Introductory Physics Laboratory, Faculty of Physics and Geosciences, University of Leipzig E 10e Transformer Tasks 1 Measure without a load resistance the dependencies of the primary current Ip, on the primary actual power Pp,W and of the secondary voltage Us on the primary voltage Up ! 2 Determine the ratio ü = Us/Up , the phase displacement ϕ as well as the loss current Iv and the magnetizing current Im as a function of Up ! Create one diagram with appropriate scales to show the influence of the primary voltage on the measurand1 ϕ, Im and Iv respectively! Discuss the dependencies! 3 Measure the quantities Up , Ip , Us and Pp,W in dependence of load resistance Rs,B across the secondary coil! 4 Create one diagram with appropriate scales, where the influence of the secondary current of Is on Im, Iv and the efficiency η is represented! Additional Tasks: Discuss the influence of the secondary current Is on phase angle ϕ and the power ratio PCu/PFe , respectively! Literature Physics, P. A. Tipler, 3rd ed.,Vol. 2, Chapt. 26-5, 26-7, 28-1, 28-6 Physikalisches Praktikum, 12. Auflage, Hrsg. D. Geschke, Elektrizitätslehre, 2.5 http://www.phys.unsw.edu.au/~jw/transformers.html Accessories AC power supply, transformer, digital multimeters, digital power meter, ohmic load resistance Keywords for preparation - Ampere′s law and Faraday′s law, Lenz rule, self-induction - mutual inductance of two electrical circuits - voltages and currents at transformer - losses of transformer, equivalent circuits Basics to transformer The transformer is a device to transform alternating voltage and current as well as for impedance matching (power matching) with only low losses in power. In principle, it consists of two coils coupled inductively by an iron core, the primary coil with number of windings np and the secondary coil with number of windings ns. In case of the ideal transformer the electrical power supplied to the primary coil is equal to the electrical power which is transferred to the secondary coil. Thus, for the ratio of effective values of secondary (Us) and primary voltage (Up) is

s s

p p

U nuU n

= − = . (1)

The minus sign indicates the phase shift of 180° in accordance to Lenz rule. The principle operation of an

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1 A particular quantity determined through measurement is called measurand.

unloaded transformer can be described in the following way whereat the description of the alternating current quantities is in analogy to DIN 5483 (small symbols for time dependent quantities, large symbols for effective values and underlined symbols for complex quantities). In case of an ideal transformer the voltage drop up across the primary coil generates the magnetizing current im in the primary coil. The current depends on the magnitude of up and on the inductive reactance of the primary windings. The current im generates a magnetic field H in the primary coil which creates the magnetic flux density B and the magnetic flux Φ in the iron core:

pp

p

( ) ( )A n

t B t A il

μΦ = = .

A is the cross section area and lp is the length of the iron core, μ = μr μ0 is the permeability of the core material and ip= im is the primary current which is equal to the magnetizing current. The time dependent quantities up and ip obey a harmonic function, e.g. ip= I0 sin (ω t). In the ideal case where the magnetic flux Φ(t) gets through the iron core completely without losses the flux induces the induction voltages up,L and us,L in the primary coil as well as in the secondary coil (the index L is related to the ideal case)

p,L pd ( )

dtu n

tΦ

= − , (2)

s,L sd ( )

dtu n

tΦ

= − . (3)

The induced voltage up,L is equal to the input voltage up across the primary coil but with different sign (up,L= -up, self-inductance). The induced secondary voltage us,L produces the voltage us across the secondary coil. Since the time dependence of up,L is described by the cosine function and the time dependence of im is described by the sinus function of the same argument. Thus, the voltage up is said to lead the current ip= im by one-fourth period or 90° (Fig. 1a). The primary current as magnetizing current is a pure ″watt-less″ current (Pp= UpIpcosϕ = 0) and its magnitude can be calculated neglecting the resistance of primary windings (inductance Lp) using the magnitude of Up and the inductive reactance ωLp according to Im= Up/ωLp. In case of the ideal loaded transformer the current is generates an additional field Hs in the secondary coil. This should reduce the magnetic flux which however is defined by the constant external voltage Up. Therefore an additional current IB (index B for load) is generated in the primary coil. IB causes a magnetic field which must compensate Hs. From the condition Hs + Hp,B = 0 and with Hs ∝ nsIs and Hp,B ∝ npIB one gets for coils of identical length and constant cross section area of the iron core

B

s p

I nI n

= s . (4)

The currents IB and Is are oppositely directed (Fig. 1b) and in the loaded case for the primary current Ip,B follows

sp,B p,L s

p

nI In

= + I . (5)

The primary active power is Pp,B = Up Ip,B cosϕ. In case of the ideal transformer where is put an ohmic load resistance across the secondary coil the active power is Ps = UsIs. The efficiency η of a transformer is defined as

s

p

PP

η = . (6)

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Fig. 1 Phasor diagram (schematic) of an ″watt-less″ transformer, a) ideal case without load and b) ohmic load In reality a series of losses appear at transformers, e.g. copper losses, iron losses, scattering losses. Copper losses (RCu): The coil turns have an ohmic resistance which causes heating due to the current. The corresponding power is proportional to the square of current. Thus, the copper losses increase squarely with the magnitude of load current. The copper losses effect like an actual resistance which is in series with the coil reactance. Scattering losses: The magnetic field is induced not only in the iron core but also outside the iron core. The magnetic flux inside the core material operates in both coils so that the scattered part of the magnetic field cannot contribute to induction. The magnetic field of the coils depend on the current and therefore on the load of the transformer. The scattering losses have the effect of an additional inductive resistance which is in series to the inductive resistance of the coil windings. Iron losses or core losses (RFe): The core losses are divided into eddy current losses and hysteresis losses. The iron core of a transformer consists normally of thin, against each other isolated sheets of soft iron in order to reduce eddy currents. The eddy currents cannot be suppressed completely. Thus the iron core is heated. The magnetic remanence produces also heat. These losses are called remagnetizing or hysteresis losses. Their magnitude is determined by the area of the hysteresis loop and by the volume of core material. In both cases the power causing the heating is taken from the outer current source. The core losses have the effect of an ohmic resistor which is parallel to the inductive reactance of the primary coil. All of the secondary quantities are converted to the primary coil so that the voltages induced in both windings are of the same magnitude and the powers are kept. According to eq.(1) the voltages have to be multiplied by ü, the currents by 1/ü and the resistivities by 1/ü2 . Using the equivalent circuit in Fig.2, it is possible to summarize the parts of both coils contributing to the common flux Φ to one inductance L. Fig. 2 Equivalent circuit of real transformer (without scattering losses)

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For the (complex) primary current one gets

( )2

p p p,Cu pFe s,B s,Cu

1 ü jI U R IR R R Lω

⎛ ⎞= − ⋅ + −⎜ ⎟⎜ ⎟+⎝ ⎠

. (7)

At the given experimental conditions is Rp,Cu << RFe and Rp,Cu << ωL and it follows

2

pp v m

s,B s,Cu

ü UI I j I

R R⎛ ⎞

≈ + −⎜ ⎟⎜ ⎟+⎝ ⎠F . (8)

The factor F is called as the voltage transfer coefficient and from eq.(7) follows

12

p,Cu

s,B s,Cu

1ü R

FR R

−⎡ ⎤

= +⎢ ⎥+⎣ ⎦ . (9)

From eq.(8) for the amount of primary current and for phase angle ϕ between current and voltage in the primary circuit one gets

( )1

2 2 2p v L mI F I I I⎡ ⎤= + +⎣ ⎦ , (10)

m

v L

arctan II I

ϕ⎛ ⎞

= −⎜ ⎟+⎝ ⎠ . (11)

In eqs.(10) and (11) the current IL (current Ip,L in Fig. 1b) with

p2L

s,B s,Cu

UI ü

R R=

+ (12)

is the additional primary current due to the secondary load. The amounts of magnetizing and of loss current can be calculated using eqs. (10) and (11):

pm sin

II

Fϕ= , (13)

pv Lcos

II

Fϕ I= − . (14)

The phase angle ϕ can be determined using the primary actual power Pp,W and the apparent power Pp,S

p,W

p,S

arccosPP

ϕ⎛ ⎞

= ⎜⎜⎝ ⎠

⎟⎟ , (15)

where Pp,S =Ip⋅Up. The approximations in the above equations are better fulfilled for small differences between the approximated factor F and the true factor F´. In complex notation it can be described by

1

2

p,CuFe s,B s,Cu

1 11 üF R jR R R Lω

−⎡ ⎤⎛ ⎞

′ = + + −⎢ ⎥⎜⎜ +⎢ ⎥⎝ ⎠⎣ ⎦⎟⎟ . (16)

Whilst the written preparation at home (laboratory record) all important equations of the specific calculations should be understood and explained in the record. 4

Hints to realize and to analyze the experiment E10e The variation of voltage Up at task 1 is made by appropriate changing the output voltage at the variable ratio transformer TST200/6. About 10 roughly equidistant primary voltages between 0 and 50 V shall be used for the measuring (U > 40 V are dangerous voltage!). At task 2 the dependences Im(Up), Iv(Up) and ϕ(Up) have to be represented in one diagram using appropriate ordinate scales and have to be discussed. The average value shall be given for the voltage transfer ü. The uncertainties of Pp,S, Iv, Im and ϕ have to be calculated for the measurement at Up ≈ 30 V. The relative error of the watt meter is 2%. At task 3 Up is adjusted to approx. 30 V. The magnitude of the secondary ohmic load resistor Rs,B shall be at transformer I 8; 10; 15; 20; 30; 50; 100; 150; 300 and ∞ ohms, at transformer II 30; 40; 50; 60; 75; 100; 150; 300; 600; ∞ ohms. The measured datasets have to be represented in one diagram with appropriate ordinate scales and have to be discussed. The resistance RE (see Fig. 3) consists of two parallel connected decade resistors (RE = 500 Ω at task 1, RE = 100 Ω at task 3). The load resistance Rs,B of the secondary side is realized using a precision measuring resistance whose maximum load may not be exceed under no circumstances. All measurements of the actual power Pp,W have to be carried out using the energy- and power-meter 53183 in the range 100 V/100 mA. The zero point of power meter has to be adjusted exactly after a warming up period of about 10 minutes. Fig. 3 Test circuit

LM energy- and power-meter

53183 TST AC power supply (variable ratio transformer) DM1 digital multimeter (Ip) DM2 digital multimeter (Up) DM3 digital multimeter (Us) RE 2 decadic variable resistors 10 x 100 Ω Rs,B load resistance (3 decades)

S: switch (cord) to connect the load resistance

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