EGR 2201 Unit 9 First-Order Circuits Read Alexander & Sadiku, Chapter 7. Homework #9 and Lab #9 due next week. Quiz next week. Review: DC Conditions in a Circuit with Inductors or Capacitors

Recall that when power is first applied to a dc circuit with inductors or capacitors, voltages and currents change briefly as the inductors and capacitors become energized. But once they are fully energized (i.e., under dc conditions), all voltages and currents in the circuit have constant values. To analyze a circuit under dc conditions, replace all capacitors with open circuits and replace all inductors with short circuits. What About the Time Before DC Conditions?

We also want to be able to analyze such circuits during the time while the voltages and currents are changing, before dc conditions have been reached. This is sometimes called transient analysis, because the behavior that were looking at is short-lived. Its the focus of Chapters 7 and 8 in the textbook. Four Kinds of First-Order Circuits

The circuits well study in this unit are called first-order circuits because they are described mathematically by first-order differential equations. Well study four kinds of first-order circuits: Source-free RC circuits Source-free RL circuits RC circuits with sources

RL circuits with sources Natural Response and Step Response The term natural response refers to the behavior of source-free circuits. And the term step response refers to the behavior of circuits in which a source is applied

at some time. So our goal in this unit is to understand the natural response of source-free RC and RL circuits, and to understand the step response of RC and RL circuits with sources. Natural Response of Source-Free RC Circuit (1 of 2) Consider the circuit shown. Assume that at time t=0, the capacitor is

charged and has an initial voltage, V0. As time passes, the initial charge on the capacitor will flow through the resistor, gradually discharging the capacitor. This results in changing voltage v(t) and currents iC(t) and iR(t), which we wish to calculate. Natural Response of Source-Free RC Circuit (2 of 2) Applying KCL,

Therefore Therefore This equation is an example of a firstorder differential equation. How do we solve it for v(t)? Math Detour: Differential Equations Differential equations arise frequently in science and engineering. Some examples:

dv 2 v 8 dt d 2v dv 5 6v 3t 2 dt dt A first-order diff. eq. A second-order diff. eq. d 4 v d 3v dv

3 4 3 8 sin t A fourth-order diff. eq. dt dt dt The equations above are all called linear ordinary differential equations with constant coefficients. Solving Differential Equations The differential equations on the previous slide are quite easy to solve. The ones shown below are more difficult. dv 2t 7v 0 Non-constant coefficient dt

2 dv 3 2 (7v) sin t dt Non-linear v v 2 2 7 t A partial differential equation t x In a later math course youll learn many

techniques for solving such equations. A Closer Look at Our Differential Equation Note that our equation, , contains two constants, R and C. It also contains two variables, v and t. Also, t is the independent variable, while v is the dependent variable. We sometimes indicate this by writing v(t) instead of just v. Our goal is to write down an equation that

expresses v(t) in terms of t, such as: (But neither of those is right!) Solving Our Differential Equation To solve our equation, , use a technique called separation of variables. First, separate the variables v and t: Then integrate both sides:

Then raise e to both sides: where A is a constant Apply the Initial Condition At this point we have: The last step is to note that if we set t equal to 0, we get:

But we assumed earlier that the initial voltage is some value that we called V0. So A must be equal to V0, and therefore: The Bottom Line I wont expect you to be able to reproduce the derivation on the previous slides. The important point is to realize that whenever we have a circuit like this the solution for v(t) is: Graph of Voltage Versus Time

Heres a graph of This curve is called a decaying exponential curve. Note that at first the voltage falls steeply from its initial value (V0). But as time passes, the descent becomes less steep. The Time Constant

The values of R and C determine how rapidly the voltage descends. The product RC is given a special name (the time constant) and symbol (): The greater is, the more slowly the voltage descends. Units of the Time Constant

What are the units of the time constant ? Since and since R is measured in ohms () and C is measured in farads (F), is measured in ohm-farads (F). Surprisingly, an ohm-farad is equivalent to a second. So is measured in seconds (s). Dont Confuse t and Since and = RC, we often write

t and are both measured in seconds. But remember that t is a variable (time), and is a constant (the time constant). Rules of Thumb After one time constant (i.e., when t = ), the voltage has fallen to about 36.8% of its initial value.

