1

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.

2

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)

3

The Differential Equation(s)

KVL around the loop:

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

R+

-Cvs(t)

+

-

vc(t)

+ -vr(t)

4

Differential Equation(s)

)()(1

)( tvdxxiC

tRi s

t

dt

tdvC

dt

tdiRCti s )()(

)(

dt

tdvRC

dt

tdvRCtv sr

r

)()()(

5

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

+

-

6

The Differential Equation(s)

KCL at the top node:

v(t)is(t) R L

+

-

)()(1)(

tidxxvLR

tvs

t

dt

tdiL

dt

tdv

R

Ltv s )()()(

7

)0();(

0,)(

211

21

vKKvK

teKKtv RC

t

)0();(

0,)(

211

21

iKKiK

teKKti L

Rt

)(ti

)0();(

0,)(

211

21

iKKiK

teKKti RC

t

Why? (Superposition)

8

Solving First Order Circuits1. Draw the circuit for t=0- and find v(0-), or i(0-)

2. Use the continuity of the capacitor voltage, or inductor current, draw the circuit for t=0+ to find v(0+), or i(0+)

3. Find v( ), or i( ) at steady state

4. Find the time constant – For an RC circuit, = RC– For an RL circuit, = L/R

5. The solution is:/)]()0([)()( texxxtx

9

The Time Constant

• For an RC circuit, = RC

• For an RL circuit, = L/R• -1/ is the initial slope of an exponential with an

initial value of 1• Also, is the amount of time necessary for an

exponential to decay to 36.7% of its initial value

10

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

11

2nd Order Circuits

• Any circuit with a single capacitor, a single inductor, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 2.

• Any voltage or current in such a circuit is the solution to a 2nd order differential equation.

12

A 2nd Order RLC Circuit

• The source and resistor may be equivalent to a circuit with many resistors and sources.

R+

-Cvs(t)

i (t)

L

13

Applications Modeled by a 2nd Order RLC Circuit

• Filters– A bandpass filter such as the IF amp for the

AM radio.– A lowpass filter with a sharper cutoff than can

be obtained with an RC circuit.

14

The Differential Equation

KVL around the loop:

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

R+

-Cvs(t)

+

-

vc(t)

+ -vr(t)

L

+- vl(t)

i (t)

15

Differential Equation (cont’d)

)()(1)(

)( tvdxxiCdt

tdiLtRi s

t

dt

tdv

Lti

LCdt

tdi

L

R

dt

tid s )(1)(

1)()(2

2

16

The Differential Equation

Most circuits with one capacitor and inductor are not as easy to analyze as the previous circuit. However, every voltage and current in such a circuit is the solution to a differential equation of the following form:

)()()(

2)( 2

002

2

tftidt

tdi

dt

tid

17

Example response: Over Damped

0

0.2

0.4

0.6

0.8

1

-1.00E-06

t

i(t)

-0.2

0

0.2

0.4

0.6

0.8

-1.00E-06

ti(t)

18

Example Response: Under Damped

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.00E-05 1.00E-05 3.00E-05

t

i(t)