RLC circuit
RLC circuit
RLC circuit is an electrical circuit with a resistor(s) (R), an inductor(s) (L), and a capacitor(s) (C), connected in series or in parallel, crucial for radio receivers and oscillators for filtering and tuning.
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capacitive = property to store electrical energy in a form of a electric charge
inductive = property to resist current change due to a magnetic field made by the current
Inductive circuit
Series circuit
Series circuit
Parallel circuit
Parallel circuit
Angles
Angles
In RLC circuits, angles arise due to the relationship of voltage and current, specifically the phase difference between them, shaped by the way the resistor, inductor, and capacitor react to AC signals.
phase: a point position in a full cycle of a wave and a measure of the timing of the wave relative to a reference (an angle or time)
Phase
Phase
A wave's phase depicts how far along a wave is in its cycle at anytime, often written in angles.
A sine wave's phase is the angle in the sine function, written as: sin (𝜔𝑡 + φ).
ω = angular frequency (how fast waves oscillate)
t = time
φ = phase (which tells you the starting point or shift in time).
Impedance phase
The impedance phase is often 90° (or 𝜋/2 radians) in purely inductive circuits is due the inductor's property. An inductor’s voltage waveform reaches its peak value 90° ahead of the current waveform, so the voltage is "ahead" of the current by a cycle's quarter.
In AC circuits, an inductor's impedance is: ZL = jωL
j or √-1 = imaginary unit (in electrical engineering)
ω = AC signal's angular frequency = 2πf
L = inductor's indutance
An inductor's impedance is an imaginary number
Leading and lagging
A purely resistive circuit's voltage and current are in phase--they reach their max and min values simultaneously.
A purely inductive circuit's current lags voltage by 90°--as voltage reaches its peak, current still increases and hasn't reached its peak yet.
A purely capacitive circuit's current leads voltage by 90°--current reaches its peak before voltage does.
Use mnemonics to remember these (E = voltage, L = inductor, I = current):
"ELI the ICEman": "ELI" says that in an inductive circuit (L), voltage (E) leads leads curent (I) by 90°. Note how E is before I in "ELI" and in "Voltage (E) leads leads curent (I)". "ICE" says that in a capacitive circuit, current (I) leads voltage (E) by 90°. Note how "I" is before "E" in this sentence and in "ICE".
Another mnemonic to recall this is "CIVIL": "CIV" says that in capacitive circuits (C), current (I) leads voltage (V) and "VIL" says voltage (V) leads current (I) in inductive circuits (L).
Phasor
Phasor
In electrical engineering, a phasor is a complex number representing a sine waveform, like voltage or currentfor its amplitude and phase angle. It simplifies AC circuits's analysis by representing time-varying sinusoidal signals as constant vectors for easier calculations.
AC circuits' voltages and currents vary sinusoidally, which are shown as sinusoidal waves, but in AC circuit analysis, it’s easier to do them in a complex form (use of complex numbers) called phasors, where:
Resistor is a real number as it doesn't cause a phase shift to current.
Voltage and current are in phase to each other.
Inductor and Capacitor both cause phase shifts between current and voltage which is where imaginary unit 'j' is used.
Impedance and reactance
Impedance and reactance
Recall that impedance (Z) is the total opposition to an AC circuit's current and includes both resistance and reactance.
And that reactance is the opposition to current caused by inductors and capacitors, and it depends on frequency.
A resistor's impedance is: ZR = R, which has no phase shift and is real (not imaginary).
An inductor's reactance is: XL = ωL, which has a shift and is purely imaginary.
To clarify, a complex number has 2 parts: the real (represents quantities that don't cause phase shifts like resistor's properties) and imaginary part (represents quantities causing phase shifts like inductors and capacitors).
Phasor diagram
Phasor diagram
A phasor diagram is drawn according to time zero (t = 0) on the horizontal axis. Phasors' lengths are proportional to voltage and current values the instant in time the phasor diagram is drawn.
Current phasor lags the voltage phasor by the angle (Φ) as the 2 phasors rotate in an anticlockwise direction as stated earlier, therefore the angle, Φ is also measured in the same anticlockwise direction.
But if the waveforms froze at time, t = 30°, the corresponding phasor diagram resembles the one above. Again the current phasor lags behind the voltage phasor as the 2 waveforms are of the same frequency.
But as the current waveform now crosses the horizontal zero axis like at this time instant, we can use the current phasor as the new reference and the voltage phasor leads the current phasor by angle Φ.
References
References
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