In-Class Problems

1. Sketch the following functions:a) x(t) = 3sin(40t) for 0≤ t ≤ 0.2 secb) z(t) = 10e-4t for 0≤ t ≤0.5 sec

2. What is ?

3. What is ?

4. What is ?

)40sin(3)( tdt

dtf

tedt

dty 410)(

tdt

dtg 6)(

Capacitor-Resistor Circuits

• In-class activity:• In pairs build this circuit in Multisim.• Look at the voltage across the capacitor on the

oscilloscope.• Describe what you see when the switch moves

between positions (let the switch stay in each position until the capacitor voltage stops changing).

V112 V

R1

1k

C11uF

J1

Key = Space

1

2

4

0

Discharge of a Capacitor Through a Resistor

• In the following circuit, when the switch moves from the battery to the wire connected to ground, the voltage across the capacitor is vc(t) = Vse-t/RC

V112 V

R1

1k

C11uF

J1

Key = Space

1

2

4

0

Vs

-vc

Class Activity

• What are the units of RC in vc(t) = Vse-t/RC ?

• Since vc(t) = Vse-t/RC and , what is i(t) in terms of Vs, R and C?

• What do vc(t) and i(t) look like on a graph?

cdvi t Cdt

Step Response: RC Time Constants

• Now, what happens when the switch moves the other way?

V112 V

R1

1k

C11uF

J1

Key = Space

1

2

4

0

The response of the capacitor voltage will be to charge up to the supply voltage.

Vs

Vc Response to Constant Voltage Vs

• The voltage across the capacitor will rise and asymptotically approach Vs

How can we describe this mathematically?

Kirchhoff’s voltage loop law S C RV v v

Ohm’s law across resistor Rv iRSubstituting for VR gives s cV v iR

Analysis of RC Circuits

V112 V

R1

1k

C11uF

J1

Key = Space

1

2

4

0

Vs

vc

Substitute in for i

From previous pageS CV v iR

CS C

dvV v RC

dt

Analysis of RC Circuits

CdvCdt

V112 V

R1

1k

C11uF

J1

Key = Space

1

2

4

0

Vs

vc

Analysis of RC Circuits

• The equation is called a differential equation.

CS C

dvV v RC

dt

• The solution is of the form:

where is defined as the time constant

1t

C Sv V e

RC

= the circuit time constant, in seconds if and only if C = the total (connected) capacitance Farads R = the total (connected) resistance Ohms

Team Activity

CS C

dvV v

dt

• Substitute

into the equation

to show that LHS = RHS

1t

C Sv V e

Show that when t is 5 times the time constant, ,the capacitor voltage is 99.33% of the peak voltage.

1t

C

S

ve

V

Team Activity

100SV V

Kirchhoff’s voltage loop law ?

Ohm’s law across resistor?

Substituting for VR gives?

Team Activity – Discharge Process

V112 V

R1

1k

C11uF

J1

Key = Space

1

2

4

0

CC

dvv

dt• From previous activity the equation

t

C Sv V e

Team Activity

and show that LHS = RHS

• Substitute

into the above equation

Rectangular Wave

• If you repeatedly switch between the battery and the short you are effectively applying a rectangular time pulse to the RC circuit.

Rectangular Wave Response

• The voltage across the capacitor will behave as below in response to such a wave: