Phasor Diagrams
In AC electrical theory every power source supplies a voltage that is either a sine wave of one particular frequency or can be considered as a sum of sine waves of differing frequencies. The neat thing about a sine wave such as V(t) = Asin(ωt + δ) is that it can be considered to be directly related to a vector of length A revolving in a circle with angular velocity ω - in fact just the y component of the vector. The phase constant δ is the starting angle at t = 0.
Since a pen and paper drawing cannot be animated so easily, a 2D drawing of a rotating vector shows the vector inscribed in the centre of a circle as indicated below. The angular frequency ω may or may not be indicated.
When two sine waves are produced on the same display, one wave is often said to be
leading or
lagging the other. This terminology makes sense in the revolving vector picture as shown in Figure 3. The blue vector is said to be leading the red vector or conversely the red vector is lagging the blue vector.
Considering sine waves as vertical components of vectors has more important properties. For instance, adding or subtracting two sine waves directly requires a great deal of algebraic manipulation and the use of trigonometric identities. However if we consider the sine waves as vectors, we have a simple problem of vector addition if we ignore ω. For example consider
Asin(ωt + φ) = 5sin(ωt + 30°) + 4sin(ωt + 140°) ;
the corresponding vector addition is:
Ax = 5
cos(30° + 4
cos(140°) = 1.26595
Ay = 5
sin(30° + 4
sin(140°) = 5.07115
Thus
and
So the Pythagorean theorem and simple trigonometry produces the result
5.23sin(ωt + 76.0°) .
Make a note not to forget to put ωt back in!
Phasors and Resistors, Capacitors, and Inductors
The basic relationship in electrical circuits is between the current through an element and the voltage across it. For resistors, the famous Ohm's Law gives
VR = IR . (1)
For capacitors
VC = q/C . (2)
For inductors
VL = LdI/dt . (3)
These three equations also provide a phase relationship between the current entering the element and the voltage over it. For the resistor, the voltage and current will be in phase. That means if I has the form I
maxsin(ωt + φ) then V
R has the identical form V
maxsin(ωt + φ) where V
max = I
maxR. For capacitors and inductors it is a little more complicated. Consider the capacitor. Imagine the current entering the capacitor has the form I
maxsin(ωt + φ). The voltage, however, depends on the charge on the plates as indicated in Equation (2). The current and charge are related by I =
dq/
dt. Since we know the form of I simple calculus tells us that q should have the form − (I
max/ω)
cos(ωt + φ) or (I
max/ω)
sin(ωt + φ - 90°). Thus V
C has the form (I
max/ωC)
sin(ωt + φ - 90°) = V
maxsin(ωt + φ - 90°). The capacitor current leads the capacitor voltage by 90°. Also note that V
max = I
max/ωC. The quantity 1/ωC is called the capacitive reactance X
C and has the unit of Ohms. For the inductor, we again assume that the current entering the capacitor has the form I
maxsin(ωt + φ). The voltage, however, depends on the time derivative of the current as seen in Equation (3). Since we assumed the form of I, then the voltage over the inductor will have the form ωLI
maxcos(ωt + φ) or V
maxsin(ωt + φ + 90°). The inductor current lags the inductor voltage. Here note that the quantity ωL is called the inductive reactance X
L. It also has units of Ohms.
The phase relationship of the three elements is summed up in the following diagram,
Note that in all three cases, resistor, capacitor, and inductor, the relationship between the maximum voltage and the maximum current was of the form
Vmax = ImaxZ . (4)
We call Z the impedance of the circuit element. Equation (4) is just an extension of Ohm's Law to AC circuits. For circuits containing any combination of circuit elements, we can define a unique equivalent impedance and phase angle that will allow us to find the current leaving the battery. We show how to do so in the next section.
Phasors and AC Circuit Problems
Phasors reduce AC Circuit problems to simple, if often tedious, vector addition and subtraction problems and provide a nice graphical way of thinking of the solution. In these problems, a power supply is connected to a circuit containing some combination of resistors, capacitors, and inductors. It is common for the characteristics of the power supply, V
max and frequency ω, to be given. The unknown quantity would be the characteristics of the current leaving the power supply, I
max and the phase angle φ relative to the power supply. To solve one needs only to follow the rules:
- Circuit elements in parallel share the same voltage.
- Circuit elements in series share the same current.
- Do one branch of the circuit at a time.
- Maintain the phase relationships given in Figure 5.
- Use Ohm's Law V = IZ where Z is the equivalent impedance of any combination of circuit elements being considered.
