Commercial electricity is generated and transmitted as three-phase electricity. Instead of a single emf ε = ε₀ cos ωt, three separate wires carry currents for the emfs ε₁ = ε₀ cos ωt, ε₂ = ε₀ cos(ωt+120°), and ε₃ = ε₀ cos(ωt−120°). This is why the long-distance transmission lines you see in the countryside have three parallel wires, as do many distribution lines within a city. Show that the potential difference between any two of the phases has the rms value 3–√ ε_rms, where ε_rms is the familiar single-phase rms voltage. Evaluate this potential difference for ε_rms = 120 V. Some high-power home appliances, especially electric clothes dryers and hot-water heaters, are designed to operate between two of the phases rather than between one phase and neutral. Heavy-duty industrial motors are designed to operate from all three phases, but full three-phase power is rare in residential or office use.

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with finding the root mean square (rms) value of the potential difference between any two phases in a three-phase system. The emfs of the three phases are given as ε1 = ε0 cos(ωt), ε2 = ε0 cos(ωt + 120°), and ε3 = ε0 cos(ωt − 120°). The goal is to show that the rms value of the potential difference between any two phases is √3 εrms, where εrms is the single-phase rms voltage.
2. Step 2: Write the expression for the potential difference between two phases. For example, the potential difference between phase 1 and phase 2 is ΔV12 = ε1 − ε2. Substituting the given expressions for ε1 and ε2, we have: ΔV12 = ε0 cos(ωt) − ε0 cos(ωt + 120°).
3. Step 3: Simplify the expression for ΔV12 using trigonometric identities. Use the identity cos(A) − cos(B) = −2 sin((A + B)/2) sin((A − B)/2). Applying this to ΔV12, we get: ΔV12 = −2ε0 sin((ωt + ωt + 120°)/2) sin((ωt − (ωt + 120°))/2). Simplify further to get: ΔV12 = −2ε0 sin(ωt + 60°) sin(−60°).
4. Step 4: Calculate the rms value of ΔV12. The rms value of a sinusoidal function is given by dividing the amplitude by √2. The amplitude of ΔV12 is 2ε0 sin(60°), where sin(60°) = √3/2. Thus, the amplitude becomes √3ε0. The rms value of ΔV12 is therefore (√3ε0)/√2 = √3 εrms, where εrms = ε0/√2.
5. Step 5: Evaluate the potential difference for εrms = 120 V. Substitute εrms = 120 V into the expression for the rms potential difference: ΔVrms = √3 εrms. This gives ΔVrms = √3 × 120 V. Perform the numerical calculation to find the final value if needed.

Key Concepts

Three-Phase Electricity

Three-phase electricity is a method of alternating current (AC) power generation and transmission that uses three separate conductors, each carrying an alternating current that is offset in time by one-third of a cycle (120 degrees). This configuration allows for a more efficient and stable power supply, as the combined power output is smoother and can deliver more energy than single-phase systems. It is commonly used in industrial applications and for long-distance power transmission.

RMS Voltage

RMS (Root Mean Square) voltage is a statistical measure of the magnitude of a varying voltage or current. It represents the equivalent direct current (DC) value that would deliver the same power to a load. For a sinusoidal waveform, the RMS value is calculated as the peak voltage divided by the square root of two, providing a useful way to compare AC voltages to DC voltages in practical applications.

Potential Difference in Three-Phase Systems

In a three-phase system, the potential difference between any two phases can be derived from the individual phase voltages. The relationship involves the use of vector addition of the phase voltages, leading to a potential difference that is √3 times the RMS voltage of a single phase. This characteristic is crucial for understanding how appliances and motors operate in three-phase systems, as it affects their efficiency and power requirements.