Energy Systems 344 Energiestelsels 344 Prof Rong-Jie Wang Department of Electrical and Electronic Engineering Stellenbosch University South Africa Introduction to AC Machines 1 / 60 Outline 1 Basic Concepts 2 4.3 – Distributed Windings 3 Multi-pole Machines 2 / 60 Basic Concepts A group coils carrying AC currents called an armature winding - AC machines: the armature is typically on the stator - DC machines: the armature is located on the rotor Usually both DC and synchronous machines have DC field windings to set up the main operating flux, which are located on; - the stator for DC machines - the rotor of AC synchronous machines (a) (b) 3 / 60 4.3 – Concentrated Windings Each phase winding consists of one coil d a is the d - or magnetic axis of phase a The magnetic axis of phase b , or d b lags d a by 120 ◦ The magnetic axis of phase c , or d c lags d a by 240 ◦ a d a a 1 a 1 b 1 b 1 c 1 c 1 4 / 60 4.3 – Concentrated Windings (cont . . . ) The (magnetic) flux in the rotor & stator caused by phase a , can be determined using the right hand grip rule See Umans 7 th Edition, Fig. 4.16 5 / 60 4.3 – Concentrated Windings (cont . . . ) The magnetic flux causes an MMF-drop which divides 50:50 across the air-gap where the flux “flows” out from the rotor to the stator on the right and where the flux “flows” back into the rotor from the stator on the left. a d a a 1 a 1 F ag F ag F ag F ag F ag F ag F ag F ag F ag F ag F ag F ag F ag F ag F ag F ag F ag F ag F ag F ag F ag F ag 6 / 60 4.3 – Concentrated Windings (cont . . . ) With the total MMF produced by phase a , equal to: F a ( t ) = Ni a ( t ) The MMF-drops can be represented by an equivalent magnetic circuit model, where we assume that the permeability of the rotor and stator is infinite. − + F a ( t )= Ni a ( t ) φ ( t ) R ag + Ni a ( t ) 2 − R ag + Ni a ( t ) 2 − 7 / 60 4.3 – Concentrated Windings (cont . . . ) The MMF distribution in the air-gap can thus be represented as follows, where the flux out is taken as positive and the flux in as negative. d a θ ae 0 ◦ 30 ◦ 90 ◦ 150 ◦ 210 ◦ 270 ◦ 330 ◦ 360 ◦ − a 1 a 1 NI max 2 − NI max 2 F ag [ At ] ˆ F ag 1 8 / 60 Fourier Analysis For reasons that will become apparent later, we are only interested in the fundamental “spacial” MMF distribution in the air-gap, F a 1 In order to calculate the fundamental spacial component of the MMF distribution, we make use of the Fourier series expansion of the MMF distribution. Because the MMF distribution is angle ( θ a ) dependant and not time ( t ) dependant, we have to work in “degrees/radians” and not in “seconds”. It is important to note that although the current in phase a might be sinusoidal, only the magnitude of the MMF will vary sinusoidally and not the MMF distribution. 9 / 60 Fourier Analysis (cont . . . ) From Nilsson & Riedel 16.2: The “time-domain” Fourier Coefficients a v = 1 T ˆ t 0 + T t 0 f ( t ) dt a n = 2 T ˆ t 0 + T t 0 f ( t ) cos( n ω 0 t ) dt b n = 2 T ˆ t 0 + T t 0 f ( t ) sin( n ω 0 t ) dt 10 / 60 Fourier Analysis (cont . . . ) With: θ = ω 0 t ∴ d θ dt = ω 0 = 2 π f = 2 π T ⇒ dt = T 2 π d θ with the integration boundaries changing from: t 0 → θ 0 & t 0 + T → θ 0 + 2 π 11 / 60 Fourier Analysis (cont . . . ) The “spacial-domain” Fourier Coefficients a v = 1 2 π ˆ θ 0 + 2 π θ 0 f ( t ) d θ a n = 1 π ˆ θ 0 + 2 π θ 0 f ( t ) cos( n θ ) d θ b n = 1 π ˆ θ 0 + 2 π θ 0 f ( t ) sin( n θ ) d θ 12 / 60 Fourier Analysis (cont . . . ) From Nilsson & Riedel, Section 16.3, - for Even-Function Symmetry: f ( θ ) = f ( − θ ) (16.13’) ∴ b n = 0 (16.16) - for Half-Wave Symmetry: f ( θ ) = − f ( θ − π ) (16.26’) ∴ a v = 0 (16.27) - for Quarter-Wave Symmetry: a n = 4 π ˆ π 2 0 f ( θ ) cos n θ d θ for n odd 0 for n even (16.36’) 13 / 60 Fourier Analysis (cont . . . ) therefore, for n uneven, the a n Fourier Coefficients for the spacial MMF distribution can be calculated as follows (with θ = θ a ): a n = 4 π ˆ π 2 0 F ag ( θ a ) cos( n θ a ) d θ a = 4 π ˆ π 2 0 Ni a ( t ) 2 cos( n θ a ) d θ a = 4 π ( Ni a ( t ) 2 ) sin ( n π 2 ) n 14 / 60 Fourier Analysis (cont . . . ) The Fourier series expansion of the spacial MMF distribution in the air-gap, can thus be written as follows: F ag ( θ a ) = 4 π ( Ni a ( t ) 2 ) ∞ ∑ n = 1 , 3 , 5 ,... sin ( n π 2 ) n · cos( n θ a ) With the fundamental component of the spacial MMF distribution in the air-gap ( n = 1 ) : F ag 1 ( θ a ) = 4 π ( Ni a ( t ) 2 ) · cos( θ a ) (4.4’) and an amplitude or peak value of: ˆ F ag 1 = 4 π ( Ni a ( t ) 2 ) (4.5’) 15 / 60 Fourier Analysis (cont . . . ) The amplitudes of the harmonic components of the spacial MMF distribution normalised with respect to the fundamental MMF component, Harmonic Amplitude 1 1 000 3 − 0 333 5 0 200 7 − 0 143 9 0 111 11 − 0 091 Negative values indicate a 180 ◦ phase shift for that specific spacial harmonic. 16 / 60 Fourier Analysis (cont . . . ) The spacial harmonic MMF components will be much smaller than the fundamental spacial component so that we can usually ignore them and use only the fundamental spacial component. a d a a 1 a 1 F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) F ag 1 ( a ) 17 / 60 Fourier Analysis (cont . . . ) The fundamental spacial MMF component can be modeled, similar to a phasor, as a space vector which indicates the magnitude and relative angular displacement of the fundamental MMF component. a d a a 1 a 1 F ag 1 = ˆ F ag 1 0 18 / 60 4.3 – Distributed Windings By making use of distributed windings, the amplitude of the spacial harmonic MMF distribution components can be drastically reduced. a d a a 2 a 2 a 1 a 1 19 / 60 4.3 – Distributed Windings (cont . . . ) Each phase winding now consists of two coils with N turns each, i.e. N ph = 2 N The MMF distribution now changes from a square wave shaped distribution to a more stepped wave distribution. d a θ ae 0 ◦ 15 ◦ 45 ◦ 75 ◦ 105 ◦ 135 ◦ 165 ◦ 195 ◦ 225 ◦ 255 ◦ 285 ◦ 315 ◦ 345 ◦ 360 ◦ − a 1 − a 2 a 1 a 2 N ph I max 2 − N ph I max 2 F ag [ At ] ˆ F ag 1 γ 20 / 60 4.3 – Distributed Windings (cont . . . ) The shape of the MMF distribution is still quarter-wave symmetrical. Consequently the Fourier series expansion of the MMF distribution will only consist of uneven a n Coefficients, and can once again be calculated using the following equation: a n = 4 π ˆ π 2 0 F ag ( θ a ) cos( n θ a ) d θ a for n odd 0 for n even A handy trick which we will later on use again, is to make use of the angle γ in the integration interval for the calculation of a n 21 / 60 4.3 – Distributed Windings (cont . . . ) For this MMF