ELECTRIC SERVO MOTOR EQUATIONS AND TIME CONSTANTS

Transcription

1 ELECIC SEO MOO EQUAIONS AND IME CONSANS George W. Younkin, P.E. Life FELLOW IEEE Industrial Controls Consulting, Div. Bulls Eye Marketing, Inc Fond du c, Wisconsin In the analysis of electric servo drive otors, the equations for the otor indicates the presence of two tie constants. One is a echanical tie constant and the other is an electrical tie constant. Coercial servo otor specifications usually list these two tie constants. However, it should be cautioned that these two tie constants as given in the specifications are for the otor alone with no load inertia connected to the otor shaft. Since these two tie constants are part of the otor block diagra used in servo analysis, it is iportant to know the real value of the tie constants under actual load conditions. here are two types of servo otors to consider. he first is the classical dc servo otor and the second is the ac servo otor often referred to as a brushless dc otor. he brushless dc otor is a three phase synchronous ac otor having a position transducer inside the otor to transit otor shaft position to the drive aplifier for the purpose of controlling current coutation in the three phases of the otor windings. A derivation of the otor equations and the electrical and echanical otor tie constants will be discussed for the dc otor followed by the ac otor. he dc otor equivalent diagra is: L e i i a J J L L a a Figure Where: e I Applied voltage (volts) i a Arature current (aps) J otal inertia of otor arature plus load (lb-in-sec )

2 e Motor voltage constant (v/rad/sec) Motor torque constant (lb-in/a) L a Motor winding inductance (Henries) a Arature resistance (ohs) L Load torque (lb-in) Motor velocity (rad/sec) α Acceleration (rad/sec ) he steady state (dc) equations are: e i i a a + e (oltage equation) () orque i a Jα (orque equation) () For the general case, the differential equations are: dia e i i a a + L a dt + e (3) d place operator S dt (4) e i a i a + L a Si a + e (5) e i ( a + L a S)i a + e (6) e i ( S + ) a i a + e (7) Also: I a J α J S (8) i a JS (9) Cobining equations gives: JS e i - e ( S + ) a (0) earranging results in: ( e) ( S + )JS his last equation can be represented in block diagra for as: ()

3 3 e i i a J S ( S + ) (-) e Figure he closed loop equation ( C G ) for the above block diagra is: + GH () JS S + e + a earranging gives: (J L a a )S + J as + e Dividing by e gives: (3) e (4) J J S + S + e e Fro the last equation, the otor echanical tie constant, t, is: t J e [sec] (5) he total inertia, J, is the su of the reflected inertia to the otor shaft plus the otor inertia. he resistance, a, is the otor winding resistance plus the external circuit resistance. hus the otor echanical tie constant is suarized as: J t e [sec] (6) Also, the otor electrical tie constant is: 3

4 4 t e [sec] (7) herefore, the closed loop otor equation can be expressed as: e (8) ttes + ts + Fro the general equation for a quadratic: S ω δs + + ω where: ω / t t e (9) he daping factor is: δ 0.5 t ω 0.5 t / t t 0.5 t / t (0) e e he echanical and electrical tie constants for a brushless dc otor have the sae basic equations with soe variations. For a brushless dc otor with a wye connected otor, the eletrical circuit is: WIING Aplifier L PHASE L-L e(l-l) PHASE e(phase) Figure 3 he echanical tie constant is: 4

5 5 PHASEJOAL t e(phase) [sec] () where: e(l L) sec e(phase) Motor voltage constant.73 AD () lb in Motor torque constant A (3) M(L-L) Motor resistance [ohs] (4) otal otor circuit resistance.35 M(L-L) [ohs] (5) M(L L) M(L L) 0.5 M(PHASE) [ohs] (6) L L-L Motor inductance [Henries] J OAL Motor arature inertia plus the reflected load inertia at the otor shaft [lb-in-sec ] Most anufacturers give the electrical paraeters in line-to-line values. hus soe of these values ust be converted to the phase values as shown above. Suarizing, the echanical tie constant can be coputed as: t L L JOAL [sec] 0.86 e(l L).73 l l Jtotalatotor e( l l) (7) he electrical tie constant for the brushless dc otor is coputed as: otal inductive path LL L te [ sec] (8) otal resistive path (L L) Another factor affecting the echanical tie constant is the teperature. Most anufacturers specify the otor paraeters at 5 o C (cold rating). his iplies that the agnet and wires are both at roo teperature. However, the otors used in industry will operate hotter which could reach a agnet teperature of 80 o C to 90 o C in a 40 o C abient. he winding teperature is considerably ore than that. Soe eans ust be used to copensate for the otor paraeters rated at 5 o C. For those anufacturers that offer the hot rating on otor specification paraeters, they should be used to calculate the tie constants. he paraeters of otor resistance, torque constant, and voltage constant should be adjusted, if needed, for the hot rating. he otor resistance will increase; the torque constant and voltage constant will decrease. However, contrary to thr iplied nae both tie constants are not of constant value. ther, they are both functions of the otor s operating teperature. 5

