Series and Parallel Circuits -------------------------------------------------- Joseph F. Alward, PhD     Department of  Physics   University of the Pacific

Applet in this eLecture: RC Circuit Animation

  Alternating Voltage Sources              

Power station provides alternating current (AC).

 
 Alternating Current     

------------------------------------------------------------------- Current surges first one way, then the other through the toaster heating filament, changing direction 120 times per second. Clockwise current is positive, counter-clockwise is negative.

 
   RMS Voltage and Current      

Peak voltage:             V0  Square peak voltage:  V02

The average, or "mean", of the square of  the peak voltage is:                    V02 / 2  Take the square root of  the mean square  peak voltage to obtain the root mean  square (rms) voltage:                    Vrms = V0 / 1.414  ---------------------------------------------------------  Similarly,                     Irms = I0 / 1.414

  Average Power                                   

The power P = IV is always positive since I and V always have the same sign.

Average power Pave = I0 V0 /2                Pave =   (I0 / 1.414) (V0 / 1.414)                       =   IrmsVrms.

 RMS Current and Voltage Problem 

I0 = V0 / R                                       P0 = I0 V0    = 34 / 8                                             = 144.5 W    = 4.25 A                                         = peak power    = peak current

Vrms =  V0 / 1.414          =  34 / 1.414          =  24 V     --------------------   Irms = Io / 1.414          =  4.25 / 1.414          =  3 A

  Series Wiring                                                                  

What is the equivalent series resistance Rs?

Rs = 9 + 6                = 15 W

I  = 30 / 15             = 2 A

    Series Wiring Example                                  

What is the power dissipated             in the 6 ohm resistor?

I  = 12 / 9     = 1.33 A ----------------  P =  I2 R       = (1.33)2 (6)     = 10.6 W

   Resistors in Series:  The Extension Cord        

Which resistor gets hotter?

I = V / (R + r)   ---------------------------- PR  =  I2 R         = [ V2 / (R+r)2 ] R ---------------------------- Pr    =  I2 r           = [ V2 /(R+r)2 ] r

    Blowing a Fuse                    

I = 9 / 0.5                      = 18 A

What is the least R which will prevent the fuse from blowing?

  Parallel Wiring                                                                         

Two pipes in parallel can be replaced by one.  The single pipe    will have less resistance than the pipe with the least resistance.

  Light Bulbs in Parallel                            

Light bulbs connected in parallel.

Equivalent resistance of this parallel combination found through the relation 1 / Rp = 1 / R1 + 1 / R2

Rp is the single resistance which is equivalent to R1 and R2 in parallel.

  Two Speakers in Parallel                  

1 / Rp =  1 / 4 + 1 / 12                                                         Rp = 12 / 4 = 3 W           = 3 / 12 + 1 / 12                                                     (Note:  less than the least.)

   House Wiring is Parallel            

Which connecting wire, A, B, C,  D, or E, will be the first to become  dangerously hot  if too many  appliances are turned on? How can overheating be prevented, even if all the appliances in the house are turned on?

   Circuit Breakers                  

Brass expands more than steel     As temperature increases, the   bimetallic strip arcs to the left,   settles into groove, and the   spring pulls the metal bar down.

   Four Resistors in Parallel                          

Find the resistance between A and B:

1 / Rp = 1 / 2 + 1 / 3 + 1 / 6 + 1 / 1    Rp = 0.50 W   ---------------------------------- VAB  = 12 V What is the current in each resistor? What is the sum of all currents?

 Three-Way Light                 

What is the resistance      of each filament?

P = V2 / R V = 120 volts for each filament ------------------------------------------ R1 = resistance of 75-W bulb: 75 = (120)2 / R1 R1 = (120)2 / 75        = 192  W ------------------------------------------ R2 = resistance of 100-W bulb: 100  = (120)2 / R2 R2 = (120)2 / 100         = 144  W

  Will Either Bird be Shocked?                         

Connecting wire has no insulation and negligible resistance.

 Equivalent Resistance of Series and
 Parallel Combination of Resistors 

Parallel combination: 1 / Rp = 1 / 6 + 1 / 3         =  0.5     Rp = 1 / 0.5            = 2 W ---------------------

    Series and Parallel Circuits                                      

What is the total resistance between Points A and B?

4 + 6 = 10 -------------------------------------- 1 / Rp = 1 / 9 + 1 / 8 + 1 / 10    Rp = 2.98 W -------------------------------------  2.98 + 3 = 5.98   ------------------------------------- 1 / Rp = 1 / 5.98 + 1 / 20    Rp = 4.60 W

   A Series-Parallel Circuit Problem:  Part One     

What is the potential difference between  Points A and B?

