Imagine two invented AC input records. Both have an effective voltage of 100 volts and an effective current of 10 amps, yet one transfers energy at an average rate of 1,000 watts and the other at 800 watts. Even the second record's main current wave is perfectly aligned with the voltage. The missing piece is the shape of the complete current wave. Power factor compares average real power with the product of RMS voltage and RMS current; an aligned main wave alone does not determine that ratio when other current harmonics are present [2]. We will explain the unfamiliar terms and calculate exactly where the two answers come from.

This matters at the AC input of an electric car's onboard charger, which converts incoming AC electricity into DC electricity for charging the traction battery [1]. Texas Instruments' July 2017 engineering account describes a power factor correction stage, or PFC, before a further conversion stage; its ideal input-current objective is a sine wave aligned with a sinusoidal supply [3]. That is a dated engineering example, not a claim about every vehicle's design today. Our records below are hypothetical mathematical waveforms. They are not charger measurements, electrical settings or instructions for working on a circuit. The question is how to read the power quantities without confusing them.

Equal RMS voltage and RMS current fix apparent power S=V_rms I_rms, but they do not fix real power P, the mean instantaneous voltage-current product. Total power factor is PF=P/S [2][4]. An original single-phase example will hold voltage at a pure sine with V_rms=100 V and hold total I_rms=10 A. Different current functions then produce P=1,000 W or 800 W, even though the lower-power case has an in-phase current fundamental. This is a distinction between waveform magnitude, harmonic composition and correlation with voltage. It cannot be resolved by assigning one phase angle to the entire distorted current waveform.

The relevant application boundary is the AC input of the onboard charger. AFDC describes the onboard unit's conversion of charge-port AC into DC for the traction battery [1]. TI's July 2017 account illustrates a PFC input stage followed by isolated DC/DC conversion and describes an ideal in-phase sinusoidal input current [3]. Neither the dated topology nor its product figures are universal present-day specifications. The calculations here use invented periodic functions, a common fundamental period and declared averaging boundaries. No vehicle, converter or instrument was tested, and no circuit implementation is proposed. DC electrical output will remain distinct from an increase in usable stored battery energy.

RMS stands for root mean square. For a changing current, square its values, average those squares over the chosen interval, then take the square root. Voltage is treated in the same way [4]. Squaring means that positive and negative parts do not simply cancel. RMS therefore gives a useful magnitude, but it does not preserve every detail of when the current is large or small. Two differently shaped waves can share that magnitude. A plain average of an alternating current can be zero while its RMS value is substantial, so those two kinds of average must not be exchanged.

Real power asks a different question: at each moment, multiply the voltage by the current, then average those products over a complete repeating cycle [4]. Here we call that result P and measure it in watts. Multiplying the separate RMS magnitudes instead gives apparent power, S, in volt-amperes, written VA [2]. Our two records both have S=100×10=1,000 VA. That statement contains no answer yet about whether P is 1,000 W, 800 W or another value. The same two magnitude summaries can conceal different joint patterns. We need the waves, or equivalent supported power information, to finish the calculation.

Let the averaging brackets mean a mean over one complete fundamental period T. Define V_rms=√⟨v²⟩, I_rms=√⟨i²⟩ and P=⟨vi⟩. OpenStax gives the instantaneous product, complete-cycle integral and RMS definitions; its native equations preserve the square roots that a flattened text extraction can lose [4]. In our finite periodic examples, the same T is used for every quantity. RMS is a norm-like magnitude of each waveform separately, whereas real power depends on their joint product. In particular, ⟨vi⟩ is not generally ⟨v⟩⟨i⟩, and neither expression should be silently substituted for V_rms I_rms.

Write θ=2πt/T. For v(θ)=100√2 sin θ V, the cycle mean of sin²θ is 1/2, so V_rms²=20,000×1/2=10,000 V² and V_rms=100 V. The factor √2 converts the chosen RMS coefficient into a pure sine's peak coefficient; it is not a generic peak-to-RMS rule for every shape [4]. Every current in the initial comparison will have I_rms²=100 A². Consequently S=1,000 VA throughout that comparison. Determining P still requires the current function, and total PF follows only after that mean product is known. The definitions already show why a pair of RMS values leaves information missing.

