Capacitive Circuits | Electricity & Magnetism - Physics PDF Download

Capacitance
Electrically, capacitance is the ability to store an electric charge. Capacitance is equal to the amount of charge that can be stored in a capacitor divided by the voltage applied across the plates
C = Q/V
where C = capacitance, F;  Q = amount of charge, C; V = voltage, V Above equation can be rewritten as follows: Q = CV  or V = Q/C.
The unit of capacitance is the farad (F). The farad is that capacitance that will store one coulomb of charge in the dielectric when the voltage applied across the capacitor terminals is one volt.

1. Capacitive Reactance
   Capacitive reactance X_C is the opposition to the flow of ac current due to the capacitance in the circuit. The unit of capacitive reactance is the ohm. Capacitive reactance can be found by using the equation
   
   where X_C = capacitive reactance, Ω ; f = frequency, Hz;  C = capacitance, F If any two quantities are known, the third can be found
   
   Voltage and current in a circuit containing only capacitive reactance can be found using Ohm’s law.  However, in the case of a capacitive circuit, R is replaced by X_C.
   
   where  I_C = current through the capacitor A; V_C = voltage across the capacitor, V X_C = capacitive reactance, Ω
2. Capacitors in Series or Parallel
   When capacitors are connected in series (as shown in figure below) the total capacitance C_T is
   
   The total capacitance of two capacitors in series is
   
   When n number of series capacitors have the same capacitance, C_T = C/n.
   Capacitances in seriesWhen capacitors are connected in parallel (as shown in figure below), the total capacitance CT is the sum of the individual capacitances.
   Parallel: C_T = C₁ + C₂ + C₃ +... + C_n
   Capacitances in parallel
3. Capacitive Circuits
   ➤ Capacitance Only
   If an ac voltage v is applied across a circuit having only capacitance (figure a), the resulting ac current through the capacitance i_c, will lead the voltage across the capacitance, v_C , by 90° (figure b and c), (Quantities expressed as lowercase letters, i_c and , v_C indicate instantaneous values.) Voltages v and v_C are the same because they are parallel. Both i_c and v_C are sine waves with the same frequency. In series circuits, the current I_C is the horizontal phasor for reference (figure d) so the voltage V_C can be considered to lag I_C by 90°.
   Circuit with C only 
   ➤ RC in Series
   As with inductive circuits, the combination of resistance and capacitive reactance    (figure a shown below) is called impedance. In a series circuit containing R and X_C, the same current I flows in X_C and R. The voltage drop across R is VR = IR, and the voltage drop across X_C is V_C = IX_C . The voltage across X_C lags the current through X_C by 90° (figure b shown below). The voltage across R is in phase with I since resistance does not produce a phase shift.
   R and X_C in seriesTo find the total voltage V_T , we add phasors V_R and V_C . Since they form a right triangle (as shown in figure below), Voltage-triangle phasorNote that the IX C phasor is downward, exactly opposite from an IX_L phasor, because of the opposite phase angle.
   The phase angle θ between V_T and V_R is expressed according to the following equation:
   
   ➤ Impedance in Series RC
   The voltage triangle corresponds to the impedance triangle because the common factor I in V_C and V_R cancels.
   Series RC impedance triangleImpedance Z is equal to the phasor sum of R and X_C 
   The phase angle θ is
   ➤ RC in Parallel
   In the RC parallel circuit (figure a shown below), the voltage is the same across the source, R, and X_C since they are all in parallel. Each branch has its individual current. The resistive branch current I_R = V_T / R is in phase with V_T. The capacitive branch current I_C = V_T /X_C leads V_T by 90° (figure b shown below). The phasor diagram has the source voltage V_T as the reference phasor because it is the same throughout the circuit. The total line current I_T equals the phasor sum of I_R and I_C (figure c shown below).
   X_C and R in parallel
   
   ➤ Impedance in Parallel RC
   The impedance of a parallel circuit equals the total voltage V_T divided by the total current I_T .
   
   ➤ Power in RC Circuits
   The power formulas given previously for RL circuits are equally applicable to RC circuits.
   Real power P = VI cosθ
   Reactive power Q = VI sinθApparent power S = VI 

Note: Capacitance, like inductance, consumes no power. The only part of the circuit consuming power is the resistance. 

Summary Table for Series and Parallel RC Circuits