1. ME 176
Control Systems Engineering
Mathematical Modeling
Department of
Mechanical Engineering

2. Mathematical Modeling: Electrical Circuits
Passive Networks
zero initial conditions.
components : resistors,
capacitors, and inductors.
no internal source of energy.
Operational Amplifier
1. Differential input, v2(t) - v1(t)
2. High input impedence, Zi is infinity
3. Low output impedence, Z0 is zero
4. High constant gain amplification,
A is infinity.
Department of
Mechanical Engineering

3. Mathematical Modeling: Electrical Circuits
Passive Networks:
where, v = voltage ; i = current ; q = charge
C = capacitance ; R = resistance ; L inductance
Department of
Mechanical Engineering

4. Mathematical Modeling: Electrical Circuits
Passive Networks:
1. Redraw the original network showing all time variables as Laplace
transforms. Example: v(t) as V(s), i(t) as I(s), and vc(t) as Vc(s).
2. Replace the component values with their respective impedance or
admittance values, depending on analysis.
Types of Analysis:
1. Mesh
2. Nodal
3. Voltage Division
Department of
Mechanical Engineering

5. Mathematical Modeling: Electrical Circuits
Passive Networks: Cramer's Rule
Simultaneous Equations:
Department of
Mechanical Engineering

6. Mathematical Modeling: Electrical Circuits
Passive Networks: Mesh Analysis
1. Replace passive elements with impedance
2. Replace sources and time variables with Laplace transforms
3. Identify mesh (loops); and each direction.
4. Apply Kirchhoff's Voltage Law around each loop.
5. Solve the resulting system of linear equations.
6. Form transfer function.
Department of
Mechanical Engineering

7. Mathematical Modeling: Electrical Circuits
Passive Networks : Nodal Analysis
1. Replace passive elements with admittance.
2. Replace sources and time variables with Laplace transforms
3. Replace transformed voltage sources with current sources.
4. Apply Kirchhoff's Current Law at each node.
5. Solve the resulting system of linear equations.
6. Form transfer function.
Department of
Mechanical Engineering

8. Mathematical Modeling: Electrical Circuits
Operational Amplifiers:
Characteristics:
1. Differential Input, v2(t) - v1(t)
2. High Input Impedence, Zi = infinity (ideal)
3. Low output Impedence, Zo=0 (ideal)
4. High constant gain amplification, A = infinity (ideal)
Types:
1. Inverting
2. Noninverting
Department of
Mechanical Engineering