After five time constants (i.e., when t = 5), the voltage has fallen to about 0.7% of its initial value. For most practical purposes we say that the capacitor is completely discharged and v = 0 after five time constants. Comparing Different Values of The greater is, the more slowly the voltage descends, as shown below for a few values of . Finding Values of Other Quantities

Weve seen that From this equation we can use our prior knowledge to find equations for other quantities, such as current, power, and energy. For example, using Ohms law we find that The Keys to Working with a Source-Free RC Circuit 1. 2. 3. 4.

Find the initial voltage across the capacitor. Find the time constant Once you know these two items, voltage as a function of time is: Once you know the voltage, use Ohms law, Kirchhoffs laws, power formula, energy equations, and so on to solve for any other circuit variables of interest. More Complicated Cases A circuit that looks more complicated at first might be reducible to a simple sourcefree RC circuit by combining resistors. Example:

Here we can combine the three resistors into a single equivalent resistor, as seen from the capacitors terminals. Where Did V0 Come From? In previous examples youve simply been given the capacitors initial voltage, V0. More realistically, you have to find V0 by considering what happened before t = 0. Example: Suppose youre told that the switch in this circuit has been closed for a long time

before its opened at t = 0. Can you find the capacitors voltage V0 at time t = 0? Natural Response of Source-Free RL Circuit (1 of 2) Consider the circuit shown. Assume that at time t=0, the inductor is energized and has an initial current, I0. As time passes, the

inductors energy will gradually dissipate as current flows through the resistor. This results in changing current i(t) and voltages vL(t) and vR(t), which we wish to calculate. Natural Response of Source-Free RL Circuit (2 of 2) Applying KVL, Therefore

Therefore This first-order differential equation is similar to the equation we had for source-free RC circuits. Solving Our Differential Equation To solve our equation, first separate the variables i and t:

Then integrate both sides: Then raise e to both sides: Apply the Initial Condition At this point we have: The last step is to note that if we set t equal to 0, we get:

But we assumed earlier that the initial current is some value that we called I0. So A must be equal to I0, and therefore: The Bottom Line I wont expect you to be able to reproduce the derivation on the previous slides. The important point is to realize that whenever we have a circuit like this the solution for i(t) is:

The Time Constant The values of R and L determine how rapidly the current decreases from its initial value to 0. Recall that for RC circuits we defined the time constant as For RL circuits, we define it as The greater is, the more slowly the current decreases from its initial value.

Units of the Time Constant What are the units of the time constant ? Since and since L is measured in henries (H) and R is measured in ohms (), is measured in henries-per-ohm (H/). Surprisingly, a henry-per-ohm is equivalent to a second. So is measured in seconds (s). Dont Confuse t and Since

and = L/R, we often write t and are both measured in seconds. But remember that t is a variable (time), and is a constant (the time constant). Graph of Current Versus Time Heres a graph of

Its a decaying exponential curve, with the current falling steeply from its initial value (I0). But as time passes, the descent becomes less steep. Rules of Thumb After one time constant (i.e., when t = ), the current has fallen to about 36.8% of its initial value. After five time constants (i.e., when t = 5),

the current has fallen to about 0.7% of its initial value. For most practical purposes we say that the inductor is completely deenergized and i = 0 after five time constants. Finding Values of Other Quantities Weve seen that From this equation we can use our prior knowledge to find equations for other quantities, such as voltage, power, and energy. For example, using Ohms law we find that The Keys to Working with a Source-Free RL Circuit

1. 2. 3. 4. Find the initial current through the inductor. Find the time constant Once you know these two items, current as a function of time is: Once you know the current, use Ohms law, Kirchhoffs laws, energy equations, and so on to solve for any other circuit variables of interest.

More Complicated Cases A circuit that looks more complicated at first might be reducible to a simple sourcefree RL circuit by combining resistors. Example: Here we can combine the resistors into a single equivalent resistor, as seen from the inductors terminals. Where Did I0 Come From? In previous examples you were simply given the inductors initial current, I0.

