Oscillatory Circuits and Lown Waveform

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Questions and Answers

What is the initial condition of the circuit at time t=0 regarding the switch?

The switch is partially open, causing intermittent current.

The switch is open, preventing any current. (correct)

The switch is closed, allowing current to flow.

The state of the switch is unknown at t=0.

Which of the following components likely affects the voltage vo(t) after the switch is opened?

The open switch itself that isolates the components.

The inductor connected in series with the voltage source.

The capacitor connected in parallel with the load. (correct)

The resistors in series with the current source.

What is the role of the inductor after t=0 once the switch is opened?

It converts electrical energy into thermal energy.

It stores energy and prevents sudden changes in current. (correct)

It instantly discharges, causing a voltage spike.

It has no effect since the switch is open.

What does the Lown waveform indicate about the circuit's damping condition?

The circuit is underdamped. Signup and view all the answers

If the circuit were to be analyzed for t>0, which method is typically utilized?

The Laplace Transform method. Signup and view all the answers

What is the expression for the voltage applied to the patient in the context described?

vo(t) = K1 e^(-␨␻o t) sin[␻t] Signup and view all the answers

What effect does opening the switch have on the circuit's current 'i(t)' for t>0?

The current changes rapidly before stabilizing. Signup and view all the answers

Which of the following represents the relationship between the angular frequency and the damped frequency?

␻ = ␻o √(1 - ␨^2) Signup and view all the answers

Given the period T of the sine function is 10 ms, what is the value of ␻?

200π rad/s Signup and view all the answers

What is the necessary inductor value given R = 50 Ω?

50.3 mH Signup and view all the answers

What is the damping factor $ heta imes eta$ calculated as?

497.0 Signup and view all the answers

How is the capacitor value calculated in relation to the inductor and resistance?

C = L / (R * (200π)^2) Signup and view all the answers

What is the calculated value of C?

31.0 μF Signup and view all the answers

Which of the following statements is true about design solutions?

Multiple design solutions can fulfill specifications. Signup and view all the answers

What does the output voltage plot confirm about the design?

It matches the Lown waveform. Signup and view all the answers

What is $ heta$ in the equation $ heta^2 + (θeta)^2 = heta_o^2$?

ω Signup and view all the answers

Which factor is crucial to create the proper output waveform for the design?

Using the correct values for R and L. Signup and view all the answers

What is the equivalent resistance across the 6$ ext{Ω}$ and 3$ ext{Ω}$ resistors in series?

9$ ext{Ω}$ Signup and view all the answers

If the voltage source provides 12V, what is the total voltage drop across the 12$ ext{Ω}$ resistor?

6V Signup and view all the answers

What would happen to the current in the circuit if the capacitance is increased to 1F?

The current would decrease Signup and view all the answers

What is the initial current flowing through the 5$ ext{Ω}$ resistor at time t=0?

6A Signup and view all the answers

In this circuit, which component primarily affects the transient response for t > 0?

The capacitors Signup and view all the answers

What is the expression for $v_R(t)$ when $t > 0$ in the given circuit?

$100e^{-400t} V$ Signup and view all the answers

If the initial voltage across the capacitor $v_C(0)$ is 10 V, what can be inferred about the circuit behavior?

The capacitor starts charging immediately. Signup and view all the answers

What is the purpose of the circuit configuration described in the problems?

To evaluate transient responses of the circuit. Signup and view all the answers

In the context of the network described, how is $i_o(t)$ expressed in the differential equation?

$rac{d^2i_o(t)}{dt^2} + 4rac{di_o(t)}{dt}$ Signup and view all the answers

For the underdamped circuit with initial conditions $i_L(0) = 1 A$ and $v_C(0) = 10 V$, what factor influences the voltage $v(t)$?

The damping ratio of the system. Signup and view all the answers

Study Notes

Lown Waveform and Underdamped Circuits

The Lown waveform indicates the circuit is underdamped, defined by the condition ( \zeta < 1 ).
The voltage applied to the patient takes the form:
( v_o(t) = K_1 e^{-\zeta \omega_o t} \sin(\omega t) ).

Definitions and Formulas

Damping ratio ( \zeta ) is calculated as:
( \zeta \omega_o = \frac{R}{2L} ).
Natural frequency ( \omega_o ) is given by:
( \omega_o = \frac{1}{\sqrt{LC}} ).
Calculated frequency ( \omega ):
( \omega = \omega_o \sqrt{1 - \zeta^2} ).

Circuit Characteristics

The period of the sine function is:
( T = 10 , \text{ms} ) which contributes to calculating ( \omega ) as:
( \omega = \frac{2\pi}{T} = 200\pi , \text{rad/s} ).

Voltage Ratio at Specific Time Intervals

At ( t = \frac{T}{4} ), ( \sin ) function equals +1; at ( t = \frac{3T}{4} ), it equals -1.
Ratio expression established:
( \frac{v_o(t/4)}{-v_o(3T/4)} = e^{\zeta \omega_o (T/2)} = \frac{3000}{250} = 12 ).
Results in ( \zeta \omega_o = 497.0 ).

Component Specifications

Given resistance ( R = 50 , \Omega ), the necessary inductor value is:
( L = 50.3 , \text{mH} ).
After solving for ( C ), the capacitance is found to be:
( C = 31.0 , \mu F ).

Design Verification

Circuit output voltage plot matches the Lown waveform, indicating design success.
Recognition that multiple design approaches could fulfill specifications, underscoring versatility in circuit design.

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Description

This quiz covers the characteristics of oscillatory circuits, particularly focusing on the Lown waveform. Participants will explore concepts like underdamped circuits and the mathematical representation of voltage applied to a patient. Test your understanding of these advanced electrical engineering principles.