Notes 1

Question 1

What are transients in circuits?

Answer 1

Transients are temporary responses in circuits when a voltage or current is applied, transitioning to a steady state response.

Question 2

What occurs during steady state behaviour?

Answer 2

Steady state behaviour occurs when all derivatives are zero.

Question 3

Why are time-domain equations important in circuit analysis?

Answer 3

The time-domain equations for resistor (R), inductors (I), and capacitors (C) are crucial for analysing circuit behaviour over time.

Question 5

What is the steady state current ( i_{0-} ) before the switch is opened?

Answer 5

6 A

The inductor is at steady state at ( t = 0- )

Question 6

What is the resistance ( R ) in the circuit?

Answer 6

8 Ω

Calculated as ( R = rac{12 + 4}{16} )

Question 7

What is the time constant ( au ) for the circuit?

Answer 7

0.25 s

( au = rac{L}{R} = rac{2}{8} )

Question 8

What is the expression for current ( i(t) ) for ( t > 0 )?

Answer 8

i(t) = 6e^{-t/τ} A

Question 9

What is the expression for voltage ( v(t) )?

Answer 9

v(t) = -48e^{-t/τ} V

Question 10

Find ( i(0.5) ).

Answer 10

Approximately 0.8 A

Question 11

Find ( t ) when ( i(t) = 3 ).

Answer 11

Approximately 0.17 s

Question 12

True or False: In an RC circuit, ( i ) can be infinite.

Answer 12

False

Question 13

What is the maximum achievable clock frequency for a microprocessor circuit with ( R = 1 Ω ) and ( C = 200 pF )?

Answer 13

50 MHz

Question 14

What is the formula for the time constant ( τ ) in an RC circuit?

Answer 14

τ = RC

Question 15

What is the minimum period ( T_{min} ) of the square wave signal for the RC transient?

Answer 15

2 × 10^{-9} s

Question 16

What is the forced response ( i_F(t) ) in the RL circuit?

Answer 16

i_F(t) = rac{V_s}{R}

Question 17

What is the natural response ( i_N(t) ) in the RL circuit?

Answer 17

i_N(t) = I_0 e^{-t/τ}

Question 18

What is the general expression for transient response?

Answer 18

y(t) = y_∞ + (y_0 - y_∞) e^{-t/τ}

Question 19

What type of equation is ODE2?

Answer 19

Non-homogeneous ODE

Question 20

What is the voltage across the capacitor ( v_c(t) ) in the RC circuit?

Answer 20

v_c(t) = V_s(1 - e^{-t/τ})

Question 21

In the RL circuit, what happens to the voltage and current at steady state?

Answer 21

Both voltage and current are constant

Question 22

What is the characteristic equation for the RL circuit?

Answer 22

s(L/R) + 1 = 0

Question 23

True or False: At steady state, the derivatives are zero.

Answer 23

True

Question 24

What is the response of the circuit ( v_{c}(t) ) at ( t > 0 ) if ( v_c(0) = 0 )?

Answer 24

v_c(t) = V_s(1 - e^{-t/τ})

Question 25

What is the forced response ( v_{cF}(t) ) in the RC circuit?

Answer 25

v_{cF}(t) = V_s

Question 26

What is the time constant ( τ ) for an RL circuit?

Answer 26

τ = L/R

Question 27

What does the term ( I_0 ) represent in the RL circuit?

Answer 27

Initial current

Question 28

What is the expression for the current ( i(t) ) in the RL circuit?

Answer 28

i(t) = I_0 e^{-t/τ} + i_F(t)

Question 29

Fill in the blank: The RC transient takes ______ time constants to decay.

Answer 29

five

Question 30

What is the formula for the time constant in an RC circuit?

Answer 30

𝜏 = 𝑅𝐶 ## Footnote The time constant represents the time it takes for a capacitor to charge to approximately 63.2% of the full voltage.

Question 31

What is the formula for the time constant in an RL circuit?

Answer 31

𝜏 = 𝐿/𝑅 ## Footnote This time constant indicates the time required for the current to reach approximately 63.2% of its final value.

