RC Circuit Analysis

1
RC Circuit Analysis

• Review of Kirchoffs rules
• Circuit with resistance and capacitance
• Charging and discharging

• ANNOUNCEMENT Hour Exam I, Thursday February 19,
  545 - 7 PM
• Rooms announced by email
• Alternative exam room sign up sheet available
  today and Monday
• Material from Chapters 15 -18 inclusive
• Exam based on required reading, pre-flights,
  lectures, homework, discussion quizzes and lab
  material.
• One page of notes (8.5 x 11) allowed, front and
  back, hand written
• 20 multiple choice questions plus test code
• Scantron will be used - bring 2 HB pencils
  calculator
• You must know your student Id and section
  number (301 - 330) fill them in on the test.

2
Resistor Circuits

• Simple EMF and load

Example Emf with ?V110V 0 - side, 110 side.
Connected to the load resistor by conductors.
Therefore 110 side and 0 on - side of the
resistor. Apply Ohms law I V/R
Simple circuits

• Resistors in series and parallel

Add sets of series and parallel resistances
reducing the problem to the one above
Complex Circuits

• Apply Kirchoffs laws

Note you could also use Kirchoffs laws on the
simple circuit
3
Example
In the circuit below your are given E1, E2, R1,
R2 and R3. Find I1, I2 and I3.
?

• Label all currents (Choose any
  direction)

?
2. Label /- for all elements (Current
goes ? - )
?
3. Choose loop and direction (Your choice!)
?

• Write down voltage drops (Second sign you hit
  is sign to use!)

?
5. Write down node equation
Loop 1 e1 - I1R1 I2R2 0
R1
I3
I1
-

I2
- I2R2 - I3R3 - e2 0

Loop 2
-

e1
R2
Loop 1
R3
-
-
Node I1 I2 I3
Loop 2

3 Equations, 3 unknowns the rest is algebra!

-
e2
4
What is the resistance of a capacitor?

• Answer depends on time!
• Uncharged
• Initially voltage drop between the plates is zero
• The charges flow in and build up on the plates
• Resistance starts off from zero, R0 V / I 0 /
  I 0
• Fully charged
• Charge on the plates builds to a maximum
• No more current flow
• Voltage reaches a maximum
• Resistance approaches infinity, R? V / I V /
  0 ?

5
RC Circuits

• A direct current circuit may contain capacitors
  and resistors, the current will vary with time
• When the circuit is completed, the capacitor
  starts to charge
• The capacitor continues to charge until it
  reaches its maximum charge (Q Ce)
• Once the capacitor is fully charged, the current
  in the circuit is zero

6
Combine RC Circuits

• Gives time dependence
• Current is not constant, I(t)
• Charge on the capacitor is not constant, q(t)
• Used for timing
• Pacemaker
• Intermittent windshield wipers
• Models nervous system
• Olympic Trivia How soon after starting gun can
  you run w/o getting a False Start?

0.1 seconds
7
RC Circuits Charging
The switches are originally open and the
capacitor is uncharged. Then switch S1 is closed.

• Kirchoffs Voltage Rule V-I(t)R-q(t)/C
  0
• Just after S1 closing qq0 0
• V - I0R0 ? I0 V /R
• Long time after I80
• V - q?/C 0 ? q? V C

R

-
V
I
-
C

QCV
-
S2
S1
8
RC Circuits Charging at any time
The switches are originally open and the
capacitor is uncharged. Then switch S1 is closed.

• Kirchoffs Loop Rule V(t) - I(t)R - q(t)/C 0
  ?V(t) I(t)R q(t)/C
• Since I(t) ?q(t)/?t ? V(t) (?q(t)/?t)R
  q(t)/C
• Solution Charging at any timeq(t)
  q?(1-e-t/RC)I(t) I0e-t/RC
• Check t 0 t ?q(0) q?(1-e-0) 0 ? I(0)
  I0e-0 I0 ? q(?) q?(1-e-?) q? ? I(?)
  I0e-? 0 ?

