Intrduction Exam. Fig . shows magnetic field control DC motor, input–voltage Uf (v），output—motor revolution ωm（t）（rad/s），write its differential equ. 。In fig. Ra(Ω)、La(H) are armuture loop resistance and inductance， Rf(Ω)、Lf(H) are forced magnetic loop resistance and inductance， ML(N·M) is total load torque。Ua=const。

Ψ Ψ0 α if

2.2 linearize the nonlinear equ. To continuous nonlinear function,we can use the tangent method or small deviation method.Within a small scope, replace non linear with a line . A non linear function y= f( x) when continuously differentiated at x0 ,can be express into Tele series.

Ignore high order item, and then have set Δy=kΔx k proportion coefficient，the slope at x0 Nonlinear function with 2 variables y=f(x1,x2)，at（x10,x20）

Ignore items more than 2-th differential，set Δy＝y-f(x10,x20) This kind of small deviation method is mostly to work for the control system.at the equilibrium point nearby, the deviation can't be generally very big, and are all small deviation point.

10≤y≤12。When x=5，y=10,solve the error of z。 Exam 2-4 linearize z=xy at 5≤x≤7 、 10≤y≤12。When x=5，y=10,solve the error of z。 Z=xy

z=11x+6y-66

Because 5≤x≤7、10≤y≤12，select x0=6，y0=11。So z0=x0y0=6×11=66. At x0=6，y0=11，z0=66,express z into tele series,and ignore high order items. Linearization equ.： z-66=11(x-6)+6(y-11) z=11x+6y-66 When x=5,y=10 ，z=xy=5×10=50 From Linearization equ. z=11x+6y-66=55+60-66=49 so，error is 50-49=1，that is

Return to the exam. above, α Ψ Ψ0 if

Write the other equation into increment form:

Math tool Laplace transformation and versa transformation ⑴ definition assume f(t) ①when t<0 f(t)=0 ② when t>0，f(t) continuous segmently , Laplace transformation of f( t) exists , ⑵basic thereom . Linear thereom Displace thereom Delay thereom Final thereom

a. F(s)has unequal poles，can express into Initial value thereom Differential thereom Integer thereom ⑶ versa Laplace transformation F(s)is written factor form： a. F(s)has unequal poles，can express into

b.F(s)has conjugate complex poles,can express into c.F(s) has many equal poles,can express into

2.3 s-domain mathematical model 2.3.1 transfer function the differential equ. is the math model describing system dynamic performance in time domain.given initial value and input , its solution can get the system respose.when parameters change,it is difficult to solve. While using Laplace transformation to solve the differential equ., we can get mathematics model of system at complex domain --- transfer function. Definition:transfer function is the ratio of tranformed output to transformed input when initial value is 0

Assume 0 initial value，set R(s)＝L[c(t)]，R(s)=L[r(t)]，we get： To a linear time-invariant system： Where c(t) output，r(t)input，ai,bj is constant which is connected with system construction and parameters。 Assume 0 initial value，set R(s)＝L[c(t)]，R(s)=L[r(t)]，we get： so，we get TF from the definition：

Property 1 TF is a fraction function of m≤n， which has all the property of complex function. Property 2 G(s) depends on system construction and parameter, independent of input.

Property 4 If G(s)known，output can be solved to various input. Property 3 G(s) Although describing the relation between input and output,it do not provide any on physical construction,Because lots of the different physical system have completely same TF. 。 Property 4 If G(s)known，output can be solved to various input. Property 5 If G(s)unknown，studying output can get Tf to given input. TF math model is an operational mode expressing input and out differentiation . Property 6 relation between TF and differential equ.。 If

property7 Laplace transform of TF is impulse response g(t) 2.3.2 affect on output of TF poles and zeros zeros poles Poles are characteristic roots of a differential equ. ，and dominate system’s motion mode.

poles far from the zeros produce more affect than these near If a zero is equal to a pole, produce no affect to system, because the numerator and denominator offset mutually.

2.3.3 solve transfer function 1.according to the definition Mass-spring-ramper

2.according to the complex impedence of electrical system

Exam.solve transfer function in the following diag.

Exam.solve transfer function in the following diag.

homework: 2-10,2-11(partly), 2-12,2-13

2.6 typical component Tf any complex system is composed of typical component。 1.proportion Potentiometer－ transform linear position or angle into voltage. Single Potentiometer used to signal transformation device.

－maximum work angle（rad） output voltage/Angle （v/rad） E-power voltage（v） －maximum work angle（rad） where K-gain characteristic：no distortion or time delay,output is proportional to input. For instance：amplifier，gear, potentiometer, etc..

K1 amplification coefficient One couple potential meter can construct an error measurement. K1 amplification coefficient Due to the load effect，in general Error angle

2. Inertial component where T-time constant characteristic： have 1 contained energy component，to a sudden input,its output Cannot respond quickly,output has no oscillation。

3.differential techometer generator － measure the speed and convert it into electric voltage. DC techometer AC techometer Rotor angle speed（rad/s） Output slope（v/rad/s） In general

characteristic： output is proportional to input integer。

-natural oscillation frequency（no-ramped oscillation frequency ） 5 oscillator where ξ－ramped ratio -natural oscillation frequency（no-ramped oscillation frequency ） characteristic： there are 2 contained energy components。 Exam. ： TF between output voltage and input voltage of RLC circuit。

6 pure delay where －delay time characteristic： Output can accurately reveal the input, but must delay a fixed time. exam.: control of Piping pressure, discharge...etc., its math model includes delay link.

Exam.1 The armature control DC servo motor Simplification differential equ. Of armature control DC servo motor is and , between Load torque, from addition principle, solve respectively TF a .set ＝0

b .set

Exam.2 Two phase servo motor : 2 stator coil,1 rotor with a high resistance.1 stator coil is stimulated Coil,the other is control coil which usually connect in the power amplifier output. 。

its torque-speed curve contains negative slope its torque-speed curve contains negative slope.Fig 2-13( b) is its mechanical characteristic curve at different control voltage. considering control system generally works in the zero speed nearby, usually extend low speed segment to high speed scope, and use low speed line to replace nonlinearity.In addition, also can use the small deviation linearization method.

Generally, two phase servo-motor mechanical characterictic can be expressed for (2-3-1) (2-3-2) If load torque is not taken into account,get torque equilibrium equ.

Substitute (2-4-2)to(2-4-1),and to (2-4-3) Take Laplace transformation Substitute (2-4-2)to(2-4-1),and to (2-4-3)

Have the same TF as DC motor. armature control DC motor often applies to compare in bigger power system, its efficiency is higher than 2 phase AC motor. 2 phase AC motor often applies at measuring devices whose power is from several watts to 100 watts.