As seen in 10.66, only two independent PRACTICAL power specifications can be made when

For each of the following sets of voltage and current, calculate the real and reactive power in the line between networks A and B in the circuit shown. In each case, state whether the power flow is from A to B or vice versa. Also state whether magnetizing vars are being transferred from A to B or vice versa. a) v = 100 cos (at - 45) V; i = 20cos(wr + 15) A. b) v = 100 cos (cot - 45) V; i = 20cos(&rf + 165) A. c) v = 100 cos (at - 45) V; i = 20 cos (w* - 105) A. d) V = 100 cos at V; i = 20 cos (art + 120) A. A i ^- + V B

Compute the power factor and the reactive factor for the network inside the box in Fig. 10.6, whose voltage and current are described in Example 10.1. Hint: Use -i to calculate the power and reactive factors.

The periodic triangular current in Example 9.4, repeated here, has a peak value of 180 mA. Find the average power that this current delivers to a 5 kQ, resistor.

The load impedance in the circuit shown is shunted by a capacitor having a capacitive reactance of -52 0,. Calculate: a) the rms phasors VL and IL, b) the average power and magnetizing reactive power absorbed by the (39 + /26) 1 load impedance, c) the average power and magnetizing reactive power absorbed by the (1 + /4) O line impedance, d) the average power and magnetizing reactive power delivered by the source, and e) the magnetizing reactive power delivered by the shunting capacitor.

The rms voltage at the terminals of a load is 250 V The load is absorbing an average power of 40 kW and delivering a magnetizing reactive power of 30 kVAR. Derive two equivalent impedance models of the load.

Find the phasor voltage Ys (rms) in the circuit shown if loads Lx and L2 are absorbing 15 kVA at 0.6 pf lagging and 6 kVA at 0.8 pf leading, respectively. Express Vv in polar form.

The source current in the circuit shown is 3.6 mH 3cos5000r A. a) What impedance should be connected across terminals a,b for maximum average power transfer? b) What is the average power transferred to the impedance in (a)? c) Assume that the load is restricted to pure resistance. What size resistor connected across a,b will result in the maximum average power transferred? d) What is the average power transferred to the resistor in (c)?

Find the average power delivered to the 100 Q resistor in the circuit shown if vg = 660 cos 5000r V.

a) Find the average power delivered to the 400 fl resistor in the circuit shown if vt = 248 cos 10,000f V. b) Find the average power delivered to the 375 H resistor. c) Find the power developed by the ideal voltage source. Check your result by showing the power absorbed equals the power developed. NOTE: Also try Chapter Problems 10.64 and 10.65. 40 mH 50 mH lOOmH l'YW^ :37512 400 O

Solve Example 10.11 if the voltage source is reduced to 146 /0 V rms and the turns ratio is reversed to 1:4.

The load impedance in Fig. PIO.IO absorbs 2500 W and generates 5000 magnetizing vars. The sinusoidal voltage source develops 7500 W. a) Find the values of inductive line reactance that will satisfy these constraints. b) For each value of line reactance found in (a), show that the magnetizing vars developed equals the magnetizing vars absorbed. Figure PIO.IO i 20 a jxn -f -VW rm-y j_ Source ~Y* Load

A dc voltage equal to Vt\c V is applied to a resistor of R il. A sinusoidal voltage equal to vs V is also applied to a resistor of R Cl. Show that the dc voltage will deliver the same amount of energy in T seconds (where T is the period of the sinusoidal voltage) as the sinusoidal voltage provided Kdc equals the rms value of vs . (Hint: Equate the two expressions for the energy delivered to the resistor.)

Find the rms value of the periodic current shown in Fig. P10.13.

The periodic current shown in Fig. P10.13 dissipates an average power of 1280 W in a resistor. What is the value of the resistor?

a) Find the rms value of the periodic voltage shown inFig.P10.15. b) If this voltage is applied to the terminals of a 411 resistor, what is the average power dissipated in the resistor? Figure P10.15 tfc(V)" 20 10 10 -20 - elc. 25 50 75 100 125 150 175 200 /(/AS) Sections 10.4-10.5