6 6 he electrical resistance of a winding, at a specified teperature, is deterined by the length, gauge and coposition (i.e, copper, aluinu, etc.) of the wire used to construct the winding. he winding in the vast ajority of industrial servootors are constructed using fil coated copper agnet wire. Based on the 93 International Electrical Coission standard, the linear teperature coefficient of electrical resistance for annealed copper agnet wire is / o C. Hence, knowing a copper winding s resistance at a specified reference or abient teperature, the windings at teperatures above or below this abient teperature is given by: () ( 0 )[ (- 0 )] (eq a) Where : Winding s eperature ( 0 C) 0 Specified Abient eperature ( 0 C). Using equation (a), a 30 0 C rise (55 0 C-5 0 C) in a copper winding s teperature increases its electrical resistance by a factor of.509. Correspondingly, the otor s echanical tie constant increases by this sae.509 factor while its electrical tie constant decreases by a factor of / In cobination, the otor s echanical to electrical tie constant ratio increases by a factor of.8 and this increase definitely affects how the servootor dynaically responds to a voltage coand. In consulting published otor data, any otor anufacturers specify thr otor s paraeter values, including resistance, using 5 0 C as the specified abient teperature. NEMA, however, recoends 40 0 C as the abient teperature in specifying otors for industrial applications, herefore, pay close attention to the specified abient teperature when consulting or coparing published otor data. Different anufacturers can, and soeties do, use different abient teperatures in specifying what can be the identical otor. In the sae published data servootors are generally rated to operate with ther a 30 0 C (Class B) or 55 0 C (Class F) continuous winding teperature. Although otors with a Class H, 80 0 C teperature rating are also available. Assuing the otor s resistance along with its electrical and echanical tie constants are specified at 5 0 C, it was just deonstrated that all three paraeters significantly change value at a 55 0 C winding teperature. If the otor s winding can safely operate at 80 0 C the resistance change is even greater because equation (7.4-4) shows that a 55 0 C rise (80 0 C-5 0 C) in winding teperature increases its electrical resistance by a factor of.609. Hence, if the servootor s dynaic otion response is calculated using the 5 0 C paraeter values then this calculation overestiates the otor s dynaic response for all teperatures above 5 0 C. In all peranent agnet otors there is an additional affect that teperature has on the otor s echanical tie constant only. As shown in eq (a) a otor s echanical tie constant changes inversely with any change in both the back EMF, e, and torque constant,. Both e and have the sae functional dependence on the otor s air gap agnetic flux density produced by the otor s agnets. All peranent agnet otors are subject to both reversible and irreversible deagnetization. Irreversible deagnetization can occur at any teperature and ust be avoided by liiting the otor s current such that, even for an instant, it does not exceed the peak current/torque specified by the otor anufacturer. Exceeding the otor s peak current rating can 6

7 7 peranently reduce the otor s e and thereby increasing the otor s echanical tie constant at every teperature including the specified abient teperature. eversible theral deagnetization depends on the specific agnet aterial bng used. Currently, there are four different agnet aterials used in peranent agnet otors. he four aterials are: Aluinu-Nickel-Cobalt (Alnico), Saariu Cobalt (SCo), Neodyiu-Iron-Boron (NdFeB), and Ferrite or Ceraic agnets as they are often called. In the teperature range, C < < 00 0 C, all four agnet aterials exhibit reversible theral deagnetization such that the aount of air gap agnetic flux density they produce decreases linearly with increasing agnet teperature. Hence, siilar to electrical resistance, the expression for the reversible decrease in both e () and () with increasing agnet teperature is given by : e, () e, ( 0) [-B(- 0 )] (eq b) In equation (b), the B-coefficient for each agnet aterial aounts to : B(Alnico) 0.000/ 0 C B(SCo) / 0 C B(NdFeB) 0.00/ 0 C B(Ferrite) 0.00/ 0 C Using equation (b) it can be calculated that a 00 0 C rise in agnet teperature causes a reversible reduction in both e and that aounts to percent for Alnico, 3.5 percent for SCo, 0 percent for NdFeB. And 30 percent for Ferrite or Ceraic agnets. Like the otor s electrical resistance, ost otor anufacturers specify the otor s e and using the sae abient teperature used to specify resistance. However, this is not always true and it is again advised to pay close attention as to how the anufacturer is specifying thr otor s paraeter values. Cobining the affects of reversible, theral deagnetization with teperature dependent resistance, the equation describing how a peranent agnetic otor s echanical tie constant increases in value with increasing otor teperature aounts to: [ ( 0 )] τ ( ) τ ( 0 ) (eq c) [( B( 0 )) ] Notice in (eq c) that the agnet s teperature is assued equal to the otor s winding teperature. Actual easureent shows that this assuption is not always correct. Motor agnets typically operate at a lower teperature copared to the winding s teperature. However this conservative approxiation is recoended and used. An exaple will be given to illustrate a change in tie constants. o raise the echanical tie constant to a 55 0 C teperature rating inside a Ferrite agnet otor, for exaple, the resistance increase will be- (55 0 C) (5 0 C) / 0 C x (55-5) x (5 0 C).509 (5 0 C) he voltage constant e and torque constant will be lowered. Since the agnet aterial is 0 0 C-5 0 C cooler than the windings the e and will be- e (40 0 C) e (5 0 C) 0.00/ 0 C x (40-5) e (5 0 C) 0.77 e (5 0 C) he echanical tie constant will increase byt (55 0 C).509/(0.77).54 t (5 0 C) 7