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ECE 3336 Introduction to Circuits & Electronics

Set #16Transformers

Fall 2012,TUE&TH 4:00-5:30 pmDr. Wanda Wosik

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InductorsInductors model the interaction

between magnetic fields and voltage and current.

- Magnetic flux in webers [Wb]

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Mutual Inductance We model this effect with what we

call “mutual inductance”, and we call it M [H]

We also rename L [H], as the “self inductance”

Magnetomotive force

vp(t)

vs(t)

vp(t)

vs(t)

v(t)=N•d/dt

turns

Voltage induced in the secondary coil in response to current changes in the primary coil.

Current direction = polarity important

http://www.allaboutcircuits.com/vol_2/chpt_9/7.html

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The Dot ConventionPhases of the voltages in the primary and secondary windings are

identified by dots.

180° phase shift between instantaneous voltages

The same phase is obtained for both instantaneous voltages v1(t) and v2(t)

If i1 and i2 are both defined as entering (or leaving) the dotted terminal, then

n2 / n1 = -i1 / i2 = N; Otherwise,

n2 / n1 = i1 / i2 = N.

The same flux

time domain

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From Time Domain to Phasor Domain

Hambley

The same flux

time domain

Voltage and current ratios

phasor domain

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Power in Ideal Transformers

Note that if the voltage increases going from one side of a transformer to the other, the current decreases by the same factor.

There is no power gain. The factor is the ratio of the number of turns. We named this as the turns ratio, N.

We can tap the secondary voltage at two (or more) points

Center-tapped transformer

240 V

120V line

120V line

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Figure 7.31, 7.32

Center-tapped transformer

Power transformers can be huge

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Other configurations of transformers

Examples of transformers

or small

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Impedance Reflection with Transformers

Transformers can be used to match loads (impedances). Note that

If we divide the first equation by the second, we get

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Impedance Reflection with Transformers

Transformers can be used to match loads (impedances).

This means that if we look at the apparent impedance seen at the primary side of a transformer (Z’) we will see the impedance at the secondary side divided by the turns ratio squared.

This can be very useful. It is often referred to as the reflected impedance.

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The maximum power transfer in AC circuits

Figure 7.35

Maximum power transfer to Rload (in DC) occurs when Rload=Rsource.

In AC circuits we will need very similar impedance matching with the source: Zs=Rs+jXs

1) The real power absorbed by Rload

Where:

2) Now, from the complex power

We calculate the real power (again)

ZL=ZS*

RL=RS and

XL=-XS

Maximum power transfer if:

When will PLMAX?

0PL=PLMAX

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Impedance transformation improves

power delivery - example

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The heaters (62.5Ω and 15.625Ω) of 1,000W power rating operate in two circuits a) and b).

If we use the heater a) with 125 V source the power will decrease P=250 W i.e. [(125V/62.5Ω)x125V] because I=2A

Now if we use a step-up (N=2) transformer: the current delivered to Rload is again I=4A and the power is restored to P=1,000 W

a)

b)

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What happens with power? How to play it loud?

Maximum power would be delivered to the load of 500 Ω. But we have 8Ω.

AC Thevenin circuit

We want to supply power from a high impedance (V high I low) amplifier to a low impedance (low V high I) speaker.

A transformer will give impedance transformation ratio 500:8 so that the delivered power will reach its maximum

http://www.allaboutcircuits.com/vol_2/chpt_9/7.html

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Figure 7.37a, b

Electric power transmission

Figure 7.37a, b, Rizzoni

(a) direct power transmission is affected by the line resistance

What will be =?We will use impedance transformation for

Reflected load here M=1/N(b) power transmission with transformers

14Figure 7.37c, d

Electric power transmission - reduction of Rline by 1/N2

Figure 7.37c, d

(c) equivalent circuit seen by generator

(d) equivalent circuit seen by load

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Figure 7.40, 7.41

Balanced three-phase Power (AC circuit)

Figure 7.41 Rizzoni

neutral

Phase voltages

Line voltagesab, bc, ca

Positive, or abc, sequence for balanced three-phase voltages (“-” acb)

All line voltages

Wye-wye (Y-Y) connection

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Figure 7.42

Balanced three-phase AC circuit (redrawn)

Figure 7.42

Three circuits are in parallel.

Constant! power

Can be eliminated

Advantage of the 3 phase also in less wiring (3)Compared to single phase (6 wires).

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Figure 7.43

Delta-connected generators

Rizzoni, Figure 7.43

Currents drawn by wye-and by delta connected loads For both currents to be the same we have to have

y

Delta draws 3 times more current than a wye load does.

V=0 I=0

Loads can be also in a delta connection

Line (-to-line) voltage

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Figure 7.50

Line voltage convention for residential circuits

red

black

white (earth ground)

This is the voltage (rms) between the hot wires

A 3-wire AC system supplied by the power company

Higher line loss will be from the 120V source. To reduce power loss (I2R) thick wires are used.

83.3A

41.7.A

Rs=0.02Ω

Power loss=69.4 W

Power loss=34.7 W

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Figure 7.52

A typical residential wiring arrangement

Figure 7.52

•Limit power dissipation by appropriate connections fro various loads.

•Avoid heat generation (safety aspects)

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Figure 7.58

Structure of an AC power distribution network (just for your curiosity)

Figure 7.58

Step-up transformer

That reduces power losses in transmission lines

Substations

Your house is carefully wired!

An electric power network =the Power grid allows for redistribution of power to various substations (various V levels obtained after stepping-down).