1. Modeling of Electrical Circuits
The electrical circuit, which will be found the mathematical model, is shown in Example 3.1.
In the electrical circuit, V1 is the input, q and q2 are the generalized charges.
The currents (qdot and q2dot) are shown on the lines. The current passing through the resistor R1 is not independent due to the first kirchoff’s law.
Example 3.1: L C V1 + - R1 R2
Current in the resistor R1 =
V1: Input
q ve q2 : Gen. charges

2. Example 3.1 (Continue) : L C V1 + - R1 R2
V1: Input
q ve q2 : Gen. charges
Current in the resistor R1 =
The magnetic energy and electrical energy equations can be stated as follows.
Then, the virtual work expression which is related to the input and resistors is written.
General voltages are found by taking parentheses of q and q2. The term in the first parenthesis is the general voltage for the charge q. The term in the second parenthesis is the general voltage for the charge q2. Qq
Qq2

3. Example 3.1 (Continue) : L C V1 + - R1 R2
V1: Input
q ve q2 : Gen. charges
Current in the resistor R1 =
When the Lagrange equation is applied to the charges to q and q2, respectively, the equations of motion are obtained.

4. Example 3.1 (Continue) : L C V1 + - R1 R2
V1: Input
q ve q2 : Gen. charges
Current in the resistor R1 =
L=3.4 mH, C=286 µF, R1=3.2 Ω, R2=4.5 Ω
This is a set of differential equations, which are similar to mathematical models of mechanical systems considered in previous lecture. These equations can be organised in matrix form.
Eigenvalue equation can be found substituting the numerical values into the matrix equation as follows. We can get the results by manual solution or Matlab.
D(s)= s s s=0
Eigenvalues: 0, ±418.70i (ξ=0.768)

5. The first kirchoff’s law is applied to the circuit in Example 3.2.
Another electrical circuit with an op-amp element, which will be found the mathematical model, is shown in Example 3.2. Op-amps are active elements that works with an external energy. C1 - + V1 V2 R1 R2 R3 C2
Input : V1
Generalized charges: q1, q2, q3
In the electrical circuit, V1 is the input, q1, q2 and q3 are the generalized charges.
The first kirchoff’s law is applied to the circuit in Example 3.2.

6. Then, the virtual work expression is written.
Example 3.2 (Contiune):
Input : V1
Generalized charges: q1, q2, q3 C1 - + V1 V2 R1 R2 R3 C2
The magnetic energy and electrical energy equations are written as follows.
Then, the virtual work expression is written.

7. Example 3.2 (Contiune): C1 R3 C2 R1 V1 V2 R2
Input : V1
Generalized charges: q1, q2, q3 C1 - + V1 V2 R1 R2 R3 C2
The Lagrange equation is applied to the charges to q1, q2 and q3. Then, the equations of motion are obtained for the electrical system.

8. Example 3.2 (Contiune): C1 R3 C2 R1 V1 V2 R2 Also, in op-amp: V+=V-=0
Input : V1
Generalized charges: q1, q2, q3 C1 - + V1 V2 R1 R2 R3 C2
Also, in op-amp: V+=V-=0
These equations can be organised in matrix form as they are linear differential equations.

9. Example 3.2 (Contiune):
Using numerical values given for Example 3.2, the eigenvalues of the circuit with an op-amp can be found as follows.
R1=15.9 kΩ, R2=837 Ω, R3=318 kΩ, C1=C2=0.005 µF
Eigenvalues: 0, ± i (ξ=0.05)

10. Circuits with op-amps are widely used instruments and control systems for multiplying, integrating, and differentiating signals. C - + V1 V2 R V2 R2 - + V1 R1 R - + V1 V2 C

11. These are examples of op-amp circuits such as a inverting-summing amplifier and a difference amplifier. - + V1 Vc R V2 V3 R V2 V1 - + Vc

12. PID control circuit: R2 - + R1 R4 C4 C3 R3 R V1 V2