Chapter 9
Sinusoids and Phasors
 Delta-to-wye and wye-to-delta conversions
 Phase Sifters.
 AC Bridges.
 Problem Solutions
Huseyin Bilgekul
Eeng224 Circuit Theory II
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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Y-Δ and Δ-Y Equivalent Circuits
 Y-Δ and Δ-Y type equivalent conversions will be most useful when considering Three
Phase circuits.
 Impedances Z1, Z2 and Z3 are are Y connected.
 Impedances Za, Zb and Zc are Δ connected.
 Y and Δ forms can be eqivalently converted from one form to the other.
 Y-Δ and Δ-Y conversions are valid for impedances as well as resistive circuits.
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Y-Δ and Δ-Y Equivalent Circuits
 Y-Δ and Δ-Y type equivalent conversions will be useful when considering Three Phase
circuits in Chapter 12.
Y   Conversion
Z Z  Z 2 Z 3  Z 3 Z1
Za  1 2
Z1
  Y Conversion
Zb Zc
Z1 
Z a  Zb  Zc
Z Z  Z 2 Z 3  Z 3 Z1
Zb  1 2
Z2
Z2 
Za Zc
Z a  Zb  Zc
Z Z  Z 2 Z 3  Z 3 Z1
Zc  1 2
Z3
Z3 
Z a Zb
Z a  Zb  Zc
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Balanced Y-Δ and Δ-Y Equivalent Circuits
 A delta (Δ) or Y (wye) circuit is balanced if it has equal impedances in all three
branches.
 Y-Δ and Δ-Y conversions is very simple for balanced circuits.
Balanced Impedance Conversions:
ZY  Z1  Z 2  Z 3
Z   Z a  Zb  Zc
Z  3ZY
1
ZY  ZY
3
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Find I in the circuit
given?
Converting Δ
Connection to Y
form, we obtain
Converting to Y form, we
can simplify the circuit by
combining the parallel and
series elements
Z
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P.P.9.12 Continued
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Phase Shifters (Leading Output)
 Phase shifting is employed to correct undesirable phase shift of the AC voltage.
Vo
R


Vi R  jX C
R
R2  X C 2
Vo Leads Vi by  = tan
1
 tan
1
XC
,
R
XC   1
C
XC
R
Phase of Vi has been changed by  = tan 1
Leading Output.
XC
R
Output Leads the input
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Phase Shifters (Lagging Output)
 Phase shifting is employed to correct undesirable phase shift of the AC voltage.
1
Vo
1
1
jC



  tan 1 ( RC )
Vi R  1
1  j RC
1  ( RC ) 2
jC
Vo Laggs Vi by  = tan 1 ( RC )
Lagging Output.
Output Lags the input
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P.P.9.13 Design an RC circuit that provide 90° lagging phase shift. Find the output
voltage if 10 Volt is applied.
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
AC Bridges
The AC bridge is Balanced when no current flows through the meter.
AC bridges are used in measuring inductance and capacitance values.
A general AC bridge circuit
V1 
Zx
Z2
Vs  V2 
Vs
Z1  Z 2
Z3  Z x
Zx
Z2

Z1  Z 2 Z 3  Z x
Zx 
 Z 2 Z 3  Z1Z x
Z3
Z2
Z1
Z x Unknown value necessary for balancing the bridge
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AC Bridges
 Unknown capacitance and inductances Cx and Lx are measured in terms of
the known standard values Cs and Ls
AC Bridge for measuring L
R2
Lx 
Ls
R1
AC Bridge for measuring C.
Cx 
R1
Cs
R2
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Chapter 9, Problem 41.
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Chapter 9, Problem 51.
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Ch 9, Prob 55. Find Z in the network given that VO  40 Volt
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Chapter 9, Problem 77
a) Calculate the phase shift at 2 Mhz.
b) Calculate the frequency at which phase shift is 45 degree.
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Chapter 9, Problem 77
a) Calculate the phase shift at 2 Mhz.
b) Calculate the frequency at which phase shift is 45 degree.
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