Question 32

What is the general form of the solution for the natural response of a first-order RC circuit?

Answer 32

𝑦(𝑡) = 𝑦_0𝑒^{−𝑡/𝜏} ## Footnote This equation describes the transient response of the circuit over time.

Question 33

What is the characteristic equation for a homogeneous ODE in an RC circuit?

Answer 33

𝑠𝑅𝐶 + 1 = 0 ## Footnote This equation is derived from the differential equation governing the circuit behavior.

Question 34

What does the damping ratio (𝜉) indicate in an RLC circuit?

Answer 34

The damping ratio indicates the nature of the circuit's response: * 𝜉 = 1: Critically damped * 𝜉 > 1: Overdamped * 𝜉 < 1: Underdamped ## Footnote Each case describes how the circuit responds to changes in current or voltage.

Question 35

Define the resonant frequency (𝜔₀) in an RLC circuit.

Answer 35

𝜔₀ = 1/√(𝐿𝐶) ## Footnote The resonant frequency is the frequency at which the circuit naturally oscillates.

Question 36

What is the form of the solution for the transient response of a second-order RLC circuit?

Answer 36

𝑦(𝑡) = 𝐴₁𝑒^{𝑠₁𝑡} + 𝐴₂𝑒^{𝑠₂𝑡} ## Footnote This represents the sum of the responses due to each pole of the circuit.

Question 37

What happens at steady-state in an inductor and capacitor?

Answer 37

Inductor: behaves as a closed circuit (𝑖 ≠ 0) Capacitor: behaves as an open circuit (𝑣 = 0) ## Footnote This indicates that inductor current remains constant while capacitor voltage remains unchanged.

Question 38

What is the equation for the voltage response of a capacitor at time t?

Answer 38

𝑣(𝑡) = 𝑉_0𝑒^{−𝑡/𝜏} ## Footnote This equation describes how the voltage across a capacitor decreases over time after a sudden change.

Question 39

Fill in the blank: The natural response of an RLC circuit is determined by solving the _______.

Answer 39

Characteristic Equation ## Footnote The characteristic equation is derived from the differential equation governing the circuit.

Question 40

What are the steps to find the response ( y(t) ) in a circuit?

Answer 40

1. Apply Kirchhoff's laws to get an ODE 2. Find ( s ) 3. Use the initial condition ( y(0) ) to find ( A ) ## Footnote These steps guide the analysis of transient responses in circuits.

Question 41

True or False: The steady-state response of a circuit is temporary.

Answer 41

False ## Footnote The steady-state response is permanent as long as the input remains constant.

Question 42

What is the relationship between the damping ratio (𝜉) and the Neper frequency (𝛼)?

Answer 42

𝜉 = 𝛼/𝜔₀ ## Footnote This relationship helps to understand how oscillations decay in an RLC circuit.

Question 43

What is the expression for the current in an inductor?

Answer 43

𝑖 = 𝐿 dfrac{𝑑𝑖}{𝑑𝑡} ## Footnote This equation describes the relationship between current and the rate of change of current in an inductor.

Question 44

What is the initial condition for a capacitor in steady-state before a switch is flipped?

Answer 44

𝑣(0−) = 𝑉_0 ## Footnote This indicates the voltage across the capacitor just before the switch operation.

Question 45

What is the condition for real and distinct poles in the characteristic equation?

Answer 45

𝜉 > 1 ## Footnote This condition results in an overdamped response where the system returns to equilibrium without oscillating.

Question 46

What is the formula for the current response in an underdamped RLC circuit?

Answer 46

𝑖(𝑡) = 𝑖_0 e^{-𝜉𝜔₀𝑡} sin(𝜔_d t + 𝜑) ## Footnote This represents a decaying sinusoidal response in the underdamped case.

Question 47

What happens to the voltage across a capacitor as time approaches infinity?

Answer 47

𝑣(𝑡) approaches steady-state value 𝑣_∞ ## Footnote This indicates that the capacitor will eventually stop charging or discharging, reaching a stable voltage.