The voltage \g in the frequency-domain circuit shown in Fig. P10.18 is 240/O^V (rms). a) Find the average and reactive power for the voltage source. b) Is the voltage source absorbing or delivering average power? c) Is the voltage source absorbing or delivering magnetizing vars? d) Find the average and reactive powers associated with each impedance branch in the circuit. e) Check the balance between delivered and absorbed average power. f) Check the balance between delivered and absorbed magnetizing vars. Figure P10.16

Find the average power, the reactive power, and the pspicE apparent power absorbed by the load in the circuit MULTISIM i n Fi g pi(U 7 i f . equal s 4 0 CQ S 5000 f m A Figure P10.17 5kH --W\> i 2H 20 nF Load

a) Find the average power, the reactive power, and PSPICE the apparent power supplied by the voltage MULTISIM source in the circuit in Fig. PI 0.18 if vv = 40 cos 10 V. b) Check your answer ^dev = Scabs' c) Check your answer Qdev = 2J Sabsill (a) by showing in (a) by showing Figure P10.18 40 fl 25 nF 80yLiH 60 n

Two 480 V (rms) loads are connected in parallel. The two loads draw a total average power of 40,800 W at a power factor of 0.8 lagging. One of the loads draws 20 kVA at a power factor of 0.96 leading. What is the power factor of the other load?

a) Find VL (rms) and 0 for the circuit in Fig. P10.20 if the load absorbs 2500 VA at a lagging power factor of 0.8. b) Construct a phasor diagram of each solution obtained in (a).

The two loads shown in Fig. PI0.21 can be described as follows: Load 1 absorbs an average power of 60 kW and delivers 70 kVAR magnetizing reactive power; load 2 has an impedance of 24 -+ /7.The voltage at the terminals of the loads is 2500V2 cos 1207rf V. a) Find the rms value of the source voltage. b) By how many microseconds is the load voltage out of phase with the source voltage? c) Does the load voltage lead or lag the source voltage? Figure P10.21 0.05 a /0.5 a - o.05 a yo.5 a

The three loads in the circuit seen in Fig. PI0.22 are described as follows: Load 1 is absorbing 7.5 kW and 2500 VAR; load 2 is absorbing 10 kVA at a 0.28 pf lead; load 3 is a 12.5 H resistor in parallel with an inductor that has a reactance of 50 fi. a) Calculate the average power and the magnetizing reactive power delivered by each source if Vgi = \g2 = 250/0^ V (rms). b) Check your calculations by showing your results are consistent with the requirements E^dev = 2fl abs SGicv - 2<2 abs* Figure P10.22 L, u

Suppose the circuit shown in Fig. PI 0.22 represents a residential distribution circuit in which the impedances of the service conductors are negligible and Vgl = \ s 2 = 120/0^ V (rms). The three loads in the circuit are Lx (a coffee maker and a frying pan); L2 (a room air conditioner, a hairdryer, and a tubetype television set); and L3 (a quick recovery water heater and a range with an oven). Assume that all these appliances are in operation at the same time. The service conductors are protected with 100 A circuit breakers. Will the service to this residence be interrupted? Explain.

The three loads in the circuit shown in Fig. PI0.24 are $\ = 5 + /1.25 kVA, S2 = 6.25 + /2.5 kVA, and 53 = 8 + /0 kVA. a) Calculate the complex power associated with each voltage source, V^ and Vg2. b) Verify that the total real and reactive power delivered by the sources equals the total real and reactive power absorbed by the network. Figure P10.24 0.05 H AMr 0.15 (2 /yW 0.05 ft ^Wv 125/0V(rms) 125ZQ!V(rras) S3

The three parallel loads in the circuit shown in Fig. PI0.25 can be described as follows: Load 1 is absorbing an average power of 7.5 kW and 9 kVAR of magnetizing vars; load 2 is absorbing an average power of 2.1 kW and generating 1.8 kVAR of magnetizing reactive power; load 3 consists of a 48 ft resistor in parallel with an inductive reactance of 19.2 ft. Find the rms magnitude and the phase angle of V,, if V = 480/(T V (rms). Figure P10.25 /0.5 ft /YV W u

The three loads in the circuit in Fig. PI 0.26 can be described as follows: Load 1 is a 240 ft resistor in series with an inductive reactance of 70 ft; load 2 is a capacitive reactance of 120 ft in series with a 160 ft resistor; and load 3 is a 30 ft resistor in series with a capacitive reactance of 40 ft. The frequency of the voltage source is 60 Hz. a) Give the power factor and reactive factor of each load. b) Give the power factor and reactive factor of the composite load seen by the voltage source. Figure PI0.26 Load 1 Load 2 Load 3