1 / Rp =  1/3 + 1/6      Rp = 2 W -------------------------  Total:  6 W -------------------------  I = 12 V / 6 W    =  2 A ------------------------- Ohm's Law: VAB = (2 A) (2 W )         = 4 V

 
A Series-Parallel Circuit Problem:  Part Two           

What is the current in the 5-W resistor?

I = V / R (Ohm's Law) ------------------------------- From Part One: VAB = 4 V ------------------------------- R = 6 W  (lower branch) ------------------------------- I = 4 V / 6 W   = 2 / 3 A This is also the current in the 1 W resistor.

 
  Series and Parallel
  Connections 

What happens when the wire   is connected?

  Internal Resistance of Batteries      

Real batteries have internal resistance.

Ideal batteries--if they existed--would have zero internal resistance.

  Internal Resistance Problem                                                    

What is VAB ?

Circuit at the left is the same as the one below:

Total resistance:    0.11  W I = 12 / 0.11   = 109.09 A ------------------------ V = I    = (109.09) (0.1)    = 10.909 V

  Kirchhoff's Junction Rule                                                  

Junctions are places where  wires are connected together.

Kirchhoff's Junction Rule: Total current into a junction is equal to the total current out of the junction.

Gustav Robert Kirchhoff              (1824-1887)

   Kirchhoff's Loop Rule                                 

A loop is a circuit, which is any   closed path with batteries and   resistors.

The sum of the potential differences encountered around a closed circuit equals zero. --------------------------------------------------- Traveling clockwise from A: -12 I  - 6 - 8 I + 24 = 0                or............. Traveling counter-clockwise from A: -24 + 8 I +6 + 12 I = 0

    Kirchhoff's Rules Problem Part One:      

     Kirchhoff's Rules Problem Part Two:              

Loop BCDB:               -4 I2 + 16 + 8 (I1 - I2)   = 0          (1)             -4 I2 + 16 +8 I1 - 8 I2    = 0          (2)              8 I1 -12 I2                     = -16       (3)

Loop ABDA:   -2 I1 -8 (I 1- I2) - 16 + 6  = 0        (4) -2 I1 -8 I 1 + 8 I2 - 10      = 0        (5) -10I1 + 8 I2                     = 10      (6) ---------------------------------------------- Solve Equations 3 and 6: I1 = 3 / 21 A I2 = 30 / 21 A

   Capacitors in Series                  

1 / Cs = 1 / C1 + 1 / C2

Example: C1 = 30  mF C2 = 60  mF 1/Cs = 1 / 30 + 1/60         = 3 / 60         = 1 / 20    Cs = 20  mF

   Series Capacitor Charge     

The charge on capacitors in series is the same.

  Voltage on Capacitors in Series                              

CA = 50 x 10-6 F                         CB = 10 x 10-6 F

Q is the same on each capacitor. ---------------------------------------------- By Kirchhoff's Loop Rule: 12 - Q / CB - Q / CA = 0  Q =  1 x 10-4 C ---------------------------------------------  VA = Q / CA        = 2 V  VB = Q / CB        = 10 V

  Capacitors in Parallel      

Cp = C1 + C2

Example: C1 = 30  mF C2 = 60  mF  Cp = 30 + 60        = 90  mF

 Charge on Capacitors in Parallel  

Each capacitor sees same voltage:   100 V C = Q/V (basic definition) Q = CV Q1 = (5)(100)      = 500 C Q2 = (2)(100)      = 200 C

  RC Circuits                                               

C = Q/V     (basic definition) Q = CV Eventual charge Q0 = CV ------------------------------------------- Define:  t = RC (t is called the "time constant") ------------------------------------------- Q = Q(t) = charge magnitude is                   a function of time Q(t) = Q0  [1 - e-t/t]         = CV [1 - e-t/t]

   Growth of Charge in an RC Circuit                             

Q(t) = Q0  [1 - e-t/t]                      = CV [1 - e-t/RC]

   Growth of Charge in an RC Circuit :  Example Problem       

Show that it takes 3.47 s for the charge on the capacitor to rise to 1/2 of its final value.

 

RC Circuit Animation See currents and voltages change as time passes; plot of current versus time.

  Washing Machine without Short   

  Impropery Grounded Washing Machine with Short  

One milliampere:    tingling sensation Ten milliamperes:   nerves and muscles overloaded 200 milliamperes:   potentially fatal; heart fibrillation 500 -1000 milliamperes:  not necessarily fatal; heat will restart One ampere or more:  burn alive

 Grounded Washing Machine