distribution, γ = 1 2 ( 360 ◦ slots ) = 15 ◦ where the number of slots, slots = 12 22 / 60 4.3 – Distributed Windings (cont . . . ) Thus: a n = 4 π ˆ π 2 − γ 0 N ph i a ( t ) 2 cos( n θ a ) d θ a = 4 π ( N ph i a ( t ) 2 ) [ sin( n θ a ) n ] π 2 − γ 0 = 4 π ( N ph i a ( t ) 2 ) sin( n π 2 − n γ ) n By making use of the following trigonometric identity: sin( α − β ) = sin α cos β − cos α sin β 23 / 60 4.3 – Distributed Windings (cont . . . ) and with a n = 0 for even values of n Definition of a n for a stepped waveform a n = 4 π ( N ph i a ( t ) 2 ) sin( n π 2 ) n cos( n γ ) The winding factor, k w n , is now defined as: k w n = cos( n γ ) 24 / 60 4.3 – Distributed Windings (cont . . . ) The Fourier series expansion of the spacial MMF distribution in the air-gap, can thus be written as follows: F ag ( θ a ) = 4 π ( N ph i a ( t ) 2 ) ∞ ∑ n = 1 , 3 , 5 ,... sin ( n π 2 ) n k w n · cos( n θ a ) With the fundamental component of the spacial MMF distribution in the air-gap ( n = 1 ) : F ag 1 ( θ a ) = 4 π ( k w 1 N ph 2 ) i a ( t ) · cos( θ a ) (4.6’) 25 / 60 4.3 – Distributed Windings (cont . . . ) and an amplitude or peak value of: ˆ F ag 1 = 4 π ( k w 1 N ph 2 ) i a ( t ) (4.7’) The fundamental winding factor for this winding configuration: k w 1 = cos( n γ ) ∣ ∣ n = 1 = cos( γ ) = cos( 15 ◦ ) = 0 966 The product k w 1 N ph is known as the effective series turns per phase 26 / 60 4.3 – Distributed Windings (cont . . . ) What this implies is that this 12–slot stator with N ph = 2 N series turns per phase, will have the same fundamental MMF distribution component as a 6–slot stator (which we analysed first) with k w 1 N ph = k w 1 2 N series turns per phase. Although the fundamental component of the MMF distribution of the distributed windings is now 0.966 times smaller, compared to the concentrated winding, - the big difference lie in the amplitudes of the spacial harmonic MMF distribution components as shown below: 27 / 60 4.3 – Distributed Windings (cont . . . ) Harmonic Amplitude (12-slot) Amplitude (6-slot) 1 0 966 1 000 3 − 0 236 − 0 333 5 0 052 0 200 7 − 0 037 − 0 143 9 0 079 0 111 11 − 0 088 − 0 091 It is especially the amplitudes of the 5 th and 7 th spacial harmonic component of the MMF distribution which were dramatically reduced in comparison to that of the 6–slot concentrated winding stator. 28 / 60 4.3 – Distributed Windings (cont . . . ) Each spacial harmonic MMF distribution component is now, k w n = cos( n γ ) times smaller than the corresponding spacial harmonic MMF distribution component of the concentrated winding configuration. 29 / 60 4.3 – Distributed Windings (cont . . . ) We would also be able to make use of MMF space vectors in order to calculate the magnitude of the fundamental component of the MMF distribution, which coil/winding a 1 and a 2 produces together. a d a a 2 a 2 a 1 a 1 ˆ F a 1 g 1 ˆ F a 2 g 1 F ag 1 30 / 60 4.3 – Distributed Windings (cont . . . ) By making use of inspection, we can say that: F a 1 g 1 = 4 π ( N 2 ) i a ( t ) − γ F a 2 g 1 = 4 π ( N 2 ) i a ( t ) γ ∴ F ag 1 = 2 · 4 π ( N 2 ) i a ( t ) cos( γ ) 0 ◦ = 4 π ( cos( γ ) 2 N 2 ) i a ( t ) 0 ◦ = 4 π ( k w 1 N ph 2 ) i a ( t ) 0 ◦ = ˆ F ag 1 0 ◦ 31 / 60 4.3 – Distributed Windings (cont . . . ) With, ˆ F ag 1 = 4 π ( k w 1 N ph 2 ) i a ( t ) (4.7’) and k w 1 = cos( γ ) as calculated earlier using the Fourier series expansion. 