Three loads are connected in parallel across a 250 V (rms) line, as shown in Fig. P10.27. Load 1 absorbs 16 kW and 18 kVAR. Load 2 absorbs 10 kVA at 0.6 pf lead. Load 3 absorbs 8 kW at unity power factor. a) Find the impedance that is equivalent to the three parallel loads. b) Find the power factor of the equivalent load as seen from the line's input terminals. Figure P10.27 < + 250 V (rms) > <> 1 <> 2 <> 3

The three loads in Problem 10.27 are fed from a line having a series impedance 0.01 + /0.08 ft, as shown in Fig. PI0.28. a) Calculate the rms value of the voltage (V5) at the sending end of the line. b) Calculate the average and reactive powers associated with the line impedance. c) Calculate the average and reactive powers at the sending end of the line. d) Calculate the efficiency (17) of the line if the efficiency is defined as V = (^load/^sending end) X 1M). Figure P10.28 0.01 n /0.08 n + v, ^ T r. Li n L2 ( L* 250.0 V (rms)

a) Find the average power dissipated in the line in Fig. P10.29. b) Find the capacitive reactance that when connected in parallel with the load will make the load look purely resistive. c) What is the equivalent impedance of the load in (b)? d) Find the average power dissipated in the line when the capacitive reactance is connected across the load. e) Express the power loss in (d) as a percentage of the power loss found in (a).

The steady-state voltage drop between the load and the sending end of the line seen in Fig. PI0.30 is excessive. A capacitor is placed in parallel with the 200 kVA load and is adjusted until the steady-state voltage at the sending end of the line has the same magnitude as the voltage at the load end, that is, 6400 V (rms). The 200 kVA load is operating at a power factor of 0.6 lag. Calculate the size of the capacitor in microfarads if the circuit is operating at 60 Hz. In selecting the capacitor, keep in mind the need to keep the power loss in the line at a reasonable level. Figure P10.30 4ft -Wv /24 ft + V, 64()().0 V (rms) 200 kVA 0.6 lag

Consider the circuit described in Problem 9.75. a) What is the rms magnitude of the voltage across the load impedance? b) What percentage of the average power developed by the practical source is delivered to the load impedance?

A factory has an electrical load of 1600 kW at a lagging power factor of 0.8. An additional variable power factor load is to be added to the factory. The new load will add 320 kW to the real power load of the factory. The power factor of the added load is to be adjusted so that the overall power factor of the factory is 0.96 lagging. a) Specify the reactive power associated with the added load. b) Does the added load absorb or deliver magnetizing vars? c) What is the power factor of the additional load? d) Assume that the voltage at the input to the factory is 2400 V (rms). What is the rms magnitude of the current into the factory before the variable power factor load is added? e) What is the rms magnitude of the current into the factory after the variable power factor load has been added?

Assume the factory described in Problem 10.32 is fed from a line having an impedance of 0.05 + /0.4 ft. The voltage at the factory is maintained at 2400 V (rms). a) Find the average power loss in the line before and after the load is added. b) Find the magnitude of the voltage at the sending end of the line before and after the load is added.

A group of small appliances on a 60 Hz system requires 20 kVA at 0.85 pf lagging when operated at 125 V (rms). The impedance of the feeder supplying the appliances is 0.01 + /0.08 0,. The voltage at the load end of the feeder is 125 V (rms). a) What is the rms magnitude of the voltage at the source end of the feeder? b) What is the average power loss in the feeder? c) What size capacitor (in microfarads) across the load end of the feeder is needed to improve the load power factor to unity? d) After the capacitor is installed, what is the rms magnitude of the voltage at the source end of the feeder if the load voltage is maintained at 125 V (rms)? e) What is the average power loss in the feeder for (d)?

a) Find the six branch currents Ia - If in the circuit in Fig. PI 0.35. b) Find the complex power in each branch of the circuit. c) Check your calculations by verifying that the average power developed equals the average power dissipated. d) Check your calculations by verifying that the magnetizing vars generated equal the magnetizing vars absorbed. Figure P10.35 lft - I c /I ft ^...-/1 ft10/0! V (rms) I, T' . /2 ft -/1 ft 1 ft

a) Find the average power delivered to the 8 ft resistor in the circuit in Fig. P10.36. b) Find the average power developed by the ideal sinusoidal voltage source. c) FindZab. d) Show that the average power developed equals the average power dissipated.

