CHAPTER 12: CAPACITORS
Chapter 12: Capacitors (Part 3)
Capacitors in AC Circuits
Overview of AC Capacitive Circuits
Study behavior of capacitive circuits in AC
Key topics include:
Familiarization with AC capacitive circuits
Phase relationship between voltage and current in capacitive circuits
Calculation of capacitive reactance and impedance
Application of Ohm's Law and circuit laws
Voltage and Current Relationship
Sinusoidal voltages produce sinusoidal currents in circuits
In a resistive circuit, voltage (V) and current (I) are in phase:
Illustration: VR = IR, both in-phase with respect to time
Phase Difference in Capacitive Circuits
In capacitive circuits, the voltage and current are out of phase
Current (I) leads voltage (V) by 90 degrees (or π/2 radians)
If purely capacitive, I leads V by 90°: VC=0, IC=max; VC=max, IC=0
Illustration:
VC = 0° at IC = max
VC=max at IC = 0°
VC = 0° at IC = max
VC=max at IC = 0°
Current Reference and Impedance Calculation
If voltage is assigned a reference phase angle of zero (0°):
Current I can be expressed as:
I = Vo / Z, where Z refers to the capacitive impedance
Expression:
Vo = V * S
I = Vo * S
I = Vo * jIA (j representing phase lead)
Capacitive Reactance (XC)
Capacitive Reactance (XC) defines a capacitor’s ability to oppose AC current
Calculation formula:
XC = 1 / (2πfC)
From XC, we can further determine the capacitive impedance (ZC)
Formula:
ZC = jXC (indicating phase lag)
Applying Ohm's Law in capacitive circuits
Impedance can be found using Ohm's Law where V and I are linked:
V = IZ
Expression:
Z = V/ I = Vo/I, providing capacitive analysis of the circuit
Example Problems
Example 12-7
Given a 1 kHz voltage applied to a capacitor of 0.0047μF:
Calculate capacitive reactance (XC)
XC = 33.9 Ohms or -33.9 kΩ
Impedance Z = j (-33.9 kΩ)
Example 12-8
Given: C = 0.0056μF and f = 10 kHz
Calculate ZC and rms current
Current leads voltage by 90°
Result: Irms = 1.76 mA
Power in AC Capacitive Circuits
Types of Power Considered:
Instantaneous Power
True Power
Reactive Power
Instantaneous Power
When positive: energy stored by capacitor from source
When negative: energy returned to the source
Instantaneous power calculation: p(t) = v(t) * i(t)
True Power
True power measures net power consumed by capacitor
Note: for ideal capacitors, true power (P) = 0 W
Only half the time it stores energy; the other half it sources power
Reactive Power
Reflects power stored in the electric field of the capacitor
Calculation method: Q = Vrms * Irms
Reactive power unit: Volt-Ampere Reactive (VAR)
Example Problems with Power
Example 12-9
Given true power P = 0 for ideal capacitor
Reactive power calculation: Q (for given parameters)
Final result calculations yield specific reactive power in VAR
Final Considerations
Capacitors block DC but allow AC
Voltage across a capacitor cannot change instantaneously
Relationships noted:
Current leads voltage by 90°
XC is inversely proportional to frequency and capacitance
True power in an ideal capacitor is zero
Reactive power represents energy stored in the dielectric