32 / 60 4.3 – Distributed Windings (cont . . . ) If we were to make use of even more distributed windings with (say) 4 coils, we can reduce the harmonic MMF distribution components even further with respect to the fundamental MMF distribution component. a d a a 4 a 4 a 3 a 3 a 2 a 2 a 1 a 1 33 / 60 4.3 – Distributed Windings (cont . . . ) The stepped MMF distribution in the air-gap will now look as follows: d a θ ae 0 ◦ 7 5 ◦ 22 5 ◦ 37 5 ◦ 52 5 ◦ 67 5 ◦ 82 5 ◦ 97 5 ◦ 112 5 ◦ 127 5 ◦ 142 5 ◦ 157 5 ◦ 172 5 ◦ 187 5 ◦ 202 5 ◦ 217 5 ◦ 232 5 ◦ 247 5 ◦ 262 5 ◦ 277 5 ◦ 292 5 ◦ 307 5 ◦ 322 5 ◦ 337 5 ◦ 352 5 ◦ 360 ◦ − a 1 − a 2 − a 3 − a 4 a 1 a 2 a 3 a 4 N ph I max 2 − N ph I max 2 F ag [ At ] ˆ F ag 1 γ 1 γ 2 In order to simplify the Fourier series expansion, we define two angles, γ 1 and γ 2 , as shown above. 34 / 60 4.3 – Distributed Windings (cont . . . ) - We can now easily calculate the Fourier Coefficients a n using superposition without have to “commit” complex integration. - The “trick” is to see that this MMF waveform: (say) F ′ ag caused by coils a 1 & a 4 , and (say) F ′′ ag caused by coils a 2 & a 3 - Furthermore, we make use of the “ Definition of a n for a stepped waveform ” as deduced earlier for coils a 1 & a 4 , γ = γ 1 for coils a 2 & a 3 , γ = γ 2 35 / 60 4.3 – Distributed Windings (cont . . . ) - The MMF waveform for coils a 1 & a 4 , looks as follow: d a θ ae 0 ◦ 7 5 ◦ 22 5 ◦ 37 5 ◦ 52 5 ◦ 67 5 ◦ 82 5 ◦ 97 5 ◦ 112 5 ◦ 127 5 ◦ 142 5 ◦ 157 5 ◦ 172 5 ◦ 187 5 ◦ 202 5 ◦ 217 5 ◦ 232 5 ◦ 247 5 ◦ 262 5 ◦ 277 5 ◦ 292 5 ◦ 307 5 ◦ 322 5 ◦ 337 5 ◦ 352 5 ◦ 360 ◦ NI max − NI max F ′ ag [ At ] − a 1 − a 4 a 1 a 4 γ 1 With an (say) a ′ n Coefficients: a ′ n = 4 π ( 2 Ni a ( t ) 2 ) sin( n π 2 ) n cos( n γ 1 ) 36 / 60 4.3 – Distributed Windings (cont . . . ) - The MMF waveform for coils a 2 & a 3 , looks as follow: d a θ ae 0 ◦ 7 5 ◦ 22 5 ◦ 37 5 ◦ 52 5 ◦ 67 5 ◦ 82 5 ◦ 97 5 ◦ 112 5 ◦ 127 5 ◦ 142 5 ◦ 157 5 ◦ 172 5 ◦ 187 5 ◦ 202 5 ◦ 217 5 ◦ 232 5 ◦ 247 5 ◦ 262 5 ◦ 277 5 ◦ 292 5 ◦ 307 5 ◦ 322 5 ◦ 337 5 ◦ 352 5 ◦ 360 ◦ NI max − NI max F ′′ ag [ At ] − a 2 − a 3 a 2 a 3 γ 2 With an (say) a ′′ n Coefficients: a ′′ n = 4 π ( 2 Ni a ( t ) 2 ) sin( n π 2 ) n cos( n γ 2 ) 37 / 60 4.3 – Distributed Windings (cont . . . ) The spacial MMF distribution of all four coils together, can thus be calculated as follows: F ag ( θ a ) = F ′ ag ( θ a ) + F ′′ ag ( θ a ) = ∞ ∑ n = 1 , 3 , 5 ,... ( a ′ n + a ′′ n ) · cos( n θ a ) = 4 π ( 2 Ni a ( t ) 2 ) ∞ ∑ n = 1 , 3 , 5 ,... sin( n π 2 ) n ( cos( n γ 1 )+ cos( n γ 2 ) ) · cos( n θ a ) = 4 π ( N ph i a ( t ) 2 ) ∞ ∑ n = 1 , 3 , 5 ,... sin( n π 2 ) n k w n · cos( n θ a ) 38 / 60 4.3 – Distributed Windings (cont . . . ) with, N ph = 4 N and the “new” winding factor, k w n = cos( n γ 1 ) + cos( n γ 2 ) 2 - As expected, the fundamental MMF distribution, F ag 1 ( θ a ) = 4 π ( k w 1 N ph 2 ) i a ( t ) · cos( n θ a ) (4.6’) 39 / 60 4.3 – Distributed Windings (cont . . . ) with amplitude of peak-value, ˆ F ag 1 ( θ a ) = 4 π ( k w 1 N ph 2 ) i a ( t ) (4.7’) - The fundamental winding factor for this winding configuration, k w 1 = cos( γ 1 ) + cos( γ 2 ) 2 = cos( 22 5 ◦ ) + cos( 7 5 ◦ ) 2 = 0 958 40 / 60 4.3 – Distributed Windings (cont . . . ) Although the fundamental component of the MMF distribution of this distributed windings is now 0.958 time smaller than that of the concentrated winding configuration (in comparison to the previous distributed winding configuration which was only 0.966 times smaller) , - the big difference again lie in the amplitudes of the spacial harmonic MMF distribution components as shown below: Harmonic (24-slot) (12-slot) (6-slot) 1 0 958 0 966 1 000 3 − 0 218 − 0 236 − 0 333 5 0 041 0 052 0 200 7 0 023 − 0 037 − 0 143 9 − 0 030 0 079 0 111 11 0 011 − 0 088 − 0 091 41 / 60 4.3 – Distributed Windings (cont . . . ) The magnitude of the fundamental component of the MMF distribution, caused by coils/windings a 1 , a 2 , a 3 & a 4 , can also once again easily be calculated using [space] vector algebra. a d a a 4 a 4 a 3 a 3 a 2 a 2 a 1 a 1 F a 1 g 1 F a 2 g 1 F a 3 g 1 F a 4 g 1 F ag 1 42 / 60 4.3 – Distributed Windings (cont . . . ) - If we zoom in a bit, we can clearly see the 4 MMF space vectors, , F a 1 g 1 , F a 2 g 1 , F a 3 g 1 & F a 4 g 1 of each of the 4 distributed windings/coils, , a 1 , a 2 , a 3 & a 4 and how they sum to the MMF space vector for phase a , F ag 1 ˆ F a 1 g 1 1 ˆ F a 2 g 1 2 ˆ F a 3 g 1 2 ˆ F a 4 g 1 1 “ F a 2 g 1 ” “ F a 3 g 1 ” “ F a 4 g 1 ” F ag 1 43 / 60 4.3 – Distributed Windings (cont . . . ) - By making use of inspection: F a 1 g 1 = 4 π ( N 2 ) i a ( t ) − γ 1 F a 2 g 1 = 4 π ( N 2 ) i a ( t ) − γ 2 F a 3 g 1 = 4 π ( N 2 ) i a ( t ) γ 2 F a 4 g 1 = 4 π ( N 2 ) i a ( t ) γ 1 ∴ F ag 1 = 2 · 4 π ( N 2 ) i a ( t ) ( cos( γ 1 ) + cos( γ 2 ) ) 0 ◦ = 4 π ( k w 1 N ph 2 ) i a ( t ) 0 ◦ 44 / 60 4.3 – Distributed Windings (cont . . . ) We once again get familiar equation: ˆ F ag 1 = 4 π ( k w 1 N ph 2 ) i a ( t ) (4.7’) but now with N ph = 4 N k w 1 = cos( γ 1 ) + cos( γ 2 ) 2 Tip for factorising k w 1 < 1 45 / 60 Multi-pole Machines Consider the following 4-pole stator with two concentrated, overlapping windings per phase a d a a 1 a 1 ́ a 1 ́ a 1 46 / 60 Multi-pole Machines (cont . . . ) The spacial MMF distribution will now have two periods, or cover 720 ◦ ( electrical ), over the 360 ◦ ( mechanical ) circumference of the stator. d a θ ae 0 ◦ 30 ◦ 90 ◦ 150 ◦ 210 ◦ 270 ◦ 330 ◦ 390 ◦ 450 ◦ 510 ◦ 570 ◦ 630 ◦ 690 ◦ 720 ◦ − a 1 − ́ a 1 a 1 ́ a 1 N ph I max 2 p − N ph I max 2 p F ag [ At ] ˆ F ag 1 47 / 60 Multi-pole Machines (cont . . . ) In order to simplify the mathematical writing work, we define the number of pole pairs, p , as: p = poles 2 Consequently we can write, θ ae = p θ a (4.1’) ∴ θ a = θ ae p = ω e t p with θ a the mechanical angle and θ ae the electrical angle measured from the d -axis of phase a 48 / 60 Multi-pole Machines (cont . . . ) Or in terms of speed, ω e = p ω m = 2 π f e (4.3’) with f e = ω e 2 π = p ω m 2 π = np 60 (4.2’) with ω e the electrical speed in [rad/s], ω m the mechanical speed in [rad/s], n the mechanical speed in [rpm] and f e the electrical frequency in [Hz]. 