61 
Figure 637 is a schematic diagram of an ideal transformer supplying a noninductive resistance R = 10 ohms. The primary winding has N_{1} = 100
turns and the secondary winding has N_{2} = 50 turns. The instantaneous voltage applied to the primary winding is v_{1} = 150 v.
(a) What is the value of the current and its direction through the resistance R?
(b) What is the value of the primary current?
(c) What is the direction of the flux in the core?
(d) What is the rate of change of
(i) The flux in the primary winding ?
(ii) The flux linkage with the primary winding?
(iii) The flux in the secondary winding?
(iv) The flux linkage with the secondary winding?
(e) The transformer and its connected load are replaced by a noninductive resistance which takes the same value of current at 150 v as that in the primary of the transformer. What is the value of the resistance in ohms?
(f) What is the ratio of the resistance in part (e) to that of the resistance R in the secondary? What is this ratio called?
(g) A polarity mark is shown near the upper terminal of the primary winding in Fig. 637. Where should the secondary polarity mark be placed?

Figure 637. Ideal transformer for Problem 61 

62 
The purpose of this problem is to show, firstly, that a sizeable reduction in frequency without a corresponding reduction in voltage leads to values of exciting current that can exceed the rated current of the transformer, and secondly, that a reduction of the voltage to a value that leads to a safe value of exciting current lowers the rating of the transformer.
A 1,500w, 240/25v, 400cycle transformer operates at a maximum flux density of 36,000 lines per sq in. and the peak value of the exciting current is 0.20 amp.
(a) Use the magnetization curve of Fig. 311 and determine the peak value of the exciting current when the primary of this transformer is connected to 120v 60cycle source. Neglect resistance and leakage flux.
(b) Suppose this transformer were to operate at a frequency of 60 cps and at magnetic flux density B_{m} = 90,000 lines per sq in., and the current density in the windings is the same as for rated 400cps operation. What would the rating of the transformer be in voltamperes?

63 
An audio output transformer operates from a 4,000ohm resistance source delivering 5 w to a noninductive load of 10 ohms. Assume the transformer to be ideal and determine
(a) The turns ratio such that the load impedance referred to the primary has a value of 4,000 ohms.
(b) The secondary and primary currents.
(c) The primary applied voltage.

64 
An ideal transformer is represented schematically in Fig. 638. Note the current directions and voltage polarities and draw a phasor diagram on a onetoone ratio basis for these directions and polarities. Assume the load current to lag the load voltage by an angle of 30°.

Figure 638. Ideal transformer for Problem 64 

65 
The secondary winding of the transformer in Example 62 is short circuited and sufficient voltage, at rated frequency, is applied to the primary winding so that rated current flows in the secondary. On the basis of the exact equivalent circuit in Fig. 611 (a) determine
(a) The secondary induced emf E_{2}.
(b) The primary induced emf E_{1}.
(c) The coreloss current on the basis that the exciting admittance is constant. (Actually, the exciting admittance is not constant; for one thing, it increases with increasing magnetization of the core.)
(d) The exciting current based on the admittance of part (c) above.
(e) The primary applied voltage V_{1}.
(f) The per unit impedance of the transformer on the basis that the per unit impedance is defined as the ratio of the voltage V_{1} with rated current on short circuit in part (e) above, to the rated primary voltage.
(g) The per unit resistance of the transformer on the basis that the ratio of
the I^{2}R losses with rated current to the voltampere rating of transformer is the per unit resistance,
(h) The mutual flux expresssed as a decimal fraction of the mutual flux at rated voltage and no load,
(i) The primary current and the secondary current when a short circuit is
applied to the secondary with full rated voltage across the primary.

66 
Repeat Problem 65, using the equivalent circuit of Fig. 614(a).

67 
(a) Determine the current, real power, and reactive power when a 60cycle emf of 240 v is applied to the secondary winding of the transformer in Example 62 when the primary winding is open.
(b) Determine the current, real power, and reactive power when a 60cycle emf of 2,400 v is applied to the primary of the transformer above with the secondary open.

68 
The results of an opencircuit test and shortcircuit test at rated frequency on a 500kva 42,000/2,400v, 60cycle transformer are as follows
Opencircuit Test  Voltage applied to Lowvoltage Side
2400 Volts
18.1 Amperes
3785 Watts
Shortcircuit Test  Lowvoltage Side Short Circuited
1910 Volts
11.9 Amperes
3850 Watts
(a) Determine the following quantities referred to the highvoltage side of the transformer.
(i) The exciting admittance, conductance, and susceptance. (ii) The equivalent leakage impedance, (iii) The equivalent resistance.
(b) Repeat part (a), referring all quantities to the lowvoltage side.
(c) Determine the leakage reactance of each winding on the basis that the same amount of equivalent leakage flux links each winding.
(d) Determine the resistance of each winding on the basis of equal amounts of copper and equal current densities in the two windings.
(e) Repeat part (d), but on the basis that the length of the mean turn in the highvoltage winding is 40 percent greater than that in the lowvoltage winding.

