# Chapter 2: transformers  (Page 4/6)

${Z}_{\text{sc}}={R}_{1}+{\text{jX}}_{{1}_{1}}+\frac{{Z}_{\varphi }\left({R}_{2}+{\text{jX}}_{{1}_{2}}\right)}{{Z}_{\varphi }+{R}_{2}+{\text{jX}}_{{1}_{2}}}$ (2.27)

${Z}_{\text{sc}}\approx {R}_{1}+{\text{jX}}_{{1}_{1}}+{R}_{2}+{\text{jX}}_{{1}_{2}}={R}_{\text{eq}}+{\text{jX}}_{\text{eq}}$ (2.28)

Typically the instrumentation will measure the rms magnitude of the applied voltage ${V}_{\text{sc}}$ , the short-circuit current ${I}_{\text{sc}}$ , and the power ${P}_{\text{sc}}$ . The circuit parameters (referred to the primary) can be found as (2.29)-(2.31).

$\mid {Z}_{\text{eq}}\mid =\mid {Z}_{\text{sc}}\mid =\frac{{V}_{\text{sc}}}{{I}_{\text{sc}}}$ (2.29)

${R}_{\text{eq}}={R}_{\text{sc}}=\frac{{P}_{\text{sc}}}{{I}_{\text{sc}}^{2}}$ (2.30)

${X}_{\text{eq}}={X}_{\text{sc}}=\sqrt{\mid {Z}_{\text{sc}}{\mid }^{2}-{R}_{\text{sc}}^{2}}$ (2.31)

• The equivalent impedance can be referred from one side to the other.
• Approximate values of the individual primary and secondary resistances and leakage reactances can be obtained by assuming that ${R}_{1}={R}_{2}=0\text{.}{5R}_{\text{eq}}$ and ${X}_{{l}_{1}}={X}_{{l}_{2}}=0\text{.}{5X}_{\text{eq}}$ when all impedances are referred to the same side.
• Note that it is possible to measure ${R}_{1}$ and ${R}_{2}$ directly by a dc resistance measurement on each winding. However, no such simple test exists for ${X}_{{l}_{1}}$ and ${X}_{{l}_{2}}$ .
• Open-Circuit Test
• The test is used to find the equivalent shunt impedance ${R}_{c}\text{//}{\text{jX}}_{m}$ .
• The test is performed with the secondary open-circuited and rated voltage impressed on the primary. If the transformer is to be used at other than its rated voltage, the test should be done at that voltage.
• An exciting current of a few percent of full-load current is obtained.
• See Fig. 2.16. Note that ${Z}_{\varphi }={R}_{c}\text{//}{\text{jX}}_{m}$ .

Figure 2.13 Equivalent circuit with open-circuited secondary. (a) Complete equivalent circuit.(b) Cantilever equivalent circuit with the exciting branch at the transformer primary.

${Z}_{\text{oc}}={R}_{1}+{\text{jX}}_{{1}_{1}}+{Z}_{\varphi }={R}_{1}+{\text{jX}}_{{1}_{1}}+\frac{{R}_{c}\left({\text{jX}}_{m}\right)}{{R}_{c}+{\text{jX}}_{m}}$ (2.32)

${Z}_{\text{oc}}\approx {Z}_{\varphi }=\frac{{R}_{c}\left({\text{jX}}_{m}\right)}{{R}_{c}+{\text{jX}}_{m}}$ (2.33)

• Typically the instrumentation will measure the rms magnitude of the applied voltage ${V}_{\text{oc}}$ , the open-circuit current ${I}_{\text{oc}}$ , and the power ${P}_{\text{oc}}$ . The circuit parameters (referred to the primary) can be found as (2.34)-(2.36).

${R}_{c}=\frac{{V}_{\text{oc}}^{2}}{{P}_{\text{oc}}}$ (2.34)

$\mid {Z}_{\varphi }\mid =\frac{{V}_{\text{oc}}}{{P}_{\text{oc}}}$ (2.35)

${X}_{m}=\frac{1}{\sqrt{\left(1/\mid {Z}_{\varphi }\mid {\right)}^{2}-\left(1/{R}_{c}{\right)}^{2}}}$ (2.36)

• The open-circuit test can be used to obtain the core loss for efficiency computations and to check the magnitude of the exciting current.
• Note the term “Voltage Regulation” which is to be discussed in Example 2.6.

§2.6 Autotransformers; Multiwinding Transformers

• Two-winding  Other winding configurations.

§2.6.1 Autotransformers

• Autotransformer connection: Fig. 2.14.

Figure 2.14 (a) Two-winding transformer. (b) Connection as an autotransformer.

• The windings of the two-winding transformer are electrically isolated whereas those of the autotransformer are connected directly together.
• In the transformer connection, winding ab must be provided with extra insulation.
• Autotransformer have lower leakage reactances, lower losses, and smaller exciting current and cost less than two-winding transformers when the voltage ration does not differ too greatly from 1:1.
• The rated voltages of the transformer can be expressed in terms of those of the two-winding transformer as

${V}_{{L}_{\text{rated}}}={V}_{{1}_{\text{rated}}}$ (2.37)

${V}_{{H}_{\text{rated}}}={V}_{{1}_{\text{rated}}}+{V}_{{2}_{\text{rated}}}=\frac{{N}_{1}+{N}_{2}}{{N}_{1}}{V}_{{L}_{\text{rated}}}$ (2.38)

• The effective turns ratio of the autotransformer is thus $\left({N}_{1}+{N}_{2}\right)/{N}_{1}$ .
• The power rating of the autotransformer is equal to $\left({N}_{1}+{N}_{2}\right)/{N}_{2}$ times that of the two winding transformer.

§2.6.2 Multiwinding Transformers

• Transformers having three or more windings, known as multiwinding or multicircuit transformers, are often used to interconnect three or more circuits which may have different voltages.

#### Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
sure. what is your question?
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!