49 / 60 Multi-pole Machines (cont . . . ) The Fourier series expansion of the spacial MMF distribution must again be slightly modified because the wavelength of the fundamental component of the spacial harmonic MMF distribution now only encompasses half of the stator’s circumference θ a = θ ae p = ω e t p ∴ d θ a dt = ω e p ∴ dt = p ω e d θ a = Tp 2 π d θ a 50 / 60 Multi-pole Machines (cont . . . ) The integration boundaries will also change to ω e t p ∣ ∣ ∣ ∣ t = 0 = 0 ω e t p ∣ ∣ ∣ ∣ t = T 4 = π 2 p Again from Nilsson & Riedel (16.15) it follows that a n = 4 p π ˆ π 2 p 0 f ( θ a ) cos( np θ a ) d θ a 51 / 60 Multi-pole Machines (cont . . . ) If we were to do the Fourier series expansion in order to find the general solution of the spacial MMF distribution for a p pole pair machine, The general solution of the spacial harmonic MMF distribution of the multi-pole machine with concentrated windings, will look as follows, F ag ( θ ae ) = 4 π ( N ph i a ( t ) 2 p ) ∞ ∑ n = 1 , 3 , 5 ,... sin ( n π 2 ) n · cos( n θ ae ) 52 / 60 Multi-pole Machines (cont . . . ) with the fundamental component, F ag 1 ( p θ a ) = 4 π ( N ph 2 p ) i a ( t ) cos( p θ a ) = 4 π ( N ph 2 p ) i a ( t ) cos( θ ae ) and the amplitude of the fundamental component, ˆ F ag 1 ( θ ae ) = 4 π ( N ph 2 p ) i a ( t ) 53 / 60 Multi-pole Machines (cont . . . ) - Although it may appear that the amplitude of the MMF distribution is now p times smaller, we must remember that there are p times more coils per phase. - For the 4-pole concentrated winding machine, N ph = 2 · N with N the number of turns per coil - In general N ph = q · N with q the number of coils per phase 54 / 60 Multi-pole Machines (cont . . . ) A 4-pole single layer distributed winding machine with 4 coils per phase, i.e. N ph = 4 ̇ N : a d a a 2 a 2 a 1 a 1 ́ a 2 ́ a 2 ́ a 1 ́ a 1 b 2 b 2 b 1 b 1 ́ b 2 ́ b 2 ́ b 1 ́ b 1 c 2 c 2 c 1 c 1 ́ c 2 ́ c 2 ́ c 1 ́ c 1 55 / 60 Multi-pole Machines (cont . . . ) Or with only phase a shown a d a a 2 a 2 a 1 a 1 ́ a 2 ́ a 2 ́ a 1 ́ a 1 56 / 60 Multi-pole Machines (cont . . . ) The MMF distribution for this single layer distributed winding machine, will look as follow: d a θ ae 0 ◦ 15 ◦ 45 ◦ 75 ◦ 105 ◦ 135 ◦ 165 ◦ 195 ◦ 225 ◦ 255 ◦ 285 ◦ 315 ◦ 345 ◦ 375 ◦ 405 ◦ 435 ◦ 465 ◦ 495 ◦ 525 ◦ 555 ◦ 585 ◦ 615 ◦ 645 ◦ 675 ◦ 705 ◦ 720 ◦ − a 1 − a 2 − ́ a 1 − ́ a 2 a 1 a 2 ́ a 1 ́ a 2 N ph I max 2 p − N ph I max 2 p F ag [ At ] ˆ F ag 1 γ 57 / 60 Multi-pole Machines (cont . . . ) The Fourier series expansion of the harmonic MMF spacial distribution will be equal to, F ag ( θ ae ) = 4 π ( N ph i a ( t ) 2 p ) ∞ ∑ n = 1 , 3 , 5 ,... sin ( n π 2 ) n k w n · cos( n θ ae ) with the winding factor, k w n = cos( n γ ) where γ is measured in electrical degrees. 58 / 60 Multi-pole Machines (cont . . . ) The fundamental component of the spacial harmonic MMF distribution will now have the general form similar to (4.6), F ag 1 ( θ ae ) = 4 π ( k w 1 N ph 2 p ) i a ( t ) · cos( θ ae ) (4.6’) with the general form of the amplitude of the fundamental component similar to (4.7), ˆ F ag 1 ( θ ae ) = 4 π ( k w 1 N ph 2 p ) i a ( t ) (4.7’) 59 / 60 Example 4.2 A four-pole synchronous ac generator with a smooth air-gap has a distributed rotor winding with 264 series turns, a winding factor of 0.935 and and air-gap length 0.7 mm. Assuming the mmf drop in the electrical steel to be negligible, find the rotor-winding current required to produce a peak, space fundamental magnetic flux density of 1.6 T in the machine air-gap. 60 / 60