Lecture 14 1

1st Order Circuits

Lecture 14 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 14 3

Important Concepts

• The differential equation

• Forced and natural solutions

• The time constant

• Transient and steady state waveforms

Lecture 14 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 14 5

Applications Modeled by a 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 14 6

More Applications

• The low-pass filter for an envelope detector in a superhetrodyne AM receiver.

• A sample-and-hold circuit for a PCM encoder:

– The capacitor is charged to the voltage of a waveform to be sampled.

– The capacitor holds this voltage until an A/D converter can convert it to bits.

Lecture 14 7

The Differential Equation(s)

KCL around the loop:

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

R+

-Cvs(t)

+

-

vc(t)

+ -vr(t)

Lecture 14 8

Differential Equation(s)

)()(1

)( tvdxxiC

tRi s

t

dt

tdvC

dt

tdiRCti s )()(

)(

dt

tdvRC

dt

tdvRCtv sr

r

)()()(

Lecture 14 9

What is the differential equation for vc(t)?

Lecture 14 10

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 14 11

Applications Modeled by a 1st Order LC Circuit

• The windings in an electric motor or generator.

Lecture 14 12

The Differential Equation(s)

KCL at the top node:

v(t)is(t) R L

+

-

)()(1)(

tidxxvLR

tvs

t

Lecture 14 13

The Differential Equation

dt

tdiL

dt

tdv

R

Ltv s )()()(

Lecture 14 14

1st Order Differential Equation

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

)()(

)( tfdt

tdvatv

Lecture 14 15

Important Concepts

• The differential equation

• Forced (particular) and natural (complementary) solutions

• The time constant

• Transient and steady state waveforms

Lecture 14 16

The Particular Solution

• The particular solution vp(t) is usually a weighted sum of f(t) and its first derivative.

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

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

Lecture 14 17

The Complementary Solution

The complementary solution has the following form:

What value must have to give a solution to

/)( tc Ketv

0)(

)( dt

tdvatv c

c

Lecture 14 18

Complementary Solution

• How do I choose the value of K?

• The initial conditions determine the value of K.

atc Ketv /)(

Lecture 14 19

Important Concepts

• The differential equation

• Forced (particular) and natural (complementary) solutions

• The time constant

• Transient and steady state waveforms

Lecture 14 20

The Time Constant

• The complementary solution for any 1st order circuit is

• For an RC circuit, = RC

• For an LC circuit, = L/R

/)( tc Ketv

Lecture 14 21

What Does vc(t) Look Like?

= 10-4

Lecture 14 22

Interpretation of

• 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 14 23

Implications of the Time Constant

• Should the time constant be large or small:

– Computer RAM

– The low-pass filter for the envelope detector

– The sample-and-hold circuit

– The electrical motor

Lecture 14 24

Important Concepts

• The differential equation

• Forced (particular) and natural (complementary) solutions

• The time constant

• Transient and steady state waveforms

Lecture 14 25

Transient Waveforms

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

Lecture 14 26

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 14 27

Computer RAM

• Voltage across a memory capacitor may look like this:

Lecture 14 28

Low Pass Filter

• Voltage in the filter may look like this:

Lecture 14 29

Sample and Hold

• The voltage in the sample and hold circuit might look like this: