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EE 100 Notes
Fundamentals of EE, Rizzoni Paul Beliveau, October, 2010
file: phase and rms.txt
Phase Angles, Lag and Lead
βˆ™ Phase angles are typically expressed in degrees, not radians, but Rizzoni uses radians.
2πœ‹ = 360∘
πœ‹ = 180∘
πœ‹/2 = 90∘
βˆ™ a positive (leading) phase angle can be expressed as a negative (lagging) phase angle
by subtracting 360∘ or 2πœ‹ π‘Ÿπ‘Žπ‘‘π‘–π‘Žπ‘›π‘ 
βˆ™ a negative (lagging) phase angle can be expressed as a positive (leading) phase angle
by adding 360∘ or 2πœ‹ π‘Ÿπ‘Žπ‘‘π‘–π‘Žπ‘›π‘ 
βˆ™ a phase angle greater than 360∘ can be reduced by multiples of 360∘ until it is less than
360∘
βˆ™ for the two waveforms π‘₯1 (𝑑) = 𝑋𝑀1 π‘π‘œπ‘ (πœ”π‘‘ + πœƒ) and π‘₯2 (𝑑) = 𝑋𝑀2 π‘π‘œπ‘ (πœ”π‘‘ + πœ™)
– if πœƒ = πœ™, the waveforms are in phase
– if πœƒ βˆ•= πœ™, the waveforms are out of phase
πœ‹
πœ‹
); sine lags cosine by 90∘ or π‘Ÿπ‘Žπ‘‘π‘–π‘Žπ‘›π‘ 
2
2
πœ‹
πœ‹
βˆ™ π‘π‘œπ‘  πœ”π‘‘ = 𝑠𝑖𝑛(πœ”π‘‘ + 90∘ ) = 𝑠𝑖𝑛(πœ”π‘‘ + ); cosine leads sine by 90∘ or π‘Ÿπ‘Žπ‘‘π‘–π‘Žπ‘›π‘ 
2
2
βˆ™ 𝑠𝑖𝑛 πœ”π‘‘ = π‘π‘œπ‘ (πœ”π‘‘ βˆ’ 90∘ ) = π‘π‘œπ‘ (πœ”π‘‘ βˆ’
βˆ™ note that the sign of the amplitude affects the phase angle by 180∘ or πœ‹ π‘Ÿπ‘Žπ‘‘π‘–π‘Žπ‘›π‘ 
βˆ™ βˆ’π΄ 𝑠𝑖𝑛(πœ”π‘‘ + πœƒ) = 𝐴 𝑠𝑖𝑛(πœ”π‘‘ + πœƒ ± 180∘ )
βˆ™ βˆ’π΄ π‘π‘œπ‘ (πœ”π‘‘ + πœƒ) = 𝐴 π‘π‘œπ‘ (πœ”π‘‘ + πœƒ ± 180∘ )
1
Average and RMS Values
We can look at the average, or DC value, of a signal.
∫
1 𝑑0 +𝑇
⟨π‘₯(𝑑)⟩ =
π‘₯(𝑑) 𝑑𝑑
average value
𝑇 𝑑0
More useful for power calculations is the root-mean-square (rms) value.
√ ∫
1 𝑑0 +𝑇 2
π‘₯ (𝑑) 𝑑𝑑
rms value
π‘₯π‘Ÿπ‘šπ‘  =
𝑇 𝑑0
(See example 4.9, Rizzoni, p. 149.)
The rms value of a sinusoid with a maximum value of 𝑋𝑀 is given by
𝑋𝑀
rms value of a sinusoid = √ = 0.707𝑋𝑀
2
The factor of 0.707 for sinusoids is useful to remember, but in general the rms value for a
signal will have a different multiplier of the peak value.
Why use rms?
The effective value of a periodic current is the constant, or DC value, which delivers the
same average power to a resistor R.
2
𝑃 = 𝐼𝑒𝑓
𝑓𝑅
The average power delivered to a resistor by a periodic current 𝑖(𝑑) is
∫
1 𝑑0 +𝑇 2
𝑖 (𝑑)𝑅 𝑑𝑑
𝑃 =
𝑇 𝑑0
Equating the two expressions
√
𝐼𝑒𝑓 𝑓 =
1
𝑇
∫
𝑑0 +𝑇
𝑖2 (𝑑) 𝑑𝑑
𝑑0
Therefore 𝐼𝑒𝑓 𝑓 is the rms value of the periodic current.
We can calculate the rms value of a voltage signal in the same way.
The power absorbed by a resistor R is
2
𝑃 = πΌπ‘Ÿπ‘šπ‘ 
𝑅=
2
2
π‘‰π‘Ÿπ‘šπ‘ 
π‘