac{V_m^2}{R}\cos^2{(\omega t)}

Always: p(t)>0p(t)\gt 0.

Average power=12×Peak power=Vm22R\text{Average power} = \frac{1}{2}\times\text{Peak power}=\cfrac{V_m^2}{2R}

Purely inductive circuit

Suppose a circuit with inductor LL is supplied a voltage of v(t)=Vmcosωtv(t)=V_m\cos{\omega t}.

Instantaneous power dissipated by the load is given by:

p(t)=Vm22ωLsin(2ωt)p(t) = \frac{V_m^2}{2\omega L}\sin{(2\omega t)}

The inductive reactive power is given by:

Q=V2ωL=I2ωLQ = \frac{V^2}{\omega L} = I^2 \omega L

Purely capacitive circuit

Suppose a circuit with capacitor CC is supplied a voltage of v(t)=Vmcosωtv(t)=V_m\cos{\omega t}.

Instantaneous power dissipated by the load is given by:

p(t)=Vm2ωC2sin(2ωt)p(t) = -\frac{V_m^2 \omega C}{2}\sin{(2\omega t)}

The capacitive reactive power is given by:

Q=V2ωC=I2ωLQ = V^2\omega C = \frac{I^2}{\omega L}

General load

Consider a general load with both resistive and reactive components. Depending on how inductive or capacitive the reactive component, the phase shift between voltage and current phasor lies between 90°90° and 90°−90°.

Suppose the circuit is supplied a voltage of v(t)=Vmcos(ωt)v(t) = V_m\cos{(\omega t)}. And the current phasor shifts in θ\theta phase angle.

i(t)=Imcos(ωtθ)i(t) = I_m\cos{(\omega t - \theta)}

This ends up with:

p(t)=12VmIm[cosθ+cos(2ωtθ)]p(t) = \frac{1}{2}V_mI_m\bigg[\cos\theta+\cos (2\omega t - \theta)\bigg]

Average over 1 cycle

Pavg=1Tt0t0+Tp(t)dt=VrmsIrmscosθP_\text{avg} =\frac{1}{T}\int_{t_0}^{t_0 + T} p(t)\,\text{d}t = V_{\text{rms}}I_{\text{rms}}\cos{\theta}

Types of power

Active power

Aka. true power, resistive power. In all electrical and electronic systems, it is the true power (the resistive power) that does the work.

P=VrmsIrmscosθP = V_{\text{rms}}I_{\text{rms}}\cos{\theta}

Reactive Power

Power delivered to/from a pure energy storage element (inductors and capacitors) is known as reactive power.

  • Average power consumed by a pure energy storage element is 00.
  • Current associated with it is not 00. Transmission lines, transformers, fuses, etc. must all be designed to withstand this current.
  • Loads with energy storage elements will draw large currents and require heavy duty wiring even though little average power is consumed.
  • Shuttles back and forth between the source and the load.
Qreactive=VrmsIrmssinθQ_\text{reactive} = V_\text{rms}I_\text{rms}\sin\theta

Apparent power

Combination of active and reactive power.

S=VrmsIrms=P2+Q2S = V_\text{rms}I_\text{rms} = \sqrt{P^2 + Q^2}

The apparent power is essentially the effective power that the source “sees”.

The Beer Analogy

  • Beer - Active power
    Liquid beer is useful power. The power that does the work.
  • Foam - Reactive power
    Wasted or lost power.
  • Mug - Apparent power
    Demand power, that is being delivered by the utility.

Power factor

If θ\theta is the phase angle difference between vv and ii, cos(θ)\cos(\theta) is called the power factor. Higher power factor indicates a more efficient use of electrical power.

Power factor appears in the equation of PavgP_\text{avg}.

cosθ=Active powerApparent power=ResistanceImpedance\cos{\theta} =\frac{\text{Active power}}{\text{Apparent power}} =\frac{\text{Resistance}}{\text{Impedance}}

Power factor is:

  • leading when II leads VV
  • lagging when II lags VV

Power triangle

A right triangle that visually represents the relationship between active, reactive and apparent power in an AC circuit.

They are represented as below:

  • Active Power (P): On the horizontal axis
  • Reactive Power (Q): On the vertical axis
  • Apparent Power (S): On the hypotenuse of the triangle
S2=P2+Q2S^2 = P^2 + Q^2

The angle θ between the active power and the apparent power represents the phase angle, and the cosine of this angle is the power factor:

cosθ=PS\cos{\theta} = \frac{P}{S}