EE 100 Notes
Fundamentals of EE, Rizzoni Paul Beliveau, October, 2010
file: phase and rms.txt
Phase Angles, Lag and Lead
∙ Phase angles are typically expressed in degrees, not radians, but Rizzoni uses radians.
2𝜋 = 360∘
𝜋 = 180∘
𝜋/2 = 90∘
∙ a positive (leading) phase angle can be expressed as a negative (lagging) phase angle
by subtracting 360∘ or 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
∙ a negative (lagging) phase angle can be expressed as a positive (leading) phase angle
by adding 360∘ or 2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
∙ a phase angle greater than 360∘ can be reduced by multiples of 360∘ until it is less than
360∘
∙ for the two waveforms 𝑥1 (𝑡) = 𝑋𝑀1 𝑐𝑜𝑠(𝜔𝑡 + 𝜃) and 𝑥2 (𝑡) = 𝑋𝑀2 𝑐𝑜𝑠(𝜔𝑡 + 𝜙)
– if 𝜃 = 𝜙, the waveforms are in phase
– if 𝜃 ∕= 𝜙, the waveforms are out of phase
𝜋
𝜋
); sine lags cosine by 90∘ or 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
2
2
𝜋
𝜋
∙ 𝑐𝑜𝑠 𝜔𝑡 = 𝑠𝑖𝑛(𝜔𝑡 + 90∘ ) = 𝑠𝑖𝑛(𝜔𝑡 + ); cosine leads sine by 90∘ or 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
2
2
∙ 𝑠𝑖𝑛 𝜔𝑡 = 𝑐𝑜𝑠(𝜔𝑡 − 90∘ ) = 𝑐𝑜𝑠(𝜔𝑡 −
∙ note that the sign of the amplitude affects the phase angle by 180∘ or 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠
∙ −𝐴 𝑠𝑖𝑛(𝜔𝑡 + 𝜃) = 𝐴 𝑠𝑖𝑛(𝜔𝑡 + 𝜃 ± 180∘ )
∙ −𝐴 𝑐𝑜𝑠(𝜔𝑡 + 𝜃) = 𝐴 𝑐𝑜𝑠(𝜔𝑡 + 𝜃 ± 180∘ )
1
Average and RMS Values
We can look at the average, or DC value, of a signal.
∫
1 𝑡0 +𝑇
⟨𝑥(𝑡)⟩ =
𝑥(𝑡) 𝑑𝑡
average value
𝑇 𝑡0
More useful for power calculations is the root-mean-square (rms) value.
√ ∫
1 𝑡0 +𝑇 2
𝑥 (𝑡) 𝑑𝑡
rms value
𝑥𝑟𝑚𝑠 =
𝑇 𝑡0
(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(𝑡) is
∫
1 𝑡0 +𝑇 2
𝑖 (𝑡)𝑅 𝑑𝑡
𝑃 =
𝑇 𝑡0
Equating the two expressions
√
𝐼𝑒𝑓 𝑓 =
1
𝑇
∫
𝑡0 +𝑇
𝑖2 (𝑡) 𝑑𝑡
𝑡0
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 = 𝐼𝑟𝑚𝑠
𝑅=
2
2
𝑉𝑟𝑚𝑠
𝑅