Multiply A's current by the same voltage throughout the cycle and its average input power is 1,000 W. For B, the aligned 8 A fundamental contributes 100×8=800 W. The third-harmonic part contributes zero to the complete-cycle average of the voltage-current product. Its contribution is positive during some parts of the cycle and negative during others, with equal totals. This cancellation concerns that component's contribution, not a claim that B's whole input power must become negative. The averaging definition comes from the source; these two results follow from our explicitly chosen waves [4].

Two snapshots help expose the different timing. One twelfth of the way through the cycle, A's instantaneous power is 500 W while B's is 1,000 W. One quarter of the way through, A's is 2,000 W while B's is 400 W. These are momentary calculated values, not the cycle averages. B can exceed A at one point and still average less across the complete cycle. Nor has 200 W of incoming energy disappeared: B's average real input was 800 W from the outset. Its 1,000 VA label was never a separate 1,000 W delivery that must be split into output and waste.

For A, p_A=2,000 sin²θ W, hence P_A=1,000 W. For B, p_B=1,600 sin²θ+1,200 sin θ sin 3θ W. The first term averages to 800 W and the second to zero, yielding P_B=800 W. The same third-harmonic coefficient that contributed 36 A² to current mean square contributes zero mean power against this voltage because the voltage has no third harmonic. These are different products and different units. Whole-period orthogonality therefore reconciles the two results without any contradiction in energy accounting. The source supplies p=vi and cycle averaging; the harmonic extension is the displayed original derivation [4].

At θ=π/6, the exact instantaneous powers are p_A=500 W and p_B=1,000 W. At θ=π/2, they are p_A=2,000 W and p_B=400 W. A few samples alone do not establish the average; they illustrate why timing matters after the complete integrals have been evaluated. The component result ⟨v i_3⟩=0 also does not imply i_3=0, a zero instantaneous product, or a universal absence of hardware losses. No resistance or converter-loss model has yet been assigned. Subtracting P_B from S would mix two differently defined quantities and would not construct a dissipation term.

The displacement factor describes the alignment of the current's fundamental with the voltage in this sine-voltage model. It is 1 for B because those two component waves line up. The distortion factor asks what fraction of B's total RMS current belongs to that fundamental: 8÷10=0.8. Multiplying the two factors gives the total power factor, 1×0.8=0.8, agreeing with 800 W divided by 1,000 VA. TI distinguishes distortion and displacement in its power-factor expression [2]. It would be misleading to describe the whole distorted current as having one simple phase angle and use that angle alone.

Another way to express B's distortion is current total harmonic distortion, abbreviated THD. For this example, divide the RMS magnitude of all the higher harmonics by the fundamental's RMS magnitude. There is just one higher harmonic, so THD=6÷8=0.75, or 75%. That percentage is relative to the 8 A fundamental, not the 10 A total. The corresponding distortion factor is 1 divided by the square root of 1+0.75², which equals 0.8. We are using a pure sine voltage, a nonzero current fundamental and no constant DC current component. Those conditions make the factorization meaningful; they should travel with the result [2][4].

For the stipulated pure fundamental voltage, let I_1 be fundamental current RMS and φ_1 its displacement from voltage. Orthogonality gives P=V_rms I_1 cos φ_1. Dividing by S yields PF=(I_1/I_rms)cos φ_1. The first factor is the distortion factor in this setting; the second is fundamental displacement factor. For B, I_1=8 A, I_rms=10 A and cos φ_1=1, so PF=0.8. TI's native equation 2 presents the related THD expression, while the sinusoidal mean-power relation is explicit in OpenStax [2][4]. Our derivation states the voltage and averaging assumptions rather than applying the expression indiscriminately.

With finite integer harmonics, no DC term and I_1>0, define THD_I=√(Σ I_h²)/I_1 for harmonics h≥2. The squared-current sum gives I_rms²=I_1²(1+THD_I²), so PF=cos φ_1/√(1+THD_I²). B has THD_I=6/8=3/4 and distortion factor 4/5. The denominator convention matters: 6/10 would be a different ratio. A zero fundamental would make this THD definition undefined, and a voltage containing harmonics would change the mean-power calculation. A single total PF also cannot reveal which factor caused it. These are boundaries of the defined calculation, not evidence about any particular charger's spectrum.