a) Find the average power dissipated in each resistor in the circuit in Fig. PI0.37. b) Check your answer by showing that the total power developed equals the total power absorbed. Figure P10.37 15ft O 100/0! V (rms) /50 ft 1400 Turns lft -WV100 Turns -j2 ft

The sinusoidal voltage source in the circuit in Fig. P10.38 is developing an rms voltage of 2000 V. The 4 ft load in the circuit is absorbing four times as much average power as the 25 ft load. The two loads are matched to the sinusoidal source that has an internal impedance of 500/0 kft. a) Specify the numerical values of a^ and a2. b) Calculate the power delivered to the 25 ft load. c) Calculate the rms value of the voltage across the 4 ft resistor. Figure P10.38 500 ft: 2000/0 / + V(rms)U ffi':l Ideal | a2:l Ideal :25ft :4ft

a) Find the average power delivered by the sinusoidal current source in the circuit of Fig. PI0.39. b) Find the average power delivered to the 20 ft resistor. Figure P10.39 5/0 A (rms) :60ft1 4:1 Ideal :40ft :20ft

Prove that if only the magnitude of the load impedance can be varied, most average power is transferred to the load when |ZL| = |ZTiJ. (Hint: In deriving the expression for the average load power, write the load impedance (ZL) in the form ZL = |ZL| cos 6 + j\ZL\ sin 6, and note that only |ZL| is variable.)

For the frequency-domain circuit in Fig. PI0.41, calculate: a) the rms magnitude of V(). b) the average power dissipated in the 9 ft resistor. c) the percentage of the average power generated by the ideal voltage source that is delivered to the 9 ft load resistor. Figure P10.41 180/0! / + V(rms)U 3ft vw/4 ft /3ftx ^ j /9 ft v<> i 9 ft

The 9ft resistor in the circuit in Fig. PI 0.41 is replaced with a variable impedance Za. Assume Z(> is adjusted for maximum average power transfer to Z0. a) What is the maximum average power that can be delivered to ZJl b) What is the average power developed by the ideal voltage source when maximum average power is delivered to Z01 c) Choose single components from Appendix H to form an impedance that dissipates average power closest to the value in part (a). Assume the source frequency is 60 Hz.

Suppose an impedance equal to the conjugate of the Thevenin impedance is connected to the terminals c,d of the circuit shown in Fig. P9.78. a) Find the average power developed by the sinusoidal voltage source. b) What percentage of the power developed by the source is lost in the linear transformer?

a) Determine the load impedance for the circuit shown in Fig. PI0.44 that will result in maximum average power being transferred to the load if a) = 5 krad/s. b) Determine the maximum average power delivered to the load from part (a) if vg = 80 cos 5000f V. c) Repeat part (a) when ZL. consists of two components from Appendix H whose values yield a maximum average power closest to the value calculated in part (b). Figure P10.44 8mH ATY V 5/ttF -1(- 40 O

The phasor voltage Vab in the circuit shown in Fig. P10.45 is 240/fT V (rms) when no external load is connected to the terminals a,b.When a load having an impedance of 90 - /30 fl is connected across a,b, the value of Vab is 115.2 - /86.4 V (rms). a) Find the impedance that should be connected across a,b for maximum average power transfer. b) Find the maximum average power transferred to the load of (a). c) Construct the impedance of part (a) using components from Appendix H if the source frequency is 60 Hz. Figure P10.45 A circuit operating in the sinusoidal steady , state a- +

The load impedance ZL for the circuit shown in Fig. PI 0.46 is adjusted until maximum average power is delivered to ZL. a) Find the maximum average power delivered to ZL.

