Power and Energy Devices and Systems (PE)

Question Set 1: Energy Conversion and Reference Frame Theory

August 2016

# Solution

## Trigonometric Identities

• Single-phase
• $\sin\left(\frac{\pi}{2} - x\right) = \cos x$
• $\sin\left(x + \pi\right) = -\sin x$
• $\cos\left(x + \pi\right) = -\cos x$
• $\cos\left(a \pm b\right) = \cos a \cos b \mp \sin a \sin b$
• $\sin\left(a \pm b\right) = \sin a \cos b \pm \cos a \sin b$
• $\cos\theta + \cos\phi = 2 \cos\left(\frac{\theta + \phi}{2}\right) \cos\left(\frac{\theta - \phi}{2}\right)$
• $\cos\theta - \cos\phi = -2 \sin\left(\frac{\theta + \phi}{2}\right) \sin\left(\frac{\theta - \phi}{2}\right)$
• $\sin\theta - \sin\phi = 2 \cos\left(\frac{\theta + \phi}{2}\right) \sin\left(\frac{\theta - \phi}{2}\right)$
• $2 \sin a \cos b = \sin(a + b) + \sin(a - b)$
• $2 \cos a \cos b = \cos(a + b) + \cos(a - b)$
• $-2 \sin a \sin b = \cos(a + b) - \cos(a - b)$
• $2 \cos^2 x = \cos 2x + 1$
• $2 \sin^2 x = 1 - \cos 2x$
• $\int_{0}^{2\pi} \cos^2 x \, dx = \int_{0}^{2\pi} \sin^2 x \, dx = \pi$
• Three-phase
• $\cos x + \cos\left(x - \frac{2\pi}{3}\right) + \cos\left(x + \frac{2\pi}{3}\right) = 0$
• $\sin x + \sin\left(x - \frac{2\pi}{3}\right) + \sin\left(x + \frac{2\pi}{3}\right) = 0$
• $\cos^2 x + \cos^2\left(x - \frac{2\pi}{3}\right) + \cos^2\left(x + \frac{2\pi}{3}\right) = \frac{3}{2}$
• $\sin^2 x + \sin^2\left(x - \frac{2\pi}{3}\right) + \sin^2\left(x + \frac{2\pi}{3}\right) = \frac{3}{2}$
• $\sin x \cos x + \sin\left(x - \frac{2\pi}{3}\right) \cos\left(x - \frac{2\pi}{3}\right) + \sin\left(x + \frac{2\pi}{3}\right) \cos\left(x + \frac{2\pi}{3}\right) = 0$
• $\sin x \cos y + \sin\left(x - \frac{2\pi}{3}\right) \cos\left(y - \frac{2\pi}{3}\right) + \sin\left(x + \frac{2\pi}{3}\right) \cos\left(y + \frac{2\pi}{3}\right) = \frac{3}{2} \sin(x - y)$
• $\sin x \sin y + \sin\left(x - \frac{2\pi}{3}\right) \sin\left(y - \frac{2\pi}{3}\right) + \sin\left(x + \frac{2\pi}{3}\right) \sin\left(y + \frac{2\pi}{3}\right) = \frac{3}{2} \cos(x - y)$
• $\cos x \cos y + \cos\left(x - \frac{2\pi}{3}\right) \cos\left(y - \frac{2\pi}{3}\right) + \cos\left(x + \frac{2\pi}{3}\right) \cos\left(y + \frac{2\pi}{3}\right) = \frac{3}{2} \cos(x - y)$
• $\sin x \cos y + \sin\left(x + \frac{2\pi}{3}\right) \cos\left(y - \frac{2\pi}{3}\right) + \sin\left(x - \frac{2\pi}{3}\right) \cos\left(y + \frac{2\pi}{3}\right) = \frac{3}{2} \sin(x + y)$
• $\sin x \sin y + \sin\left(x + \frac{2\pi}{3}\right) \sin\left(y - \frac{2\pi}{3}\right) + \sin\left(x - \frac{2\pi}{3}\right) \sin\left(y + \frac{2\pi}{3}\right) = -\frac{3}{2} \cos(x + y)$
• $\cos x \cos y + \cos\left(x + \frac{2\pi}{3}\right) \cos\left(y - \frac{2\pi}{3}\right) + \cos\left(x - \frac{2\pi}{3}\right) \cos\left(y + \frac{2\pi}{3}\right) = \frac{3}{2} \cos(x + y)$

## Alumni Liaison

has a message for current ECE438 students.

Sean Hu, ECE PhD 2009