Unit 6A: Capacitance

Reading: Sections 15.1 and 15.2

Objectives.  Be able to:

• Define capacitance and Farad.
• Describe the construction of a capacitor.
• Explain how a capacitor works in a DC circuit
  • Describe the response of a capacitor in a DC circuit when the switch is closed.
  • Explain capacitor charging and discharging
• Describe the effects of combining resistance and capacitance in a circuit
• Explain how a capacitor works in an AC circuit.

15.1 Capacitance and Capacitors

Capacitor

• A device that temporarily stores electrical charges.
• Schematic symbols: 

• Construction:  2 plates of a conductive material, separated by an insulating material called a dielectric.
• Capacitors store energy as separated charges.
  • What happens when we separate charges?
    • We get voltage!
  • We say that a capacitor stores energy in an electric field (an electric field occurs whenever we have separated charges.)
• Capacitor ratings are capacitance value and maximum voltage.

Electrostatic Field

• Lines of force that are created between separated charges.

Dielectric constant

• A measure of the ability to support an electric field. 
• A larger dielectric constant means more ability to support an electric field.
• Some typical values of dielectric constants:

Air	1.0
Glass	8.0
Paper	3.5
Mica	6.0
Ceramic	100

Capacitance (C)

• A measure of the ability of a device to store electrical charge in an electrostatic field.
• A property that opposes any change in voltage in a circuit.
• C = # electrons stored (charge) / increase in potential (voltage)
• C = Q / V
• Unit:  Farad.  1 Farad = 1 coulomb/volt
  • commonly used units for capacitance:  microfarads (µF) and Pico farads (pF)
  • One farad is developed when one coulomb of charge develops a potential of 1 volt between two points in a circuit.
  • Capacitance is the amount of charge required to produce 1 volt of potential difference.
• Factors that affect capacitance
  • dielectric material used (directly proportional) .  dielectric constant increases  =>  capacitance increases
  • area of plates used (directly proportional).  area  increases => capacitance increases
  • distance between plates used (inversely proportional).  distance increases => capacitance decreases
  • Formula:

  C µ K A (n - 1) / d

  K = dielectric constant
  A = area of one side of a plate in m²
  d = distance in m²
  n = number of plates

  The capacitance of a capacitor will go up if the area of the plates is increased, or if the distance between the plates is decreased.

Working voltage (WV) - maximum voltage the capacitor can handle.  If exceeded, the dielectric material breaks down and arcing results.  This can cause a short circuit and ruin other components in the circuit.

Types of capacitors:

• electrolytic capacitors
  • Polarized with terminals marked as + or -.
  • Have low voltage ratings and can blow up if the voltage rating is exceeded.
  • Uses an electrolytic chemical gel as the dielectric.
  • Has a very high dielectric constant but polarity must be observed.
  • Can have very large capacitance values (up to 1 farad or larger).
• ceramic capacitors
  • Disk or tubular shape
  • Smaller capacitance(pF) with large voltage ratings
  • not polarized
  • Most ceramic capacitors are numbered with 3 digits. 
    • A capacitor marked 503 would have a capacitance value of 5 x 10³ picofarads.
    • The first two digits give the magnitude of the capacitance value.
    • The third digit is the multiplier.
    • The unit for capacitance is picofarads.
• Mica capacitors
  • constructed of tinfoil separated by mica as the dielectric material.
  • large breakdown voltages but small capacitance values.
• Tantalum
  • electrolytic
  • uses tantalum electrodes
  • smaller size, long shelf life, less leakage current.
• Mylar
  • constructed of tinfoil separated by mylar.

15.2 Capacitance in DC Circuits

• The response of current and voltage in a circuit immediately after a change in applied voltage is called the transient response.
  • Description of transient response in a DC circuit.
    
    
    
    
    DC Resistance-Capacitance (RC) Circuit
     
  • For the above diagram, the switch is closed at time t = 0.  The transient response of the circuit that happens immediately after the switch closes is described below.
    • Initially, V_C = 0.  V_C is the voltage dropped across the capacitor.
    • There is an immediate spike in current and I_C assumes its maximum value, which is determined by the source voltage of the circuit divided by the resistance R.
      
