# Phasors and power

1. Apr 22, 2012

### Centurion

This is not necessarily a HW problem but more me just trying to understand a concept. Why can you multiply phasors together to find Voltage using V=IZ but you can't multiply a current phasor with a voltage phasor to find a power phasor. Instantaneous power is Voltage and Current functions multiplied together, right? So why the seeming discrepancy? Why does it seem to work for some things but not for other things? It's driving me crazy. For example, find the instantaneous power if v(t) = 4cos(pi*t/6) is applied across an impedance Z = 2∠60°. So V = 4∠0° so I = V/Z = 2∠-60°. Just multiplying those phasors together gives me a totally different answer than if I was to convert the phasors to sinusoidal functions and use angle formulas to arrive at an answer. With multiplying phasors, I get 8*cos(pi*t/6 - 60). With angle formulas, I of course get instantaneous power which is 2 + 4*cos(pi*t/3-60) [W]. I'm sorry for my ignorance in this matter. It may be a stupid question but I'm just trying to understand why you can use phasors sometimes but not other times.

2. Apr 22, 2012

### Staff: Mentor

Hi Centurion! http://img96.imageshack.us/img96/5725/red5e5etimes5e5e45e5e25.gif [Broken]
The impedance of reactive elements is not sinusoidal. If you want to work in the time domain, then you must use functions of time and solve the applicable differential equation.

Example: apply v(t) = 4cos(Ѡt) to an inductor of L henries.

v(t) = L · di/dt
4cos(Ѡt) = L · di/dt
di/dt = 1/L · 4cos(Ѡt)

∴ i(t) = 4/L · ∫cos(Ѡt)
= 4/(ѠL) · sin(Ѡt)

For an inductive impedance of 2Ω, reactance ѠL = 2Ω
→ i(t) = 2 sin(Ѡt)

This is identical to the result using phasors.

To determine how the inductor's impedance varies with time, for a sinusoidal excitation,

z(t) = v(t) / i(t)

= 4cos(Ѡt) / (2 sin(Ѡt))

= 2 / tan(Ѡt)

https://www.physicsforums.com/images/icons/icon2.gif [Broken] This shows for an inductor z(t) swings from -∞ through 0 to ˖∞ periodically and with period half that of the sinusoids' period.

Last edited by a moderator: May 5, 2017
3. Apr 23, 2012