69 
Determine the efficiency of the transformer in Problem 68 when delivering at 2,400 secondary v.
(a) Rated load at unity power factor.
(b) At 1/4, 1/2, 3/4, 1 and 5/4 rated load all at 0.80 power factor.

610 
Determine the regulation of the transformer in Problem 68 when delivering rated load at 0.80 power factor
(a) Current lagging.
(b) Current leading.

611 
The transformer in Problem 68 is a core type and has its highvoltage winding surrounding the lowvoltage winding, which is next to the core. If the lowvoltage winding were left unchanged, and the highvoltage winding were
rewound for 69,000 v without a change in the amount of copper and without appreciable change in the space, as to both amount and configuration, what would be the
(a) Equivalent resistance of the transformer referred to
(i) The lowvoltage side? (ii) The highvoltage side?
(b) Equivalent leakage reactance of the transformer referred to
(i) The lowvoltage side? (ii) The highvoltage side?
(c) The exciting current at normal excitation (69,000/2,400v operation) on
(i) The lowvoltage side? (ii) The highvoltage side?

612 
A 240/120v autotransformer delivers a current of 180 amp to a load connected to the low side. Neglect the exciting current and determine the current in each of the windings.

613 
A 2,400/240v transformer has an efficiency of 0.94 at rated load 0.80 power factor when operating as a 2circuit or 2winding transformer. This transformer is reconnected to operate as an autotransformer with the windings carrying their rated currents and feeding a 0.80 power factor load. Show a diagram of connections and determine the efficiency when the autotransformer is connected for operation
(a) 2,640/2,400 v.
(b) 2,400/2,640 v.
(c) 2,400/2,160 v.
(d) 2,640/240 v.

614 
The data for an output transformer are as follows
Resistance of primary winding R_{1} = 225 ohms
Resistance of secondary winding R_{2} = 0.52 ohms
Primary shortcircuit inductance L_{sc1} = 0.10 h
Primary opencircuit inductance L_{oc1} = 5.85 h
Turns ratio a = 20.
This transformer is connected between a source and the voice coil of a loud speaker. The source has an internal impedance of 2,000 ohms and the voice coil has an impedance of 8 ohms. Both of these impedances are practically noninductive. The power output of the transformer is 6 w. Assume the frequency to be of such a value that the exciting current and the leakage reactance of the transformer are negligible, and determine
(a) The secondary current in the transformer.
(b) The ac component in the primary of the transformer.
(c) The internal ac emf of the source.

615 
Suppose that this transformer in Problem 614 were redesigned without any changes in the core and secondary winding, but that the turns in the primary windings were changed, keeping the amount of copper the same as before, so that the impedance of the voice coil exactly matches that of the source and the transformer. If the voltage and frequency were the same as in Problem 614
(a) What would be the new power supplied to the loud speaker voice coil?
(b) What percentage increase is this over the corresponding amount in Problem 614?

616 
Determine for the transformer in Problem 614
(a) The upper and lower halfpower frequencies f_{h} and f_{l}.
(b) The geometric mean frequency f_{0}.
(c) The ratio f_{h}/f_{l}.

617 
Repeat Problem 616 for a transformer having an identical core, the same amount of copper in the windings, but in which the number of turns in the primary and secondary is 20 percent greater than in the corresponding windings of the transformer in Problem 616.

618 
Three singlephase transformers are to be connected for 4,000/440v, 3phase operation. The voltages are line to line. The 3phase rating is to be 750 kva. There are four ways of connecting this transformer bank. Show a diagram of connections for each of the four ways and specify the voltage and current ratings of both windings in each transformer for each of the connections.

619 
Figure 639 shows a bank of three singlephase, onetoone ratio transformers with one side connected in wye to a 416v, 3phase source with the neutral grounded. The other side of the bank is connected in delta with the center tap of one phase grounded. The source is completely ungrounded.

Figure 639. Wyedelta transformer bank for Problem 619 
Determine
(a) The linetoground voltages on the high and low sides of the transformer bank.
(b) The voltages Aa, Ab, and Ac.
(c) The voltages Aa, Ab, and Ac if the polarity of both windings in the middle transformer of Fig. 639 is reversed.
(d) Repeat parts (a) and (b) if Cphase terminal of the 416v side is grounded instead of the neutral,

620 
A 12,000v, 3phase bus supplies (1) a bank of three singlephase transformers connected deltadelta and delivering a balanced 3phase load of 5,000 kva, 0.90 power factor, current lagging, at 4,000 v and (2) a deltawye connected, 3phase transformer delivering a balanced 3phase load of 2,500 kva, 0.71 power factor, current lagging, at 2,300 v. Determine the total load supplied to the two transformer arrangements by the 12,000v bus. Neglect the exciting current and leakage impedance of the transformers.