The peak amplitude of the sinusoidal voltage PSPICE source in the circuit shown in Fig. P10.47 is 180 V, MULTISIM an d its frequeriCy j s 5000 rad/s. The load resistor can be varied from 0 to 4000 ft, and the load capacitor can be varied from 0.1 ^F to 0.5 /xF. a) Calculate the average power delivered to the load when R0 = 2000 ft and C0 = 0.2 /iF. b) Determine the settings of R0 and C0 that will result in the most average power being transferred to R0. c) What is the most average power in (b)? Is it greater than the power in (a)? d) If there are no constraints on R() and Ca, what is the maximum average power that can be delivered to a load? e) What are the values of R0 and Ca for the condition of (d)? f) Is the average power calculated in (d) larger than that calculated in (c)?

a) Assume that R6 in Fig. P10.47 can be varied PSPICE between 0 and 10 kXl. Repeat (b) and (c) of MULTISIH p rob i e m 10.47. b) Is the new average power calculated in (a) greater than that found in Problem 10.47(a)? c) Is the new average power calculated in (a) less than that found in 10.47(d)?

The variable resistor in the circuit shown in Fig. P10.49 is adjusted until the average power it absorbs is maximum. a) FindR. b) Find the maximum average power.c) Find a resistor in Appendix H that would have the most average power delivered to it. Figure P10.49 6ft -/18X1 20 O J60 n

The variable resistor R() in the circuit shown in Fig. P10.50 is adjusted until maximum average power is delivered to Ra. a) What is the value of R() in ohms? b) Calculate the average power delivered to Rt}. c) If Ra is replaced with a variable impedance Z0% what is the maximum average power that can be delivered to Z? d) In (c), what percentage of the circuit's developed power is delivered to the load Zal

The values of the parameters in the circuit shown in Fig. P10.51 are Lt = 8mH; L2 = 2mH; k = 0.75; Rg = 1 O; and RL = 7 0 . If vs = 54 V2 cos 1000* V, find a) the rms magnitude of vQ b) the average power delivered to RL c) the percentage of the average power generated by the ideal voltage source that is delivered to RL.

Assume the coefficient of coupling in the circuit in Fig. PI0.51 is adjustable. a) Find the value of k that makes va equal to zero. b) Find the power developed by the source when k has the value found in (a).

Assume the load resistor (RL) in the circuit in Fig. PI0.51 is adjustable. a) What value of RL will result in the maximum average power being transferred to RjJI b) What is the value of the maximum power transferred?

The sending-end voltage in the circuit seen in Fig. PI0.54 is adjusted so that the rms value of the load voltage is always 4000 V. The variable capacitor is adjusted until the average power dissipated in the line resistance is minimum. a) If the frequency of the sinusoidal source is 60 Hz, what is the value of the capacitance in microfarads? b) If the capacitor is removed from the circuit, what percentage increase in the magnitude of V, is necessary to maintain 4000 V at the load? c) If the capacitor is removed from the circuit, what is the percentage increase in line loss?

Find the impedance seen by the ideal voltage source in the circuit in Fig. PI 0.55 when Z() is adjusted for maximum average power transfer to Z0.

The impedance ZL in the circuit in Fig. P10.56 is adjusted for maximum average power transfer to ZL- The internal impedance of the sinusoidal voltage source is 4 + /7 O. a) What is the maximum average power delivered toZL ? b) What percentage of the average power delivered to the linear transformer is delivered to ZL?

a) Find the steady-state expression for the currents PSPICE MULTISIM /g and iL in the circuit in Fig. PI 0.57 when vg = 200 cos 10,000; V. b) Find the coefficient of coupling. c) Find the energy stored in the magnetically coupled coils at t - 50TT /xs and t = 100-77 /is. d) Find the power delivered to the 15 H resistor. e) If the 15X1 resistor is replaced by a variable resistor i?L, what value of RL will yield maximum average power transfer to i?L? f) What is the maximum average power in (e)? g) Assume the 15 H resistor is replaced by a variable impedance ZL. What value of ZL will result in maximum average power transfer to ZL? h) What is the maximum average power in (g)? Figure P10.57 0.5 mH lmH 0 15 ft

The sinusoidal voltage source in the circuit in Fig. P10.58 is operating at a frequency of 20 krad/s. The variable capacitive reactance in the circuit is adjusted until the average power delivered to the 100 H resistor is as large as possible. a) Find the value of C in microfarads. b) When C has the value found in (a), what is the average power delivered to the 100 H resistor? c) Replace the 100 11 resistor with a variable resistor R(). Specify the value of R() so that maximum average power is delivered to Ra. d) What is the maximum average power that can be delivered to R,,? Figure P10.58 15 ar A(r%Uj2iHl} Vl0il 1-5 L Ideal 100 Q

a) Find the turns ratio NJN2 for the ideal transformer in the circuit in Fig. PI0.59 so that maximum average power is delivered to the 400 a load. b) Find the average power delivered to the 400 a load. c) Find the voltage \ { . d) What percentage of the power developed by the ideal current source is delivered to the 400 11 resistor?

