CH 7_BOOK_FIRST ORDER CIRCUITS Flashcards
1) The current through a capacitor is directly proportional to what?
2) The voltage across a capacitor is directly proportional to what?
3) The current through a capacitor is ________ unless the voltage is changing.
4) The voltage across a capacitor can/cannot change instantly?
1) The time rate of change of the voltage across it.
2) The time integral of the current through it.
3) Zero.
4) Cannot (but current can change instantaneously).
1) The voltage across an inductor is directly proportional to what?
2) The current through an inductor is directly proportional to what?
3) The voltage across an inductor is ________ unless the current is changing.
4) The current through an inductor can/cannot change instantly?
1) The time rate of change of the current through it.
2) The time integral of the voltage across it.
3) Zero.
4) Cannot (but voltage can change instantaneously).
Applying Kirchhoff’s laws to purely resistive circuits results in algebraic equations, while applying the laws to RC and RL circuits produces ___________ equations.
Differential
A first-order circuit is characterized by a first-order __________ equation.
Differential
Regarding the 4 possible situations studied in this chapter: What are the 2 types of first-order circuits? What are the 2 ways to excite them?
RC and RL.
- Initial conditions of the storage elements in the circuits (source-free).
- Independent sources.
How does energy behave in source-free circuits?
Without independent sources, we assume that energy is initially stored in the capacitive or inductive element. The energy causes current to flow in the circuit and is gradually dissipated in the resistors.
T/F: Source-free circuits are free of independent sources but they may have dependent soruces.
True
How do you define a circuit response?
A circuit response is the manner in which the circuit reacts to an excitation.
For Source-free RC circuits, what is the formula for voltage?
Define the “natural response” of a Source-free RC circuit.
The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation.
What is the circuit response in a Source-free RC circuit? What kind of response is it?
The voltage across the capacitor.
A natural response (due to the initial energy stored and the physical characteristics of the circuit, not due to some external voltage or current source).
VIEW: The voltage response of the RC circuit.
7.2. Note that at t = 0, we have the correct initial condition as in Eq. (7.1). As t increases, the voltage decreases toward zero. The rapidity with which the voltage decreases is expressed in terms of the time constant, denoted by τ, the lowercase Greek letter tau.
What is the time constant of a Source-free RC circuit?
The time constant of a circuit is the time required for the response to decay to a factor of 1/e or 36.8 percent of its initial value.
For Source-free RC circuits, what is the formula for time constant?
In Source-free RC circuits, it is customary to assume that the capacitor is fully discharged after how many time constants?
5
VIEW: Values of v(t)/V(0)
VIEW: Graphical determination of the time constant from the response curve.
For Source-free RC circuits, what is the formula for the current through the resistor?
For Source-free RC circuits, what is the formula for the power dissipated in the resistor?
For Source-free RC circuits, the formula for the energy absorbed by the resistor up to time t is?
In Source-free RC circuits, what happens as t -> infinity with regards to energy?
What are the 2 keys to working with a Source-Free RC circuit?
What can be obtained from these 2 values?
- Finding the initial voltage v(0) across the capacitor.
- Finding the time constant.
When finding the time constant in Source-free RC circuits, R is often the?
The Thevenin equivalent resistance at the terminals of the capacitor (take out the capacitor C and find R = R_th at its terminals.
What is the circuit response in a Source-free RL circuit? What kind of response is it?
The current through the inductor.
A natural response (the exponential decay of the initial current).
For Source-free RL circuits, what is the formula for current?
For Source-free RL circuits, what is the formula for the time constant?
For Source-free RL circuits, what is the formula for the voltage across the resistor?
For Source-free RL circuits, what is the formula for the power dissipated in the resistor?
For Source-free RL circuits, the formula for the energy absorbed by the resistor up to time t is?
In Source-free RL circuits, what happens as t -> infinity with regards to energy?
What are the 2 keys to working with a Source-Free RL circuit?
What can be obtained from these 2 values?
- The initial current i(0) through the inductor.
- The time constant of the circuit.
When finding the time constant in Source-free RL circuits, R is often the?
The Thevenin equivalent resistance at the terminals of the inductor.
The smaller the time constant τ of a circuit, the __________ the rate of decay of the response. The larger the time constant, the __________ the rate of decay of the response.
Faster, slower
When does a circuit reach steady state?
The response decays to less than 1 percent of its initial value (i.e., reaches steady state) after 5τ.
What is a singularity function?
Singularity functions (aka switching functions) are functions that either are discontinuous or have discontinuous derivatives.
What is the unit step function for?
What graphs and equations are associated with the unit step function?
VIEW: Unit step function: voltage source and its equivalent circuit
VIEW: Unit step function: current source and its equivalent circuit
What is the unit impulse function?
The unit impulse function (aka the delta function) is the derivative of the unit step function.
What graphs and equations are associated with the unit impulse function?
For the unit impulse function, define the unit impulse and the unit area.
What is the unit ramp function?
The unit ramp function is the integral of the unit step function.
What graphs and equations are associated with the unit ramp function?
How are the 3 singularity functions related?
What is the step response of a circuit?
The step response of a circuit is its behavior when the excitation is the step function, which may be a voltage or a current source. It is the response of the circuit due to a sudden application of a dc voltage or current source.
What is the formula for the complete response of an RC circuit to a sudden application of a dc voltage source? (Assuming the capacitor is initially charged).
Hint: Piecewise function
What are the two ways of decomposing the complete response of an RC circuit?
The first is to break it into a “natural response and a forced response” and the second is to break it into a “transient response and a steady-state response.”
Starting with the natural response and forced response, v_f is known as the forced response because it is produced by the circuit when an external “force” (a voltage source in this case) is applied. It represents what the circuit is forced to do by the input excitation. The natural response eventually dies out along with the transient component of the forced response, leaving only the steady-state component of the forced response.
Next is to see the two components as one being temporary and the other permanent. The transient response v_t is temporary; it is the portion of the complete response that decays to zero as time approaches infinity. Thus, the steady-state response v_ss is the portion of the complete response that remains after the transient response has died out. Thus, the first decomposition of the complete response is in terms of the source of the responses, while the second decomposition is in terms of the permanency of the responses. Under certain conditions, the natural response and transient response are the same. The same can be said about the forced response and steady-state response.
This is the same as saying that the complete response is the sum of the transient response and the steady-state response.
What is the main formula for the complete response of an RC circuit?
Where v(0) is the initial voltage at t = 0+ and v(∞) is the final or steady-state value.
What is the transient response?
What is the steady-state response?
What are the 3 things needed to find the step response of an RC circuit?
We obtain item 1 from the given circuit for t<0 and items 2 and 3 from the circuit for t > 0.
What is the general formula for the natural response?
Where x represents current through (or voltage across) a resistor, a capacitor, or an inductor, and x(0) is the initial value of x.
What is the general formula for the step response?
Finding the step response of a firstorder circuit requires the initial value x(0+), the final value x(∞), and the time constant τ.
What is the main formula for the complete response of an RL circuit?
Where i(0) and i(∞) are the initial and final values of i.
What are the 3 things needed to find the step response of an RL circuit?
We obtain item 1 from the given circuit for t < 0 and items 2 and 3 from the circuit for t > 0.
What is the formula for the complete response of an RL circuit to a sudden application of a dc voltage source? (Assuming the capacitor is initially uncharged).
Hint: Piecewise function