621 
Two identical 4,000/240v, singlephase transformers are connected open delta and have a 3phase rating of 11.53 kva. What is the 3phase rating if another similar transformer is added for deltadelta operation?

622 
Suppose that windings A and C in the 3phase, coretype transformer in Fig. 629(a) were connected in parallel to a 100v ac source and windings B, a, b, and c were open circuited. How much voltage would be induced in the B winding if
(a) The A and C windings have their marked (polarity) terminals connected together and their unmarked terminals connected together?
(b) The marked terminal of A is connected to the unmarked terminal of C and the unmarked terminal of A is connected to the marked terminal of C? Assume the transformer to be ideal.

623 
Describe the effect of exciting windings A, B, and C in parallel with
(a) Like polarities of all three windings connected together, at rated voltage and frequency from a singlephase source.
(b) The polarity of B winding reversed.

624 
Repeat Problem 622 for the 3phase shelltype transformer in Fig. 629(b), assuming the reluctance of the vertical components in the flux paths to equal that of the horizontal components. For example, in following around the flux path through the windings B and b in the center of the core, each vertical section marked y and each outer horizontal section has twice the reluctance of the horizontal center portion between points marked x.

625 
Repeat Problem 623 for the 3phase shelltype transformer in Fig. 629(b).

626 
Is it possible to operate one or both of the transformers in Fig. 629 at rated voltage and frequency as a singlephase transformer without excessive exciting current by connecting all three primary windings in parallel with each other and all three secondary windings in parallel with each other?

627 
Three singlephase transformers, each rated 2,400/240/120 v and lOkva, have their primaries connected in delta and the center taps on their secondaries to form a neutral for 6phase operation, as shown in Fig. 630. Determine the lineline voltages on the 6phase side and the rated current in the 6phase line wire.

628 
A 6phase forkedwye connection, with the primary connected 3phase delta, converts 13,800v 3phase to 500 v line to line on the 6phase side. The secondaries of each phase are divided into three equal sections. Determine the ratio of primary turns to all the turns in the secondary winding.

629 
Three identical singlephase, 10,000kva transformers are connected delta on the 22kv side and wye on the 132kv side with the neutral on the 132kv side isolated. Balanced 3phase voltages are applied to the 22kv side such that
and the exciting current in the delta between A and B lines is
Express as functions of time
(a) The voltages v_{BC} and v_{CA} on the 22kv side.
(b) The exciting currents i_{excBC} and i_{excCA} m the delta.
(c) The noload current i_{excA}, i_{excB}, i_{excC} the 22kv lines.

630 
The three singlephase transformers of Problem 629 are operating without load and are excited from the 132kv side with the neutral connected to the source. Assume aphase, linetoneutral voltage on the 132kv side to be in phase with the linetoline voltage between A and B phases on the 22kv side. If the delta connection is opened
(a) How do the exciting currents i_{exca} i_{excb} and i_{excc} vary as functions of time?
(b) How does the neutral current i_{n} vary as a function of time ?

631 
The three singlephase transformers of Problems 629 and 630 are operating without load and are excited from the 132kv side with the neutral isolated and the delta closed. Express the exciting currents i_{exca} i_{excb} and i_{excc} as functions of time.

632 
The distribution transformer in Fig. 635 delivers 30 amp at 120 v, 1.00 power factor a to n; 40 amp at 120 v, 0.707 power factor, current lagging b to n; 25 amp at 240 v, 0.90 power factor, current lagging a to b. Neglect transformer impedance and admittance and calculate the primary current and the primary power factor.

633 
The transformer bank in Example 68 delivers a load of 60,000 kva, 0.80 power factor, current lagging to a balanced 3phase load on the 11kv side. There is no load on the 2.2kv side. Neglect resistance and determine the voltage on
(a) The 66kv side.
(b) The 2.2kv side.
if the load voltage is 11kv line to line.

634 
Determine for the transformer of Example 62 the following base quantities
(a) On the high side.
(b) On the low side using the rating of the transformer for the base power,
(i) Current, (ii) Voltage, (iii) Impedance

635 
Determine the per unit values of
(a) Exciting current
(b) Resistance
(c) Impedance
(d) Leakage reactance
of the transformer in Example 62, using the rating of the transformer as the base power.

636 
Repeat Problem 635, but for a base kva of 450.

637 
Determine the per unit resistance, reactance, and exciting current of the transformer in Problems 68 and 611, using the transformer rating as the base.

638 
A 3phase transformer rated at 150,000 kva, 138/13.8 kv has an impedance of 0.10 per unit and is operating at 132/13.2 kv. Calculate the per unit impedance for a 1,000,000kva base and 132kv base.