Tlie load impedance ZL in the circuit in Fig. P10.60 is adjusted until maximum average power is transferred to ZL. a) Specify the value of ZL if N\ = 3600 turns and N2 = 600 turns. b) Specify the values of I L and VL when ZL is absorbing maximum average power.

a) If TVi equals 1000 turns, how many turns should be placed on the N2 winding of the ideal transformer in the circuit seen in Fig. P10.61 so that maximum average power is delivered to the 6800 11 load? b) Find the average power delivered to the 6800 H resistor. c) What percentage of the average power delivered by the ideal voltage source is dissipated in the linear transformer?

The variable load resistor RL in the circuit shown in PSPICE Fig. PI0.62 is adjusted for maximum average power MULTISIM transfer to i?L. a) Find the maximum average power. b) What percentage of the average power developed by the ideal voltage source is delivered to RL when RL is absorbing maximum average power? c) Test your solution by showing that the power developed by the ideal voltage source equals the power dissipated in the circuit. Figure P10.62 80 ft 20 ft 40 ft 1 1:2 i 500/0 f + V (rms) 360 ft

Repea t Problem 10.62 for the circuit shown in PSPICE Fig. P10.63.

Find the average power delivered to the 10 ft resistor in the circuit of Fig. P10.64. Figure P10.64 2.5 ft a l :201 - 7|30:1 1 ( i i \ ) \ Ideal I [ideal] 10ft

The ideal transformer connected to the 10 ft load in Problem 10.64 is replaced with an ideal transformer that has a turns ratio of a:\. a) What value of a results in maximum average power being delivered to the 10 ft resistor? b) What is the maximum average power?

The hair dryer in the Practical Perspective uses a piwrecnvi 6 0 H z smusoida l voltage of 120 V (rms). The heater element must dissipate 250 W at the LOW setting, 500 W at the MEDIUM setting, and 1000 W at the HIGH setting. a) Find the value for resistor R2 using the specification for the MEDIUM setting, using Fig. 10.31. b) Find the value for resistor Ri using the specification for the LOW setting, using the results from part (a) and Fig. 10.30. c) Is the specification for the HIGH setting satisfied?

As seen in Problem 10.66, only two independent PRACTICAL power specifications can be made when two resisPERSPECTIVE * r PSPICE tors make up the healing element of the hair dryer. a) Show that the expression for the HIGH power rating (PH) is PH = Pi PA pr where PM = the MEDIUM power rating and PL = the LOW power rating. b) If PL = 250 W and PM = 750 W, what must the HIGH power rating be?

Specify the values of R\ and R2 in the hair dryer cirpRAcncAt cu i t in Fig. 10.29 if the low power rating is 240 W PERSPECTIVE b r 0 PSPICE and the high power rating is 1000 W Assume the MULTISIM supply voltage is 120 V (rms). (Hint: Work Problem 10.67 first.)

If a third resistor is added to the hair dryer circuit in PRACTICAL Fig. 10.29, it is possible to design to three independPSPICE ent power specifications. If the resistor R$ is added MULTISIM j n ser i e s w i t h the Thermal fuse, then the corresponding LOW, MEDIUM, and HIGH power circuit diagrams are as shown in Fig. P10.69. If the three power settings are 600 W, 900 W, and 1200 W, respectively, when connected to a 120 V (rms) supply, what resistor values should be used? Figure P10.69 LOW MEDIUM HIGH

You have been given the job of redesigning the hair PRACTICAL dryer described in Problem 10.66 for use in 'ERSPECTIVE J DESIGN England. The standard supply voltage in England is PROBLEM M. -mf i~J f PSPICE 220 V (rms). What resistor values will you use in MULTISIM your design to meet the same power specifications?

Repeat Problem 10.68 using single resistor values from Appendix H. Calculate the resulting low, medium, and high power ratings.

Repeat Problem 10.70 using single resistor values from Appendix H. Calculate the resulting low, medium, and high power ratings.