      I_C = 10v/2KW = 5ma
       
    • The resistance of the capacitor R_C is obtained by Ohm's Law:  R_C = V_C/I_C = 0 W
       
    • The voltage drop across the resistor is also at the maximum, since there is no voltage drop across the capacitor.  So, all the voltage in the circuit is dropped across the resistor.
      
      V_R = 10v
       
    • The source voltage, V_S = 10 volts and remains constant throughout the transient.
      
      
       
    • Over a period of 5 time constants (5 * R * C), the capacitor stores electrical potential and the voltage dropped across the capacitor increases until it equals the source voltage.
      
      
       
    • As the V_C increases, I_C decreases (exponentially) since current is inversely proportional to resistance accoring to Ohm's Law.  As the capacitor charges the current in the circuit decreases to 0, so the opposition to current flow (resistance) of the capacitor increases.
      
      
       
    • Also, as V_C increases, V_R decreases, since by Kirchoff's Voltage Law the total voltage dropped in the circuit adds up to the source voltage.  V_R = V_S - V_C
      
      
       
    • As the capacitor charges, values of V_C, V_R, I, and R_cap are as follows:
      
       

      VC	VR	I	Rcap
      0v	10v	5ma	0W
      2v	8v	4ma	.5KW
      4v	6v	3ma	1.33KW
      6v	4v	2ma	3KW
      8v	2v	1ma	8KW
      10v	0v	0ma	¥ W (open)

    • Observations:
      • The capacitor charges quickly initially (V_C increases rapidly), then levels off.
      • R_cap increases as the capacitor charges.
      • I is largest initially, then drops to zero.
         
• RC Time constant(t).  The amount of time needed for the capacitor to charge or discharge 63.2% is the time constant of the circuit.

t = R  × C

• A capacitor is fully charged or discharged after 5 time constants have elapsed.

• Square wave example:
  
  
   
• For the square wave, the voltage source V_S cycles ON and OFF
  
  
   
• V_C alternately charges and discharges each time V_S cycles ON and OFF.  The capacitor is charging when V_C increases.  The capacitor is discharging when V_C decreases.
  
  
   
• V_R = V_S - V_C
  
  
   
• The current is positive when the capacitor is charging.  When the capacitor is discharging, the current reverses direction and flows out of the capacitor back into the circuit.
  
  
   
• The time to fully charge or discharge the capacitor
  =  5RC  =  5 (2KW) (.5mF)
  =  5 (2000W) (0.5 x 10^-6 mF)
  =  5 x 10^-3 sec = 5 msec
   
  • The time constant  t = RC = 1 msec  (approximately 63.2% of the change in voltage occurs during the first time constant).
  • After 5 time constants, the capacitor acts like an open circuit!
     
• The capacitor opposes any change in voltage!  It smooths out voltage waveforms by blocking an increase in voltage and by returning voltage to the circuit when the source voltage cycles off.
• The capacitor blocks DC!  It acts like an open circuit after it is charged, preventing DC current flow.
   
• I_C = C (DV)/(Dt)   Current flows only when the capacitor voltage is changing --- i.e., when it's charging or discharging.
   

Series and Parallel Capacitance

Current through a capacitor:  I_c = C × (DV/Dt)      D denotes a change in a value

• Capacitors in Parallel
  
         

  I_s = I_C1 + I_C2 + I_C3

  C_t(DV/Dt) = C₁(DV/Dt) + C₂(DV/Dt) + C₃(DV/Dt)

  C_t = C₁ + C₂ + ... + C_N

• Capacitors in Series
  
         

V_s = V₁ + V₂ + V₃

I_s(Dt)/C_t = I₁(Dt)/C₁ + I₂(Dt)/C₂ + I₃(Dt)/C₃

1/C_t = 1/C₁ + 1/C₂ + ... + 1/C_N

C_eq = 1/[1/C₁ + 1/C₂ + ... + 1/C_N]

15.3 Capacitance in AC Circuits

Capacitive Reactance

• The resistance to the flow of alternating current produced by capacitance.
• The capacitor's opposition to AC (current) signals.
• X_C = 1/(wC), where w is the angular frequency in radians = 2pf
  X_C = 1/(2p fC)
• unit: ohms
• Note: X_C is inversely proportional to frequency. 
  • As f increases, X_C decreases and as f decreases, X_C increases.
  • For very low frequencies (like DC),  X_C is very large (infinite or open for DC).
• There is also a phase shift between voltage and current.
  • Current occurs first (current leads voltage) charging or discharging the capacitor, then the voltage occurs.  There must be movement of charges (current) before voltage occurs.

Power in Capacitive Circuits

• Power is stored temporarily in the capacitor and returned to the circuit when the capacitor discharges.
• True Power = 0 (since the power is returned to the circuit in a pure capacitive circuit)
• Apparent Power --- the power that appears to be provided.
• P_apparent = V_eff x I_eff
• Power Factor = True Power / Apparent Power = cos q, where q is the angle of the phase displacement between current and voltage.

Resistance and Capacitance in an AC Circuit (supplemental material to explain impedance).

• In a pure resistance circuit (no capacitance), voltage and current are in-phase.  Thus, the phase difference is 0. q = 0 degrees.
• PF = cos 0 = 1  and True Power = V_eff x I_eff
   
• In a circuit with resistance and capacitance, the capacitive reactance causes a 90 degree phase difference and the total resistance to the flow of alternating current in the circuit is the vector sum of resistance and capacitive reactance.  This total resistance to alternating current is called impedance (Z).
   
• Impedance is calculated using the Pythagorean Theorem: 
  
  
  
  Z = (R² + X_C²)^1/2
  
  Note: X_C is shown at -90 degrees relative to R, since V_C lags V_R in a capacitor.
   
• The angle (q) between R and Z in the diagram is the same as the angle of phase displacement between current and voltage in the RC circuit.  We can find this angle using the cosine function:
  
  cos (q) = R/Z,  or  q = COS^-1(R/Z)
  
  which can be evaluated using either the INV COS or 2ND COS on a scientific calculator.
   
• If R = 300 ohms, X_C = 400 ohms, then Z = 500 ohms (using the Pythagorean Theorem).
  
  P_apparent = I_eff X V_eff
  
  P_true = P_apparent x cos(q) = 0.6 x P_apparent
   
• Note: Ohm's law for AC circuits uses impedance (Z) instead of resistance.
  
  I = V/Z
  
  If V_s = 100 volts in the given example, then I = 100 volts/500 ohms = 0.2A = 200 mA.

Applications of Capacitors

• In series circuits, capacitors are used as coupling devices to block DC signals and pass AC signals.  In other words, any DC signals are removed from the signal and only the AC portion of the signal is allowed to pass through the capacitor.  This is a result of the fact that the capacitive reactance in an RC circuit is very high for low frequency signals and DC has a frequency of 0.
• In parallel circuits, capacitors are used as filter or bypass devices to ground or filter out high frequencies.  As a bypass or filter device, the  capacitor opposes changes in voltage and will smooth out the voltage waveform.
• Capacitors are also used in timing circuits (such as the 555 timer) to provide a timing mechanism based on the time constant for RC circuits.
• Backup storage of electricity.  Capacitors are frequently used to provide temporary storage for appliances such as microwave ovens, and to prevent damages from voltage spikes.

Testing Capacitors

In order to test a capacitor, it must first be discharged.  If you are testing a capacitor in a circuit make sure the power is off to the circuit.  If at all possible, a capacitor should be removed from the circuit to test it.  A capacitor can be discharged by placing a jumper wire across it.  For larger capacitors, it is recommended to discharge the capacitor with a 100 ohm resistor. 

There are two ways to test a capacitor:

• If you have access to a LCR meter, simply use the meter to see if you capacitor has a capacitance value that is close to it's rated value.  Normally if a LCR meter can read the capacitor's rated value the capacitor is okay.
• If you don't have access to a LCR meter, you can use an ohmmeter.  Set the ohmmeter to a large resistance setting. (Approximately 1 Mohm)  Place the ohmmeter across the capacitor.  If the capacitor has polarity, match the polarity with the ohmmeter.  A good capacitor should start off with low resistance and then as the capacitor charges up, the resistance value should rise until the capacitor is fully charged.  When the capacitor is fully charged, the ohmmeter should read an open circuit (infinity).  When checking a large capacitor the resistance read on the ohmmeter will rise slowly whereas when checking a small capacitor the resistance value will rise faster.  This is because it takes longer to charge a large capacitor than a small capacitor because a large capacitor requires more charge per volt.