Phasor Diagrams for electrical circuits

7/27/2019 Phasor Diagrams for electrical circuits

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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. In Figure 1, an animated

GIF shows this relation [you may need to click on the image for it to

animate].

Figure 1

Since a pen and paper drawing cannot be animated so easily, a 2Ddrawing of a rotating vector shows the vector inscribed in the centre

of a circle as indicated in Figure 2 below. The angular frequency ω

sor Diagrams http://www.kwantlen.ca/science/physics/faculty/mcoombes/P2421_Not...

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may or may not be indicated.

Figure 2

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 bluevector is said to be leading the red vector or conversely the red vector

is lagging the blue vector.


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Figure 3

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 ignoreω. For example consider

Asin(ωt + φ) = 5 sin(ωt + 30°) + 4 sin(ωt + 140°) ;

the corresponding vector addition is:


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Ax = 5cos(3

Ay = 5 sin(30

Thus

and

Figure 4

So the Pythagorean theorem and simple trigonometry produces the

result

5.23 sin(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


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VC = q/C . (2)

For inductors

VL = Ld I/d t . (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

Imax sin(ωt + φ) then VR has the identical form Vmax sin(ωt + φ) where

Vmax = ImaxR. For capacitors and inductors it is a little more

complicated. Consider the capacitor. Imagine the current entering the

capacitor has the form Imax sin(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 = d q/d t. Since we know the form

of I simple calculus tells us that q should have the form −

(Imax/ω)cos(ωt + φ) or (Imax/ω) sin(ωt + φ - 90°). Thus VC has the

form (Imax/ωC) sin(ωt + φ - 90°) = Vmax sin(ωt + φ - 90°). The

capacitor current leads the capacitor voltage by 90°. Also note that

Vmax = Imax/ωC. The quantity 1/ωC is called the capacitive reactanceXC and has the unit of Ohms. For the inductor, we again assume that

the current entering the capacitor has the form Imax sin(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 ωLImaxcos(ωt + φ) or

Vmax sin(ωt + φ + 90°). The inductor current lags the inductor voltage.

Here note that the quantity ωL is called the inductive reactance XL. It

also has units of Ohms.

The phase relationship of the three elements is summed up in the

following diagram, Figure 5.


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Figure 5

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 justan 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, Vmax and frequency ω, to be given. The unknown

quantity would be the characteristics of the current leaving the power


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supply, Imax 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.1.

Circuit elements in series share the same current.2.

Do one branch of the circuit at a time.3.Maintain the phase relationships given in Figure 5.4.

Use Ohm's Law V = IZ where Z is the equivalent impedance of

any combination of circuit elements being considered.

5.

Consider the following example.

The emf for the circuit in Figure 6 is ε = 10sin(1000t). Find the current delivered to the circuit. Find the

equivalent impedance of the circuit. Find the equation

of the current and voltage drop for each element of the

circuit.

Figure 6

Step 1. Determine the reactances and assign a current to each

branch of the circuit.

First we determine the reactances:


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XC = 1/ωC = 1/[1000 rad/s× 10 µF] = 100 Ω ,

and

XL = ωL = 1000 rad/s× 40 mH = 40 Ω .

Next we assign a current to each branch of the circuit.

Figure 7

Step 2. Use the phase relationships to determine the impedance of

the circuit one piece at a time.

The branch with current I2 is easy. Since current and voltage drop arein phase for resistors, we have

V2 = (20 Ω)I2 .

The branch carrying I1 needs more work. Since the current is

common we draw a diagram that indicates the appropriate phase

relationships. We need to find the equivalent impedance Z1 and the phase angle φ1 that we can use to replace the capacitor/resistor

combination.


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Figure 8

Using the Pythagorean Theorem and trigonometry, we find the

impedance of the branch

and the phase angle

As we see from Figure 8, the current I1 leads V1 by 63.4°.

Now these two branches containing the 20 Ω resistor and Z1 are in

parallel, that is V1 = V2 = V. Since the voltage is common we draw a

diagram like the following to find the equivalent impedance Z12 and

phase angle φ12 .


