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Lecture #12 EGR 272 – Circuit Theory II

Read: Chapter 11 in Electric Circuits, 6th Edition by Nilsson

Generator and line impedancesSometimes impedances in the generator and the lines are considered in 3-phase circuits. This will result in power losses in each line and in reduced load voltages.

c Cj3

j3

+1

+

-

+2

j10

j2 j3

j2

+2

n

+5

A

j10 j10

+

-

+

-

+2

+5

N

+5

b

+1

B+

1j2

a

Generator Line Load

Example: The 4-wire Y-Y system below has a balanced generator with 720 V RMS and an acb phase sequence. Determine IaA, Van, VAN, and the power loss in each line.

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Lecture #12 EGR 272 – Circuit Theory II- system

Recall for a generator that VL = Vp. So analyzing the - system is similar to

analyzing the Y- system except that the line voltages are more easily found.Example: Determine the three line currents for a - system that has a balanced generator with 240 V RMS and a negative phase sequence. The loads are as follows: ZAB = 3+j4, ZBC = 3-j4, and ZCA = 2+j2.

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Lecture #12 EGR 272 – Circuit Theory II Load with line impedances

Including line impedances with a load makes the analysis more difficult. A

good way to approach the problem is to use a -Y conversion to change the load

into a Y load.

Delta-to-Wye (-Y) and Wye-to-Delta (Y-) Transformations

In EGR 271 -Y and Y- transformations were used with resistive circuits. These transformations can also be used with circuits consisting of AC impedances. Recall that the transformation equations are derived based on a specific labeling of the impedances, so the equations below are somewhat useless without the corresponding figures.

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Lecture #12 EGR 272 – Circuit Theory II

Delta-to-Wye (-Y) and Wye-to-Delta (Y-) Transformations

Z2

a b

c Wye Circuit

a b

c

Delta Circuit

Z1

Z3

Zc

Za Zb

3

133221c

2

133221b

1

133221a

Z

Z Z Z Z

Z

Z Z Z Z

Z

Z Z Z

:Equations Conversion -Y

ZZZ

ZZZ

ZZZZ

Z Z Z

Z Z

Z Z Z

Z Z

Z Z Z

Z Z

:Equations Conversion Y-

cba

ba3

cba

ac2

cba

cb1

Z

Z

Z

load) (balanced Z3 Z Y load) (balanced 3

Z ZY

Special case: If the load is balanced, these equations reduce to:

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Lecture #12 EGR 272 – Circuit Theory II

Example: Determine the three line currents in a 3 Y- circuit described as follows:

• The Y generator is balanced with an abc phase sequence and Van = 480 V

• Each of the 3 lines (between source and load) has an impedance of 2 + j4 ohms• The load is balanced where each of the three loads have an impedance of 60 +

j90 ohms

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Lecture #12 EGR 272 – Circuit Theory II

Power Calculations in 3 CircuitsTotal power delivered = (power delivered to each phase)Or

) - (or impedance theof angle phase theis and

phaseeach for angle phase and current, voltage, theare and , I , V and

) cos(IV P whereP PTotal

IV

Note: Power can be calculated, as it would be for any AC circuit. For example, total power could be found by finding the power to the resistive portion to each load.

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Lecture #12 EGR 272 – Circuit Theory II

Example: Find the total power delivered to the Y-Y circuit analyzed last class (4-wire Y-Y system has a balanced generator with Van = 480 V and a positive

phase sequence with ZAN = ZBN = 2 + j2 and ZCN = 2 - j2).

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Lecture #12 EGR 272 – Circuit Theory II

Example: Find the total power delivered to the Y-Y circuit analyzed last class (balanced system with Van = 240 V, a negative phase sequence, and with

impedances as follows: ZAB = 6 + j8, ZBCN = 6 – j8, and ZCA = 6).

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Lecture #12 EGR 272 – Circuit Theory II

Measuring Power with Wattmeters:

A wattmeter is a piece of equipment that measures average power, P, in watts. A wattmeter has connections for both current and voltage, as shown below on the left (Electric Circuits, 9th Ed., by Nilsson). Note that the positive side of the current coil and the positive side of the voltage coil are labeled + or +. The wattmeter shown below on the right shows how a wattmeter might be connected in a circuit.

n

+

-

a+ +

W1

Van

IaA

Van

+

-

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Lecture #12 EGR 272 – Circuit Theory II

The 2-wattmeter method and the 3-wattmeter method:

Two common methods for measuring power are the 2-wattmeter method and the 3-wattmeter method. In the 3-wattmeter method, all negative voltage connection on each of the wattmeters is common (typically on the neutral line). In the 2-wattmeter method, the positive voltage terminal on two wattmeters is connected to any two of the lines and both negative terminals are connected to the third line. It can be proven that total power is the sum of the wattmeter readings in either method.

Both methods are illustrated on the following page.

+ + W1

I

V

+

-

Wattmeter reading:

I and Vbetween difference phase theis where),cos(IV W P

or

IVRe W P

so

. terminals wattmeterat thecurrent and voltage theare I and V where

IV S power,complex theofpart real thereadsIt

(W). in watts P, power, average)(or real readsr A wattmete

11

*

11

*

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Lecture #12 EGR 272 – Circuit Theory II

The 3-wattmeter method:

The 2-wattmeter method:

*

aAan1

*

bBbn2

*

cCcn3

T 1 2 3

W Re V I

W Re V I

W Re V I

P W W W

W W P

IVRe W

IVRe W

BAT

*bBbcB

*aAacA

c C

+

-

n

A

+

-

+

-

N

b B

a + + W1

+ + W2

+ + W3

Vbn Vcn

VanZAN

ZBN ZCN

IaA

IbB

IcC

Van

+

+

+

-- -

Vbn

Vcn

c C

+

-

n

A

+

-

+

-

N

b B

a + + WA

+ + WB

Vbn Vcn

Van ZAN

ZBN ZCN

IaA

IbB

Vac

+

+

-- Vbc

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Lecture #12 EGR 272 – Circuit Theory II

c C

+

- n

A

+

-

+

-

N

b B

a + +

WA

+ +

WB

Vbn Vcn

Van

ZAN

ZBN ZCN

IaA

IbB

Vac

+

+

-- Vbc

Example: Determine the reading for each wattmeter below and the total power absorbed by the load if the circuit has a balanced generator with Van = 480V, a

positive phase sequence, and impedances ZAN = 6+j8, ZBN = 8+j6, and ZCN = 5-j5.