【正文】 is its phase angle defined by φ = arctan(ω 1L1/R) . (16) Equations (12) and (13) indicate that multiple equilibria exist, which means that these equilibria can never be globally stable. One can see that there are two groups of equilibria as shown in Equations (12) and (13). The first group represented by Equation (12) corresponds to the real 24 operatingconditions of the motor. The second group represented by Equation (13) is always unstable and does not relate to the real operating conditions. In the following, we will concentrate on the equilibria represented by Equation (12). 25 開始 初始化 紅燈，等待 掃描鍵盤 是否有按鍵被按下 參數(shù)按鍵 轉(zhuǎn)向 開始鍵 是 數(shù)據(jù)鍵 轉(zhuǎn)向 處理 啟動 步進(jìn)電機 輸入下一組數(shù)據(jù) 確認(rèn)？ 改變速度？ 處理速度 步進(jìn)電機工作 一步結(jié)束 顯示 否 否 是 否 是 。 d frame can be obtained as vq = Riq + L1*diq/dt + NL1idω + Nλ 1ω , vd=Rid + L1*did/dt ? NL1iqω , (5) Figure 2. a, b, c and d, q reference frame. where L1 D L CM, and ! is the speed of the can be shown that the motor’s torque has the following form [2] T = 3/2Nλ 1iq The equation of motion of the rotor is written as J*dω /dt = 3/2*Nλ 1iq ? Bfω – Tl , where Bf is the coefficient of viscous friction, and Tl represents load torque, which is assumed to be a constant in this paper. In order to constitute the plete state equation of the motor, we need another state variable that represents the position of the rotor. For this purpose the so called load angle _ [8] is usually used, which satisfies the following equation Dδ /dt = ω ?ω 0 , where !0 is steadystate speed of the motor. Equations (5), (7), and (8) constitute the statespace model of the motor, for which the input variables are the voltages vq and vd. As mentioned before, stepper motors are fed by an inverter, whose output voltages are not sinusoidal but instead are square 22 waves. However, because the nonsinusoidal voltages do not change the oscillation feature and instability very much if pared to the sinusoidal case (as will be shown in Section 3, the oscillation is due to the nonlinearity of the motor), for the purposes of this paper we can assume the supply voltages are sinusoidal. Under this assumption, we can get vq and vd as follows vq = Vmcos(Nδ ) , vd = Vmsin(Nδ ) , where Vm is the maximum of the sine wave. With the above equation, we have changed the input voltages from a function of time to a function of state, and in this way we can represent the dynamics of the motor by a autonomous system, as shown below. This will simplify the mathematical analysis. From Equations (5), (7), and (8), the statespace model of the motor can be written in a matrix form as follows ? = F(X,u) = AX + Fn(X) + Bu , (10) where X D Tiq id ! _UT , u D T!1 TlUT is defined as the input, and !1 D N!0 is the supply frequency. The input matrix B is defined by The matrix A is the linear part of , and is given by , and is given by The input term u is independent of time, and therefore Equation (10) is 23 autonomous. There are three parameters in 。 c reference, only two variables are independent (ia C ib C ic D 0)。 d reference are given by In the a。 c frame to the q。 d transformation, the frame of reference is changed from the fixed phase axes to the axes moving with the rotor (refer to Figure 2). Transformation matrix from the a。 附件 2：外文原文 The Stepper motor control circuit be based on Single chip microputer The AT89C51 is a lowpower, highperformance CMOS 8bit microputer with 4K bytes of Flash programmable and erasable read only memory (PEROM). The device is 12 manufactured using Atmel’s highdensity nonvolatile memory technology and is patible with the industrystandard MCS51 instruction set and pinout. The onchip Flash allows the program memory to be reprogrammed insystem or by a conventional nonvolatile memory programmer. By bining a versatile 8bit CPU with Flash on a monolithic chip, the Atmel AT89C51 is a powerful microputer which provides a highlyflexible and costeffective solution to many embedded control applications. Function characteristic The AT89C51 provides the following standard features: 4K bytes of Flash, 128 bytes of RAM, 32 I/O lines, two 16bit timer/counters, a five vector twolevel interrupt architecture, a full duplex serial port, onchip oscillator and clock circuitry. In addition, the AT89C51 is designed with static logic for operation down to zero frequency and supports two software selectable power saving modes. The Idle Mode stops the CPU while allowing the RAM, timer/counters, serial port and interrupt system to continue functioning. The Powerdown Mode saves the RAM contents but freezes the oscillator disabling all other chip functions until the next hardware reset. Pin Description VCC： Supply voltage. GND： Ground. Port 0： Port 0 is an 8bit opendrain bidirectional I/O port. As an output port, each pin can sink eight TTL inputs. When 1s are written to port 0 pins, the pins can be used as highimpedance 0 may also be configured to be the multiplexed loworder address/data bus during accesses to external program and data memory. In this mode P0 has internal 0 also receives the code bytes during Flash programming,and outputs the code bytes during programverification. External pullups