The electrical engineering handbook CH009

The electrical engineering handbook

Dorf, R.C., Wan, Z “T-∏ Equivalent Networks”

The Electrical Engineering Handbook

Ed Richard C Dorf

Boca Raton: CRC Press LLC, 2000

9.1 Introduction 9.2 Three-Phase Connections 9.3 Wye Û Delta Transformations

9.1 Introduction

Two very important two-ports are the T and P networks shown in Fig 9.1 Because we encounter these two geometrical forms often in two-port analyses, it is useful to determine the conditions under which these two networks are equivalent In order to determine the equivalence relationship, we will examine Z-parameter equations for the T network and the Y-parameter equations for the Pnetwork

For the T network the equations are

V1 = ( Z1 + Z3) I1 + Z3I2

V2 = Z3I1 + ( Z2 + Z3) I2

and for the P network the equations are

I1 = ( Ya+ Yb) V1 – YbV2

I2 = – YbV1 + ( Yb + Yc) V2

Solving the equations for the T network in terms of I1 and I2, we obtain

where D1 = Z1Z2 + Z2Z3 + Z1Z3 Comparing these equations with those for the P network, we find that

Z V D

D

Z Z

1

1 1

3 2 1

1

1 2

è

ö ø

è

ö ø

–

–

Zhen Wan

University of California, Davis

Richard C Dorf

University of California, Davis

or in terms of the impedances of the P network

If we reverse this procedure and solve the equations for the P network in terms of V1 and V2 and then compare the resultant equations with those for the T network, we find that

(9.1)

D

D

D

a

b

c

=

=

=

2 1 3 1 1 1

Z

Z

Z

a

b

c

=

=

=

1 2 1 3 1 1

D

D

D

1 2

2 2

3 2

=

=

=

c

a

b

where D2 = YaYb + YbYc + YaYc Equation (9.1) can also be written in the form

The T is a wye-connected network and the P is a delta-connected network, as we discuss in the next section

9.2 Three-Phase Connections

By far the most important polyphase voltage source is the

bal-anced three-phase source This source, as illustrated by Fig 9.2,

has the following properties The phase voltages, that is, the

voltage from each line a, b, and cto the neutral n, are given by

Van = Vp Ð 0 °

Vcn= Vp Ð +120 °

An important property of the balanced voltage set is that

From the standpoint of the user who connects a load to the balanced three-phase voltage source, it is not

important how the voltages are generated It is important to note, however, that if the load currents generated

by connecting a load to the power source shown in Fig 9.2 are also balanced, there are two possible equivalent

configurations for the load The equivalent load can be considered as being connected in either a wye (Y) or a

delta (D) configuration The balanced wye configuration is shown in Fig 9.3 The delta configuration is shown

in Fig 9.4 Note that in the case of the delta connection, there is no neutral line The actual function of the

1

2

3

=

=

=

a b

b c

a c

Van

+ –

Balanced three-phase

Vcn

a

b

c

n

phase a

phase b

phase c + +

a

b

c

n

ZY

ZY

ZY

a

b c

ZD

ZD

ZD

neutral line in the wye connection will be examined and it will be shown that in a balanced system the neutral

line carries no current and therefore may be omitted

9.3 Wye Û Delta Transformations

For a balanced system, the equivalent load configuration

may be either wye or delta If both of these configurations

are connected at only three terminals, it would be very

advantageous if an equivalence could be established

between them It is, in fact, possible characteristics are

the same Consider, for example, the two networks

shown in Fig 9.5 For these two networks to be

equiva-lent at each corresponding pair of terminals it is necessary that the input impedances at the corresponding

terminals be equal, for example, if at terminals a and b, with c open-circuited, the impedance is the same for

both configurations Equating the impedances at each port yields

(9.4)

Solving this set of equations for Za, Zb, and Zc yields

(9.5)

and Delta Load Configurations Parameter Wye Configuration Delta Configuration

3

3

Z Z Z Z Z Z

ab a b

bc b c

ca c a

a

b

c

=

=

=

1 2

1 3

2 3

Similary, if we solve Eq (9.4) for Z1, Z2, and Z3, we obtain

(9.6)

Equations (9.5) and (9.6) are general relationships and apply to any set of impedances connected in a wye or

delta configuration For the balanced case where Za = Zb = Zc and Z1 = Z2 = Z3, the equations above reduce to

(9.7)

and

Defining Terms

Balanced voltages of the three-phase connection: The three voltages satisfy

Van + Vbn + Vcn = 0

where

Van = Vp Ð0°

Vbn = Vp Ж120°

Vcn = Vp Ð+120°

T network: The equations of the T network are

V1 = (Z1 + Z3)I1 + Z3I2

V2 = Z3I1 + (Z2 + Z3)I2

P network: The equations of P network are

I1 = (Ya + Yb)V1 – YbV2

I2 = –YbV1 + (Yb + Yc)V2

T and P can be transferred to each other

Related Topic

3.5 Three-Phase Circuits

Z Z Z Z Z Z Z

Z

Z Z Z Z Z Z Z

Z

Z Z Z Z Z Z Z

Z

1

2

3

c

b

a

ZY = 1 Z

3

J.D Irwin, Basic Engineering Circuit Analysis, 4th ed., New York: MacMillan, 1995 R.C Dorf, Introduction to Electric Circuits, 3rd ed., New York: John Wiley and Sons, 1996.

Further Information

IEEE Transactions on Power Systems

IEEE Transactions on Circuits and Systems, Part II: Analog and Digital Signal Processing