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In resistive elements the current and voltage are in phase.
In capacitive circuits, the current leads the voltage by 90°.
In inductive circuits the current lags the voltage by 90°.
Related: Inductance
Because the voltages and the currents are not in phase in capacitive and inductive AC circuits, these devices have impedance (Z) and not resistance (R) and therefore impedance and resistance cannot be directly added. Consider the following circuit where a resistor, capacitor and inductor are connected in in series:
The same current will flow through all the three devices, but the voltage in the capacitor and inductor will be 180° out of phase, and 90° out of phase with the voltage in the resistor.
These voltages can be combined using vectors as illustrated in figure 4(c) and 4(d) to give:
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Where,
E = supply
VR =voltage across the resistor
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VL = voltage across the inductor
VC = voltage across the capacitor
The vector addition of the voltages is shown in figure 4(c) where the relationship between VR, VL and VC are shown. In figure 4(d) the VC – VL vector and VR are shown vectors are shown with the resulting E vector which from the trigonometry function gives equation (a) above.
The impedance triangle is illustrated below:
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Or if VL is greater the VC
The current flowing through the circuit can be obtained from ohm’s law as follows:
XL and XC are frequency dependent and as the frequency increases XL increases and XC decreases. A frequency is reached where XL and XC are equal and the voltages across each component are equal and opposite and cancel each other. At this frequency Z = R, E =IR and the current is maximum. This frequency is called the resonant frequency of the circuit.
At resonance
When the input frequency is below the resonant frequency, XC is larger than XL and the circuit is capacitive, and above the resonant frequency, XL is greater than XC and the circuit is inductive. Plotting the input current against the frequency shows a peak in the input current at the resonant frequency as illustrated below:
Related: Capacitors, Capacitance and Charge
Consider the circuit below:
In a parallel circuit shown in figure 4(h), each component receives the same voltage but not the same current as illustrated by the waveforms below:
The source current is the sum of the currents in each component and is given by
The impedance of the circuit Z as seen by the input is given by:
At resonant frequency, IL and IC become equal and cancel each other so that E = IR. This can be noted in equation (b).
Below the resonant frequency, the circuit is inductive, and above the resonant frequency the circuit is capacitive. Plotting the current against frequency shows that the current is at minimum at the resonant frequency as shown in figure 4(g) above.
The frequency at resonance is given by
Current through the circuit is given by
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