TLO 12 Define the capacitor as a storage element for energy in the electric field, and develop the current and voltage behaviour in an R-C circuit.

ELO 12.1 Describe the electric field and the terminology used to define electric field strength.

ELO 12.2 Define capacitance and relate it to voltage and charge.

ELO 12.3 State the formula for the capacitance between two oppositely charged parallel plates.

ELO 12.4 Define the dielectric strength and breakdown voltage of various materials.

ELO 12.5 Distinguish between an ideal and a practical capacitor by defining leakage current.

ELO 12.6 Describe the physical construction of various types of capacitors including polarized devices.

ELO 12.7 Describe the physical characteristics of a capacitor while charging, and establish the expressions i_c(t) and v_c(t).

ELO 12.8 Describe the physical characteristics of a capacitor while discharging, and establish the expressions i_c(t) and v_c(t).

ELO 12.9 Define the term "time constant" for an R-C circuit, and explain its role in design.

ELO 12.10 Describe the relationship of capacitor current as being proportional to the rate of change of capacitor voltage with respect to time (derivate and slope).

ELO 12.11 Work at least four examples in class utilizing the concepts developed for R-C circuits.

ELO 12.12 Derive the expression for the equivalent capacitance of series-connected and of parallel-connected capacitors.

ELO 12.13 Describe the method for finding the equivalent capacitance of a series-parallel circuit of capacitors.

ELO 12.14 State the expression for energy stored in the electric field of a capacitor.

ELO 12.15 Briefly describe the term "stray capacitance" and the implication of stray capacitance in electronic and power circuits.

THE ELECTRIC FIELD

• The electric field that exists around a charged body is represented by electric flux lines

• Electric flux lines begin on charges, so: = Q coulombs, C

ELECTRIC FIELD STRENGTH

• Coulomb's Law says that if we bring a test charge Q_T within the field created by the charge Q, then Q_T will experience a force:

where k = 9 x 10⁹ N m² / C²

• A basic expression says that the force on the test charge Q_T is:

F = Q_T E

where E is the electric field strength in newtons/coulomb

• By definition, electric field strength at a point is the force acting on a unit positive charge at the point:

In our diagram at left E is greater at b than at a, since a > b

• Electric flux lines always extend from the positively charged to the negatively charged body, always terminate perpendicular to the charged surfaces, and never intersect.

CAPACITANCE

• It is one thing to move charges around in free space



• If we move charges in a circuit containing a particular device, we can send them to that device and store them

• Electric dipoles, which naturally occur in the material, line up due to the charges on the plates



• We say that the material has become polarized

• The aligned dipoles of the dielectric create an electric field
• It opposes the electric field set up by free charges on the parallel plates.

• Effect of dielectric is to increase the charge storage capability, that is to increase the capacitance of the parallel plates



• This effect is expressed by the relative permittivity, also called the dielectric constant

• This can also be written in terms of the air capacitor

DIELECTRIC STRENGTH

• Even though a dielectric is an insulator, all insulators can be broken down and made to conduct



• The voltage required per unit length to establish conduction in a dielectric is an indication of its dielectric strength and is called breakdown voltage.



• Table 10.2 shows breakdown voltages in volts/mil, or volts/one one-thousandths of an inch, for some dielectrics

LEAKAGE CURRENT

• A capacitor is supposed to store charge

• Some current leaks across the capacitor (of the order of 10^-6 amps). We model this current flow with a resistor

• If we charge a capacitor and place it on the shelf, it will discharge through the leakage resistor



• Leakage discharge may take days or weeks

TYPES OF CAPACITORS

• Fixed



• Variable

TRANSIENTS IN CAPACITIVE NETWORKS:

• To study transients in a resistive - capacitive circuit, we need to use four concepts (two of which we have used many times)

• KVL

• Ohm's law

• Current flow to a capacitor



• The fourth concept is very important.

• There is energy stored in the electric field

• If we tried to collapse the field in zero time:

• Physically, infinite power cannot exist.




Or in the practical application sense:

• Voltage across a capacitor v_C cannot jump, or change instantaneously in zero time, otherwise we would have

CHARGING PHASE

• Capacitor is initially uncharged



• Capacitor charges when the switch is at position 1



• KVL says: E - v_R - v_C = 0

• Substituting v_R = R I, and then substituting

• We can argue a solution to this equation



• What time function v_C has a derivative related to itself by a constant ?

Answer: An exponential form

• v_C cannot jump so at t = 0+, v_C = 0 still. Since the RHS is a constant, at this time in the LHS the slope dv_C/dt is a maximum
• So while v_C is still zero, it is set to undergo a rapid increase



• Later as v_C starts to go up, the slope dv_C/dt must decrease



• When v_C gets very large, dv_C/dt must get very small



• When dv_C/dt goes to zero, v_C E (the source voltage)

• When the switch is closed, the current instantaneously jumps up to E/R (current can jump, but v_C can not jump)

• The current decays exponentially, and after a long time the current goes to zero (when the capacitor gets fully charged).

TWO USEFUL PICTURES AT EXTREME TIMES

• At t = o+ just after closing the switch:
• Capacitor initially uncharged.
• v_C can not jump, so v_C = 0 still.
• Capacitor appears as a short circuit.

• The transient has completely finished in 5 Time Constants



• Remember that T = RC, and may range from nanoseconds,

(10 ^-9 seconds) to milliseconds (10 ^-3 seconds), to minutes.

VARIATION IN TIME CONSTANT

• R can be fixed, and C can vary (shown on the graph below)



• C can be fixed, and R can vary.

• From the basic expression q_C = C v_C and the above v_C expression, the charge placed on the capacitor is:

• From Ohm's Law, the voltage across the resistor is:

• We return to the original circuit from which we started our analysis of charging, except that now the switch is moved to position 2

• Three points to note:
• The discharge current will flow opposite to the charging current. So we assume the opposite direction, rather than use a minus sign



• For this reason, we also reverse the polarity signs on v_R

• We assume that the capacitor has been charged up to E volts
• We could write KVL, put v_R = R I, and substitute I = C dv_C/dt to get a differential equation.



• Solving the differential equation, we can get:

• From KVL:

• From Ohm's Law:

• The charging and discharging equations which we have developed apply only to single loop problems



• If we have more than one loop, we must reduce the multiple:

• Voltage sources

• The + side of the source attracts electrons to it, leaving the left plate of C₁ positive



• This draws electrons to the right plate of C₁, leaving the same amount of positive charges on the left plate of C₂

• This continues through all three capacitors, so that:

Q_T (from source) = Q₁ = Q₂ = Q₃

• From the basic expression relating charge and voltage:

• For two capacitors in parallel:

• So capacitors in series add like resistors in parallel.

CAPACITORS IN PARALLEL

• Law of Conservation of Charge (or KCL really):