So far we held total current equal and compared the resulting power. We can ask the reverse question: what happens if B's entire current wave is scaled until its real input power matches A's 1,000 W? Multiplying B by 1,000÷800=1.25 does that. The 8 A fundamental becomes 10 A, the 6 A third harmonic becomes 7.5 A, and the total RMS current becomes 12.5 A. The shape ratios stay the same, so the power factor stays at 0.8. This is a new comparison condition: equal real power now requires unequal current for these two waveform families.

Suppose the mathematical exercise separately imposes a 10 A RMS ceiling. A can deliver 1,000 W within that invented constraint, while an unchanged B-shaped waveform delivers at most 800 W at the same ceiling. Scaling B to 1,000 W would fall outside the permitted model. This follows from the definition P=V_rms I_rms PF [2]; it is not a recommendation to raise a charging setting. A real current rating, supported operating state and waveform would require their own documentation. The useful distinction is between holding current constant and holding power constant. Statements about 'more current' need to say which of those comparisons they mean.

For a fixed voltage function and positive scalar k, scaling the current by k scales I_rms and P by k, while leaving PF, THD_I and fundamental displacement unchanged. Apply k=5/4 to B: i_B,scaled=√2(10 sin θ+7.5 sin 3θ) A. Its squared RMS is 100+56.25=156.25 A², so I_rms=12.5 A, S=1,250 VA and P=1,000 W. The total PF remains 1,000/1,250=0.8. These results follow independently from the mean-product and mean-square definitions [2][4], rather than by interpreting an apparent-power difference as lost real power.

Under a separately stipulated I_rms≤10 A constraint and fixed 100 V sine supply, family A reaches 1,000 W at the boundary; the same B shape reaches 800 W there. The scaled B record is outside that feasible set. This is conditional algebra, not a circuit rating, thermal assessment or guarantee about a charger's controller. If waveform shape, voltage or operating conditions change with load, the fixed-PF scaling premise may cease to apply. The comparison therefore retains the waveform family as an input. It demonstrates precisely why equal-current and equal-real-power comparisons answer different questions, without importing an unstated hardware model.

Return to the original B record: 100 V RMS, 10 A RMS, 1,000 VA apparent power and 800 W real input power. Now add a separate invented assumption: a converter delivers 90% of that real input power as mean DC electrical output. Its output is 0.90×800=720 W. If its stored energy is unchanged over the averaging interval, the remaining 80 W belongs in this model's conversion-loss account. The power factor of 0.8 did not supply the 90% assumption. The two ratios have different jobs: one describes the AC input waveforms, and the other connects real input with electrical output.

An onboard charger sits between the incoming AC and the DC supplied for traction-battery charging [1][3]. That physical boundary is why input power factor and conversion efficiency must stay separate. Calling the difference between 1,000 VA and 800 W '200 watts of heat' would invent a loss that the waveform calculation never established. Calling 720 W an increase in usable battery energy would skip another boundary as well. We have calculated mean DC electrical output, not battery chemistry or a vehicle's usable-energy display. A clearer record keeps the AC waveform quantities, converter output and any later storage account in separately named columns.

Introduce an independent conversion model with η=P_dc/P_ac=0.90 and P_ac=P_B=800 W. Here P_dc denotes mean DC electrical output, and the averaging interval is assumed to have no net change in stored converter energy. The stipulated balance gives P_dc=720 W and P_loss=P_ac−P_dc=80 W. None of these values is inferred from PF_B=0.8. The latter ratio uses P_ac and V_rms I_rms at the same AC boundary; η compares real powers across two boundaries. The onboard AC/DC role is documented, while the efficiency value and balance are explicitly hypothetical [1][3].

Apparent power S=1,000 VA is a product of RMS magnitudes, so S−P_ac is not an energy-conservation residual. Treating it as a watt loss would assign the wrong meaning to the operands. The mean DC output also does not establish the rate of increase of usable stored battery energy without a further storage and auxiliary-load model. The earlier vehicle-inverter and charging-schedule topics discuss their own energy boundaries; the new point here is that harmonic current can change total PF before any such conversion model is introduced. Preserving the boundaries prevents waveform distortion from being misreported as a measured conversion-efficiency loss.