Ch. 8: Basic RL and RC Circuits

Question 1

Homogeneous Linear

Differential Equations:

Definition

Answer 1

A simple type of Differential Equation.

Every term is either of the first degree in the independent variable, or is one of its derivatives.

(y’ or lower, no y’’ or above terms)

Question 2

Homogeneous Linear Differential Equations:

a df/dt + bf = 0

Finding the Characteristic Equation

and General Solution

(Steps)

Answer 2

• Start w/ general form:
  
  • adf/dt + bf = 0
• Substitute s and s(0)=1 for df/dt and f
  
  • as + b = 0
  • This is the Characteristic Equation
• Obtain the root
  
  • s = -b/a
• The General Solution is given by
  
  • f = Ae^st
• Therefore:
  
  • f = Ae^(-b/a)t

Question 3

Source Free RL

Circuit Diagram

and Differential Equation derivation

Answer 3

Assume that initial current exists: i(0) = I_o

Start with the KVL Equation:

v_r + v_L = 0

Ri + Ldi/dt = 0

Rearrange to obtain the differential equation

di/dt + (R/L)i = 0

Question 4

Source Free

Series RL Circuit

Time Constant

𝝉_RL

Answer 4

𝝉_{RL = L/R}

Question 5

RL Circuits with multiple Inductors:

Number of negative exponential terms

Answer 5

Corresponds to the number of inductors that remain after all possible inductor combinations have been made.

The circuit is as simplified as possible.

Question 6

Source Free RL Circuit:

Natural Response

(Equation)

Answer 6

i(t) = I_oe^-t/𝝉

= I_oe^-Rt/L

where 𝝉 = L/R

Question 7

Two Response Components

in RLC Circuits

Answer 7

Natural/Transient Response

Response due to the nature of the circuit, from a starting point (initial conditions) with no external forces.

Can be found by analyzing the source-free circuit.

Forced/Steady-State Response

The response due to independent sources acting on a circuit.

Complete Response = Natural Response + Forced Response

Question 8

Source Free RC Circuit:

Circuit

Differential Equation

General solution

Answer 8

Starting with the KCL Equation:

i_c + i_R = 0

Cdv/dt + v/R = 0

Rearrange to get the differential equation

dv/dt + v/RC = 0

With a general solution:

v(t) = V₀e^-t/RC

Question 9

Source Free RC Circuit:

Time Constant

𝝉_RC

Answer 9

𝝉_{RC = RC}

Question 10

Time Constant:

Definition

Answer 10

The time that it would take to discharge an element, if the discharge continued at the initial rate.

It is designated by 𝝉 (tau)

𝝉_RL = L/R 𝝉_RC = RC

And is part of the Exponential Response (charging/discharging).

It takes approximately 5 time constants for a circuit to actuall discharge.

Question 11

Time Constant:

How the number of resistors in a circuit alters the time constant.

Answer 11

It doesn’t, assuming the Equivalent Resistance is the same.

Question 12

Unit Step Function

u(t)

Mathematical Definition

Answer 12

u(t) = 0, when t <0

= 1, when t > 0

Question 13

Unit Step Function:

Shifted Step function

u(t - t₀)

definition

Answer 13

u(t-t0) = 0 , t < t₀

= 1 , t > t₀

Question 14

Source Free RC Circuit:

Natural Response

(Discharging)

Answer 14

v(t) = V₀e^-t/𝝉

v(t) = V₀e^-t/RC

Question 15

Source Free RC Circuit:

Discharge

Energy absorbed by Resistor

w_R(t)

Answer 15

w_R(t) = (1/2)C V₀²( 1 - e^-2t/𝝉 )

Question 16

Source Free RC Circuit:

Discharge:

Power p(t)

dissipated by the Resistor

p_R(t)

Answer 16

Instantaneous steady-state power is

p_R(t) = V²/R

p_R(t) = (1/R) V₀² e^-2t/𝝉

Question 17

Source Free RC Circuit:

Discharge:

Current
i_R

Answer 17

Steady State is:

i_R(t) = v(t)/R

So discharging:

i_R(t) = (V₀/R) e^-t/𝝉

Question 18

Source Free RC Circuit:

Time to discharge

until

Capacitor Voltage < 1 % ?

Answer 18

Takes about 5 time constants

Question 19

RC Circuit:

Step Response

Answer 19

v(t) = v0 when t < 0

= V_s + (V₀ - V_s)e^-t/𝝉 when t > 0