lecture 17 1

First-Order Circuits (6.1-6.2)

Prof. Phillips

March 24, 2003

lecture 17 2

1st Order Circuits

• Any circuit with a single energy storage element, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 1.

• Any voltage or current in such a circuit is the solution to a 1st order differential equation.

lecture 17 3

Important Concepts

• The differential equation

• Forced and natural solutions

• The time constant

• Transient and steady-state waveforms

lecture 17 4

A First-Order RC Circuit

• One capacitor and one resistor• The source and resistor may be equivalent

to a circuit with many resistors and sources.

R

Cvs(t)

+

–

vc(t)

+ –vr(t)

+–

lecture 17 5

Applications Modeled bya 1st Order RC Circuit

• Computer RAM

– A dynamic RAM stores ones as charge on a capacitor.

– The charge leaks out through transistors modeled by large resistances.

– The charge must be periodically refreshed.

lecture 17 6

The Differential Equation(s)

KVL around the loop:

vr(t) + vc(t) = vs(t)

R

Cvs(t)

+

–

vc(t)

+ –vr(t)

+–

lecture 17 7

Differential Equation(s)

)()(1

)( tvdxxiC

tRi s

t

dt

tdvCti

dt

tdiRC s )(

)()(

dt

tdvRCtv

dt

tdvRC s

rr )(

)()(

lecture 17 8

What is the differential equation for vc(t)?

lecture 17 9

A First-Order RL Circuit

• One inductor and one resistor• The source and resistor may be equivalent

to a circuit with many resistors and sources.

v(t)is(t) R L

+

–

lecture 17 10

Applications Modeled by a 1st Order LC Circuit

• The windings in an electric motor or generator.

lecture 17 11

The Differential Equation(s)

KCL at the top node:

)()(1)(

tidxxvLR

tvs

t

v(t)is(t) R L

+

–

lecture 17 12

The Differential Equation

dt

tdiL

dt

tdv

R

Ltv s )()()(

lecture 17 13

1st Order Differential Equation

Voltages and currents in a 1st order circuit satisfy a differential equation of the form

)()()(

tftvadt

tdv

lecture 17 14

Important Concepts

• The differential equation

• Forced (particular) and natural (complementary) solutions

• The time constant

• Transient and steady-state waveforms

lecture 17 15

The Particular Solution

• The particular solution vp(t) is usually a weighted sum of f(t) and its first derivative.– That is, the particular solution looks like the

forcing function

• If f(t) is constant, then vp(t) is constant.

• If f(t) is sinusoidal, then vp(t) is sinusoidal.

lecture 17 16

The Complementary Solution

The complementary solution has the following form:

Initial conditions determine the value of K.

/)( ttac KeKetv

lecture 17 17

Important Concepts

• The differential equation

• Forced (particular) and natural (complementary) solutions

• The time constant

• Transient and steady-state waveforms

lecture 17 18

The Time Constant ()

• The complementary solution for any 1st order circuit is

• For an RC circuit, = RC

• For an RL circuit, = L/R

/)( tc Ketv

lecture 17 19

What Does vc(t) Look Like?

= 10-4

lecture 17 20

Interpretation of

• The time constant, is the amount of time necessary for an exponential to decay to 36.7% of its initial value.

• -1/ is the initial slope of an exponential with an initial value of 1.

lecture 17 21

Implications of the Time Constant

• Should the time constant be large or small:

– Computer RAM

– A sample-and-hold circuit

– An electrical motor

– A camera flash unit

lecture 17 22

Important Concepts

• The differential equation

• Forced (particular) and natural (complementary) solutions

• The time constant

• Transient and steady-state waveforms

lecture 17 23

Transient Waveforms

• The transient portion of the waveform is a decaying exponential:

lecture 17 24

Steady-State Response

• The steady-state response depends on the source(s) in the circuit.

– Constant sources give DC (constant) steady-state responses.

– Sinusoidal sources give AC (sinusoidal) steady-state responses.

lecture 17 25

LC Characteristics

Element V/I Relation DC Steady-State

Resistor V(t) = R I(t) V = I R

Capacitor I(t) = C dV(t)/dt I=0; open

Inductor V(t) = L dI(t)/dt V=0; short

lecture 17 26

Class Examples