State Space circuit model

Op Amp circuit example Note: ip1=0, ∴vp1=vo=vA & vB=vp2=0 Let vC1 & vC2 be s.v., vo output.

KCL at A: vo is not s.v. nor input, use vo=vC2

KCL at B: vo1 not s.v. nor input, vo1=vA+vC1=vn1+vC1 =vp1+vC1=vo+vC1 =vC2+vC1

Output eq:

Physical Laws for Mechanical

Example: car suspension

Car suspension: simplified Ignore tire deformation. Suppose y1(t) is measured from equilibrium position when gravity has set in. So gravity is canceled by spring force at eq. pos. ∴There are two forces on m: y1(t) x(t)

Newton’s Law: or num= den= T.F.=H(s)=

State Space Model For linear motion For angular motion Define two state variables for each mass x1=position, x2 = velocity; x1-dot = x2 x2-dot is acc and solve for it from Newton’s For angular motion Define two state variables for each rotating inertia x1= angle, x2 = angular velocity; x1-dot = x2 x2-dot is angular acc and solve for it from Euler’s law

Quarter car suspension

u

Electromechanical systems Motors DC motors Induction motors Variable reluctance motors Generators Angular position sensors Encoders Tachometers

For field control with constant armature current For armature control with constant field current

Armature controlled motor in feedback

Get TF from wd to w and Td to w.