# Electric Circuits Discussion 1

date post

25-Jan-2022Category

## Documents

view

0download

0

Embed Size (px)

### Transcript of Electric Circuits Discussion 1

Electric Circuits Discussion 1Contents

2

• Homework 6

1. Second-Order Circuit

And there are THREE cases you should know

First solving the Eigen-function of and Eigenvalues of a second order formula

Case 1: Overdamped (α>ω0)

2 2 2 2 1 2o os sα α ω α α ω= − + − = − − −0

1 2 R L LC

α ω= =

() = 1 + 2 −

6

1 = − + 2 − 02 = − + − 02 − 2 = − +

2 = − − 2 − 02 = − − − 02 − 2 = − −

where = −1 and = 02 − 2.

• ω0 is often called the undamped natural frequency. • ωd is called the damped natural frequency.

The natural response

= − 1cos + 2sin 7

= − ± 2 − 02

Recall Euler’s formula

• Exponential − * Sine/Cosine term • Exponentially damped, time constant =

1/ • Oscillatory, period = 2

• Behavior captured by damping • Gradual loss of the initial stored

energy • determines the rate of damping

• > 0 (i.e., > 2

), overdamped

), underdamped

= 11 + 22

• Series

10

• Parallel

( ) 1 2 1 2

( ) ( )2 1 tv t A At e α−= +

( ) ( )1 2cos sint d dv t e A t A tα ω ω−= +

Finding Initial and Final Values

• Working on second order system is harder than first order in terms of finding initial and final conditions.

• You need to know the derivatives, dv/dt and di/dt as well.

• Capacitor voltage and inductor current are always continuous.

• For capacitor, 0+ = 0− ; • For inductor, 0+ = 0− .

General Second-Order Circuits

• The principles of the approach to solving the series and parallel forms of RLC circuits can be applied to general second order circuits, by taking the following four steps: 1. First determine the initial conditions, x(0) and dx(0)/dt. 2. Turn off the independent sources and find the form of the

transient response by applying KVL and KCL. • Depending on the damping found, the unknown constants will be found.

3. We obtain the steady-state response as:

where x(∞) is the final value of x obtained in step 1. 4. The total response = transient response + steady-state response.

( ) ( )ssx t x= ∞

13

14

2. Homework 6

Case 2: Critically Damped (α=ω0)

Case 3: Underdamped (α<ω0)

Case 3: Underdamped (α<ω0)

Properties of Series RLC Network

Series vs. Parallel (Source-Free RLC Network)

Finding Initial and Final Values

General Second-Order Circuits

2

• Homework 6

1. Second-Order Circuit

And there are THREE cases you should know

First solving the Eigen-function of and Eigenvalues of a second order formula

Case 1: Overdamped (α>ω0)

2 2 2 2 1 2o os sα α ω α α ω= − + − = − − −0

1 2 R L LC

α ω= =

() = 1 + 2 −

6

1 = − + 2 − 02 = − + − 02 − 2 = − +

2 = − − 2 − 02 = − − − 02 − 2 = − −

where = −1 and = 02 − 2.

• ω0 is often called the undamped natural frequency. • ωd is called the damped natural frequency.

The natural response

= − 1cos + 2sin 7

= − ± 2 − 02

Recall Euler’s formula

• Exponential − * Sine/Cosine term • Exponentially damped, time constant =

1/ • Oscillatory, period = 2

• Behavior captured by damping • Gradual loss of the initial stored

energy • determines the rate of damping

• > 0 (i.e., > 2

), overdamped

), underdamped

= 11 + 22

• Series

10

• Parallel

( ) 1 2 1 2

( ) ( )2 1 tv t A At e α−= +

( ) ( )1 2cos sint d dv t e A t A tα ω ω−= +

Finding Initial and Final Values

• Working on second order system is harder than first order in terms of finding initial and final conditions.

• You need to know the derivatives, dv/dt and di/dt as well.

• Capacitor voltage and inductor current are always continuous.

• For capacitor, 0+ = 0− ; • For inductor, 0+ = 0− .

General Second-Order Circuits

• The principles of the approach to solving the series and parallel forms of RLC circuits can be applied to general second order circuits, by taking the following four steps: 1. First determine the initial conditions, x(0) and dx(0)/dt. 2. Turn off the independent sources and find the form of the

transient response by applying KVL and KCL. • Depending on the damping found, the unknown constants will be found.

3. We obtain the steady-state response as:

where x(∞) is the final value of x obtained in step 1. 4. The total response = transient response + steady-state response.

( ) ( )ssx t x= ∞

13

14

2. Homework 6

Case 2: Critically Damped (α=ω0)

Case 3: Underdamped (α<ω0)

Case 3: Underdamped (α<ω0)

Properties of Series RLC Network

Series vs. Parallel (Source-Free RLC Network)

Finding Initial and Final Values

General Second-Order Circuits