R

-
V
I
-
C

QCV
-
S2
S1
RC
2RC
q?
q
0
t
9
Charging in an RC Circuit

• The charge on the capacitor varies with time
• q Q(1 e-t/RC)
• The time constant, ?RC
• The time constant representsthe time required
  for the chargeto increase from zero to 63.2of
  its maximum
• In a circuit with a large timeconstant, the
  capacitor chargesvery slowly
• The capacitor charges veryquickly if there is a
  small timeconstant
• After t 10 ?, the capacitor is over 99.99
  charged

10
RC Circuits Discharging
After a long time, capacitor is fully charged.
Switch S1 is opened and Switch S2 is closed
restart the clock at t 0.

• Kirchoffs Loop Rule I(t)R q(t)/C 0
• Since I(t) ?q(t)/?t ? (?q(t)/?t)R q(t)/C 0
• Solution Discharging at any time q(t)
  q0e-t/RC I(t) I0e-t/RCNote in this I0 is
  negative(clockwise)!
• where q0 q? VC from charging case I0 V/R
• and since V(t)q(t)/CV(t) (q0/C)e-t/RC

R

-
V
I
-
QCV
C

S2
-
S1
11
Discharging in an RC Circuit

• When a charged capacitor is placed in the
  circuit, it can be discharged
• q Qe-t/RC
• The charge decreases exponentially
• At t ? RC, the charge decreases to 0.368 Qmax
• In other words, in one time constant, the
  capacitor loses 63.2 of its initial charge

12
Preflight Short Term

• Both switches are initially open, and the
  capacitor is uncharged. What is the current
  through the battery just after switch S1 is
  closed?

2R

-

• Ib 0
• Ib V/(3R)
• 3) Ib V/(2R)
• 4) Ib V/R

Ib


C
R
V
-
-
S2
S1
Loop V - I(2R) - q/C 0 q 0 ? I V/(2R)
13
Preflight Long Time

• Both switches are initially open, and the
  capacitor is uncharged. What is the current
  through the battery after switch 1 has been
  closed a long time?

2R

-

• Ib 0
• Ib V/(3R)
• Ib V/(2R)
• Ib V/R

Ib


C
R
V
-
-
S2
S1

• Long time ? current through capacitor is zero
• Ib0 because the battery and capacitor are in
  series.
• Loop V - 0 - q?/C 0 ? q? VC

14
Preflight Discharge

• After being closed for a long time, the switch 1
  is opened, while switch 2 is closed
  simultaneously. What is the current through the
  resistor, R?

2R

-

• IR 0
• IR V/(3R)
• IR V/(2R)
• IR V/R

IR
Ib


C
R
V
-
-
S2
S1

• Long time ? capacitor is fully charged to V
• Switch 1 open, switch 2 closed means C
  discharges through R.
• Loop q?/C - IRR 0 ? V IRR q?/C

15
Charging Intermediate Times
Calculate the charge on the capacitor 3?10-3
seconds after switch 1 is closed.
R 10 W, V 50 Volts, C 100mF
q?(1-e-3?10-3 /(20?100?10-6))) q?
(0.78) Recall q? V C (50)(100x10-6)
(0.78) 3.9 x10-3 Coulombs
q(t) q?(1-e-t/RC)
V
16
Electrical Safety

• Electric shock can result in fatal burns
• Electric shock can cause the muscles of vital
  organs (such as the heart) to malfunction
• The degree of damage depends on
• the magnitude of the current
• the length of time it acts
• the part of the body through which it passes

17
Effects of Various Currents

• 5 mA or less
• can cause a sensation of shock
• generally little or no damage
• 10 mA
• hand muscles contract
• may be unable to let go a of live wire
• 100 mA
• if passes through the body for 1 second or less,
  can be fatal

18
Ground Wire

• Electrical equipment manufacturers use electrical
  cords that have a third wire, called a ground
• Prevents shocks

19
Ground Fault Interrupts (GFI)

• Special power outlets
• Used in hazardous areas
• Designed to protect people from electrical shock
• Senses currents (of about 5 mA or greater)
  leaking to ground
• Shuts off the current when above this level

20
Electrical Signals in Neurons

• Specialized cells in the body, called neurons,
  form a complex network that receives, processes,
  and transmits information from one part of the
  body to another
• Parts of the neuron act as a capacitor making an
  RC circuit.
• Three classes of neurons
• Sensory neurons
• Receive stimuli from sensory organs that monitor
  the external and internal environment of the body
• Motor neurons
• Carry messages that control the muscle cells
• Interneurons
• Transmit information from one neuron to another

21
Diagram of a Neuron