Formula Sheets: Bridges and Potentiometers | Electrical and Electronic Measurements - Electrical Engineering (EE) PDF Download

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Electrical and Electronic Measuremen ts: Bridges &
P oten tiometers F orm ula S heet for Electrical GA TE
DC Bridges
• Wheatstone Bridge : F or medium resistance measuremen t (1 ? to 1 M? ).
R
1
R
2
=
R
3
R
x
Unkno wn resistance:
R
x
= R
3
·
R
2
R
1
where R
1
,R
2
,R
3
are kno wn resistances, R
x
is unkno wn resistance (? ).
• Kelvin Bridge : F or lo w resistance measuremen t (< 1 ? ).
R
x
= R
s
·
R
2
R
1
where R
s
is standard lo w resistance, accoun ts for lead and con tact resistances.
A C Bridges
• General A C Bridge Balance : Imp edance balance condition.
Z
1
Z
4
= Z
2
Z
3
where Z
1
,Z
2
,Z
3
,Z
4
are complex imp edance s.
• Maxw ell’s Bridge : F or inductance measuremen t.
L
x
= R
2
R
3
C
4
, R
x
=
R
2
R
3
R
4
where L
x
is unkno wn inductance (H ), R
x
is unkno wn resistance (? ), C
4
is kno wn
capacitance (F ).
• Ha y’s Bridge : F or high-Q inductance measuremen t.
L
x
=
R
2
R
3
C
4
1+?
2
C
2
4
R
2
4
, R
x
=
R
2
R
3
R
4
where ? is angular frequency (rads
-1
).
• Sc hering Bridge : F or capacitance measuremen t.
C
x
= C
s
·
R
4
R
3
, Dissipation F actor= ?C
s
R
s
where C
x
is unkno wn capacitance (F ), C
s
is standard capacitance, R
s
is standard
resistance.
• Wien Bridge : F or frequency measuremen t.
f =
1
2p
v
R
1
R
2
C
1
C
2
where f is frequency (Hz ), R
1
,R
2
are resistances (? ), C
1
,C
2
are capacitances (F ).
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Electrical and Electronic Measuremen ts: Bridges &
P oten tiometers F orm ula S heet for Electrical GA TE
DC Bridges
• Wheatstone Bridge : F or medium resistance measuremen t (1 ? to 1 M? ).
R
1
R
2
=
R
3
R
x
Unkno wn resistance:
R
x
= R
3
·
R
2
R
1
where R
1
,R
2
,R
3
are kno wn resistances, R
x
is unkno wn resistance (? ).
• Kelvin Bridge : F or lo w resistance measuremen t (< 1 ? ).
R
x
= R
s
·
R
2
R
1
where R
s
is standard lo w resistance, accoun ts for lead and con tact resistances.
A C Bridges
• General A C Bridge Balance : Imp edance balance condition.
Z
1
Z
4
= Z
2
Z
3
where Z
1
,Z
2
,Z
3
,Z
4
are complex imp edance s.
• Maxw ell’s Bridge : F or inductance measuremen t.
L
x
= R
2
R
3
C
4
, R
x
=
R
2
R
3
R
4
where L
x
is unkno wn inductance (H ), R
x
is unkno wn resistance (? ), C
4
is kno wn
capacitance (F ).
• Ha y’s Bridge : F or high-Q inductance measuremen t.
L
x
=
R
2
R
3
C
4
1+?
2
C
2
4
R
2
4
, R
x
=
R
2
R
3
R
4
where ? is angular frequency (rads
-1
).
• Sc hering Bridge : F or capacitance measuremen t.
C
x
= C
s
·
R
4
R
3
, Dissipation F actor= ?C
s
R
s
where C
x
is unkno wn capacitance (F ), C
s
is standard capacitance, R
s
is standard
resistance.
• Wien Bridge : F or frequency measuremen t.
f =
1
2p
v
R
1
R
2
C
1
C
2
where f is frequency (Hz ), R
1
,R
2
are resistances (? ), C
1
,C
2
are capacitances (F ).
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P o ten t iometers
• DC P oten tiometer : Measures v oltage b y comparing with a standard cell.
V
x
= V
s
·
l
x
l
s
where V
x
is unkno wn v oltage (V ), V
s
is standard v oltage, l
x
,l
s
are lengths of p oten-
tiometer wire.
• Resistance Measuremen t with P oten tiometer :
R
x
= R
s
·
l
x
l
s
where R
x
is unkno wn resistance (? ), R
s
is standard resistance.
• A C P oten tiometer : Measures A C v oltage magnitude and phase.