More realistically, you have to find I0 by considering what happened before t = 0. Example: Suppose youre told that the switch in this circuit has been closed for a long time before its opened at t = 0. Can you find the inductors current I0 at time t = 0? Where We Are Weve looked at: Source-free RC circuits

Source-free RL circuits We still need to look at: RC circuits with sources RL circuits with sources Before doing this, well look at some mathematical functions called singularity functions (or switching functions), which are widely used to model electrical signals that arise during switching operations.

Three Singularity Functions The three singularity functions that well study are: The unit step function The unit impulse function The unit ramp function

The Unit Step Function The unit step function u(t) is equal to 0 for negative values of t and equal to 1 for positive values of t: 0, <0 ( )= 1, > 0 { Shifting and Scaling the Unit Step Function

We can obtain other step functions by shifting the unit step function to the left or right 0, <2 s ( 2)= 1, >2 s { or by multiplying the unit step function by a scaling constant:

0, <0 3 ()= 3, >0 { Flipping the Unit Step Function Horizontally As with any mathematical function, we can flip the unit step function horizontally by replacing t with t: 1, <0

()= 0, >0 { Adding Step Functions By adding two or more step functions we can obtain more complex steplike functions, such as the one shown below from the books Practice Problem 7.6. Using Step Functions to Model Switched Sources Step functions are useful for modeling

sources that are switched on (or off) at some time: The Unit Impulse Function The unit step functions derivative is the unit impulse function (t), also called the delta function. The unit impulse function is 0 everywhere except at t =0, where it is undefined: 0, <0 ( ) = Undefined , =0 0, >0

{ Its useful for modeling spikes that can occur during switching operations. Shifting and Scaling the Unit Impulse Function We can obtain other impulse functions by shifting the unit impulse function to the left or right, or by multiplying the unit impulse function by a scaling constant:

We wont often use impulse functions in this course, but theyre used in more advanced courses. The Unit Ramp Function The unit step functions integral is the unit ramp function r(t). The unit ramp function is 0 for negative values of t and has a slope of 1 for positive values of t: 0, <0

()= , 0 { Shifting and Scaling the Unit Ramp Function We can obtain other ramp functions by shifting the unit ramp function to the left or right 0, < 0.2 s

( 0.2 ) = , 0.2 s { or by multiplying the unit ramp function by a scaling constant: 0, < 0 4 ( )= 4 , 0 {

Adding Step and Ramp Functions By adding two or more step functions or ramp functions we can obtain more complex functions, such as the one shown below from the books Example 7.7. Step Response of a Circuit A circuits step response is the circuits behavior due to a sudden application of a dc voltage source

or current source. We can use a step function to model this sudden application. t = 0 versus t = 0+ We distinguish the following two times: t = 0 (the instant just before the switch closes) t = 0+ (the instant just after the switch closes)

Since a capacitors voltage cannot change abruptly, we know that v(0) = v(0+) in this circuit. But on the other hand, i(0) i(0+) in this circuit. (A capacitors current can change abruptly.) Step Response of RC Circuit (1 of 2)

Assume that in the circuit shown, the capacitors initial voltage is V0 (which may equal 0 V). As time passes after the switch closes, the capacitors voltage will gradually approach the source voltage VS. This results in changing voltage v(t) and current i(t), which we wish to calculate. Step Response of RC Circuit (2 of 2) Applying KCL, for t>0,

Therefore Separating variables, Integrating both sides, Solution of Our Differential Equation Raising e to both sides and rearranging:

Applying the initial condition and letting = RC: So, finally, ( )= { 0 , <0 +( 0 )

, 0 The Bottom Line I wont expect you to be able to reproduce the derivation on the previous slides. The important point is to realize that for a circuit like this: the solution for v(t) is: ( )=

{ 0 , <0 +( 0 ) , 0 Graph of Voltage Versus Time Depending on whether V0 < VS or V0 > VS, the

0 , <0 graph of ( )= { +( 0 ) , 0 will have one of these shapes: A saturating exponential curve.

A decaying exponential curve. Rules of Thumb, etc. We can repeat many of the same remarks as for source-free circuits, such as: The greater is, the more slowly v(t) approaches its final value. For most practical purposes, v(t) reaches its final value after 5. Knowing v(t), we can use Ohms law to find current, and we can use other familiar formulas to find power, energy, etc. Another Way of Looking At It

We can rewrite the complete response as: Here v(0) is the initial value and v() is the final, or steady-state, value. This same equation works for source-free RC circuits too, since setting v() to 0 gives . The Keys to Finding an RC Circuits Step Response

1. 2. 3. 4. 5. Find the capacitors initial voltage . Find the capacitors final voltage . Find the time constant Once you know these items, voltage is: Once you know the voltage, solve for any other circuit variables of interest. More Complicated Cases A circuit that looks more complicated at first might be reducible to a simple RC

circuit by combining resistors. Example: Here, for t > 0 we can combine the resistors into a single equivalent resistor, as seen from the capacitors terminals. Two Ways of Breaking It Down There are two useful ways of looking at the response 1. 2.

As the sum of a natural response and a forced response. As the sum of a transient response and a steady-state response. This way is of more interest to us. Natural Response Versus Forced Response We can think of the complete response as being the sum of: 1. A natural response 2.

that depends the capacitors initial charge. Plus a forced response that depends on the voltage source. Transient Response Versus Steady-State Response We can think of the complete response as being the sum of: 1. A transient response that dies away as time passes. 2.

Plus a steady-state response that remains after the transient response has died away. Under DC Conditions = SteadyState Recall that earlier we used the term under dc conditions to refer to the time after an RC or RL circuits currents and voltages have settled down to their final values. This is just another way of referring to what were now calling steady-state values.

So way can say that in the steady state, capacitors look like open circuits and inductors look like short circuits. Step Response of RL Circuit (1 of 2) Assume that in the circuit shown, the inductors initial current is I0 (which may equal 0 A). As time passes, the inductors current will gradually approach

a steady-state value. This results in changing current i(t) and voltage v(t), which we wish to calculate. Step Response of RL Circuit (2 of 2) Using the same sort of math we used previously for RC circuits, we find 0 , < 0 ( )= + 0

, >0 { ( ) where = L/R, just as for source-free RL circuits. Another Way of Looking At It We can rewrite

as: Here i(0) is the initial value and i() is the final, or steady-state, value. This same equation works for source-free RL circuits too, since setting i() to 0 gives . Graph of Current Versus Time Depending on whether i(0) < i() or i(0) > i(), the graph of

will have one of these shapes: A saturating exponential curve. A decaying exponential curve. Rules of Thumb, etc. We can repeat many of the same remarks as for previous circuits, such as: The greater is, the more slowly i(t) approaches its final value. For most practical purposes, i(t) reaches its final value after 5. Knowing i(t), we can use Ohms law to

find voltage, and we can use other familiar formulas to find power, energy, etc. The Keys to Finding an RL Circuits Step Response 1. 2. 3. 4. 5. Find the inductors initial current . Find the capacitors final current . Find the time constant Once you know these items, current is: Once you know the current, solve for any other circuit variables of interest.

More Complicated Cases A circuit that looks more complicated at first might be reducible to a simple RL circuit by combining resistors. Example: Here we can combine the resistors into a single equivalent resistor, as seen from the inductors terminals. A General Approach for First-Order Circuits (1 of 3) As noted in the margin note on page

274 of the textbook: Once we know , , and , almost all the problems in this chapter can be solved using the formula where x is a current or voltage in a first-order circuit. A General Approach for First-Order Circuits (2 of 3) 1. 2. 3. 4. So our general approach for finding x(t) in

a first-order circuit is: Find the quantitys initial value . Find the quantitys final value . Find the time constant: for an RC circuit. for an RL circuit. Once you know these items, solution is: A General Approach for First-Order Circuits (3 of 3) The equation from the previous slide, always graphs as either:

A decaying exponential curve if the initial value x(0) is greater than the final value x(). Or a saturating exponential curve if the initial value x(0) is less than the final value x().