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Figure 9

To do the vector addition, we will treat the voltage vector as the

x-axis. Then

and


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Thus

Hence the equivalent impedance of the two arms together is Z12 =

18.319 Ω. The phase angle is

As we see from Figure 9, the current leads the voltage.

Next the current I12 equals I and this current passes through the 100

Ω resistor, the impedance Z12, and XL. Figure 10 shows the

appropriate diagram for determining the total circuit's equivalent

impedance Zeq and phase angle φf.

Figure 10

To do the vector addition, we will treat the current vector as the

x-axis. Then


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and

Thus

Hence the equivalent impedance of the circuit together is Zeq = 123.9

Ω. The phase angle is

As can be seen from Figure 10, the voltage leads the current. Since

εmax = 10 Volts, we have Imax = εmax/Z = 10/123.875 A = 80.73 mA.The requested equation for the current is

I = (80.73 mA) sin(ωt − 17.53°) .

Step 3. Find the current or voltage for each piece using the phaserelationship and Ohm's Law.


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The battery current passes through the 100 Ω resistor and the

inductor. Using Ohm's Law, the voltage drop across the 100 Ω resistor

is

V100 = IR = (8.073 V) sin(ωt − 17.53°) .

For the inductor

VLmax = ImaxXL = 80.73 mA × 40 Ω = 3.229 Volts .

The phase relation between VL and I yields

VL = VLmax sin(ωt − 17.53° + 90°) = (3.229 V) sin(ωt + 72.47°) .

The maximum voltage drop across Z12 is

Vmax = ImaxZ12 = 80.73 mA × 18.319 Ω = 1.479 Volts.

Since the voltage lags I by φ12, we find

V = Vmax sin(ωt − 17.53° - φ12) = (1.479 V) sin(ωt − 25.96°) .

From here on we reverse the steps we took to find Z12 in the first

place.

By definition, the voltage drop across Z12 is also the voltage across

the 20 Ω resistor. The maximum current through the resistor will be

I2max = Vmax/R = 1.479 V / 20 Ω = 73.95 mA .

The equation for this current is

I2 = (73.95 mA) sin(ωt − 25.96°) .

The voltage V is also the potential drop across Z1. The maximum

current in this branch is

I1max = Vmax/Z1 = 1.479 V / 111.803 Ω = 13.23 mA .


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Recalling the phase information we derived for Z1, the current

formula will be

I1 = I1max sin(ωt − 25.96° + f 1 = (13.23 mA) sin(ωt + 37.48°) .

This in turn is the current through the capacitor and 50 Ω resistor. Themaximum voltage drop over the capacitor is

VCmax = I2maxXC = 13.23 mA × 100 &Ohms; = 1.323 Volts .

We know that VC must lag I1 by 90°. Hence the equation for the

voltage will be

VC = VCmax sin(ωt + 37.48° - 90°) = (1.323 V) sin(ωt − 52.52°) .

Finally the maximum voltage drop over the 50 Ω resistor will be

V50max = I2maxR 50 = 13.23 mA × 50 &Ohms; = 0.661 Volts.

Current and voltage are in phase for a resistor, so the equation will be

V50 = (0.661 V) sin(ωt + 37.48°) .

Summarizing

Element Voltage Current

Power Supply (10 V) sin(ωt) (80.73

mA) sin(ωt −

17.53°)

100 Ω Resistor (8.073 V) sin(ωt −

17.53°)

(80.73

mA) sin(ωt −17.53°)


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40 mH Inductor (3.229 V) sin(ωt +

72.47°)

(80.73

mA) sin(ωt −

17.53°)

Z12 (1.479 V) sin(ωt −25.96°)

(80.73mA) sin(ωt −

17.53°)

20 Ω Resistor (1.479 V) sin(ωt −

25.96°)

(73.94 mA)

sin(ωt −

25.96°)

Z1 (1.479 V) sin(ωt −

25.96°)

(13.23 mA)

sin(ωt +

37.48°)

10 µF Capacitor (1.323 V) sin(ωt −

52.52°)

(13.23 mA)

sin(ωt +

37.48°)

50 Ω Resistor (0.661 V) sin(ωt +

37.48°)

(13.23 mA)

sin(ωt +

37.48°)

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Phasor Diagrams for electrical circuits