V
x
= V
s
·
l
x
l
s
??
where ? is phase angle, adjusted using in-phase and quadrature comp onen ts.
• Crompton’s DC P oten tiometer : Standardized v oltage measuremen t.
V
x
= IR
std
·
l
x
l
std
where I is constan t curren t, R
std
is standard resistance, l
x
,l
std
are wire lengths.
Sensitivit y and Errors
• Bridge Sensitivit y : Galv anometer deflection p er unit un balance.
S =
?
(?R/R)
where ? is galv anometer deflection, ?R is resistance c h ange.
• P oten tiometer Sensitivit y : V oltage resolution p er unit length.
S =
V
l
where V is v oltage across wire, l is wire length.
• Error in Bridge Measuremen t :
Error=± A ccuracy· F ull Scale V alue
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Page 3


Electrical and Electronic Measuremen ts: Bridges &
P oten tiometers F orm ula S heet for Electrical GA TE
DC Bridges
• Wheatstone Bridge : F or medium resistance measuremen t (1 ? to 1 M? ).
R
1
R
2
=
R
3
R
x
Unkno wn resistance:
R
x
= R
3
·
R
2
R
1
where R
1
,R
2
,R
3
are kno wn resistances, R
x
is unkno wn resistance (? ).
• Kelvin Bridge : F or lo w resistance measuremen t (< 1 ? ).
R
x
= R
s
·
R
2
R
1
where R
s
is standard lo w resistance, accoun ts for lead and con tact resistances.
A C Bridges
• General A C Bridge Balance : Imp edance balance condition.
Z
1
Z
4
= Z
2
Z
3
where Z
1
,Z
2
,Z
3
,Z
4
are complex imp edance s.
• Maxw ell’s Bridge : F or inductance measuremen t.
L
x
= R
2
R
3
C
4
, R
x
=
R
2
R
3
R
4
where L
x
is unkno wn inductance (H ), R
x
is unkno wn resistance (? ), C
4
is kno wn
capacitance (F ).
• Ha y’s Bridge : F or high-Q inductance measuremen t.
L
x
=
R
2
R
3
C
4
1+?
2
C
2
4
R
2
4
, R
x
=
R
2
R
3
R
4
where ? is angular frequency (rads
-1
).
• Sc hering Bridge : F or capacitance measuremen t.
C
x
= C
s
·
R
4
R
3
, Dissipation F actor= ?C
s
R
s
where C
x
is unkno wn capacitance (F ), C
s
is standard capacitance, R
s
is standard
resistance.
• Wien Bridge : F or frequency measuremen t.
f =
1
2p
v
R
1
R
2
C
1
C
2
where f is frequency (Hz ), R
1
,R
2
are resistances (? ), C
1
,C
2
are capacitances (F ).
1
P o ten t iometers
• DC P oten tiometer : Measures v oltage b y comparing with a standard cell.
V
x
= V
s
·
l
x
l
s
where V
x
is unkno wn v oltage (V ), V
s
is standard v oltage, l
x
,l
s
are lengths of p oten-
tiometer wire.
• Resistance Measuremen t with P oten tiometer :
R
x
= R
s
·
l
x
l
s
where R
x
is unkno wn resistance (? ), R
s
is standard resistance.
• A C P oten tiometer : Measures A C v oltage magnitude and phase.
V
x
= V
s
·
l
x
l
s
??
where ? is phase angle, adjusted using in-phase and quadrature comp onen ts.
• Crompton’s DC P oten tiometer : Standardized v oltage measuremen t.
V
x
= IR
std
·
l
x
l
std
where I is constan t curren t, R
std
is standard resistance, l
x
,l
std
are wire lengths.
Sensitivit y and Errors
• Bridge Sensitivit y : Galv anometer deflection p er unit un balance.
S =
?
(?R/R)
where ? is galv anometer deflection, ?R is resistance c h ange.
• P oten tiometer Sensitivit y : V oltage resolution p er unit length.
S =
V
l
where V is v oltage across wire, l is wire length.
• Error in Bridge Measuremen t :
Error=± A ccuracy· F ull Scale V alue
2
Key Notes
• Standard units: Resistance (? ), Capacitance (F ), Inductance (H ), V oltage (V ), F re-
quency (Hz ).
• Wheatstone and Kelvin bridges are for DC; Maxw ell, Ha y’s, Sc hering, and Wien
bridges are for A C.
• P oten tiometers require standardization with a standard cell for accurate measure-
men ts.
• Ensure bridge balance (no curren t through galv anometer) b efore applying form ulas.
• Use SI units for GA TE calculations.
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