# Waveforms for Series AC Circuits

Applet

The best way to understand the applet may be to play with the applet a bit before reading how it works.

## Controls:

Animation shows the intantaneous waveforms for the six types of Series AC Circuits:

• R - resistor.
• L - ideal inductor.
• C - capacitor.
• RL - resistor and ideal inductor.
• RC - resistor and capacitor.
• RLC - resistor, ideal inductor and capacitor.

Select type of circuit from Circuit dropdown. When selected, the circuit is drawn on the lower left. The two analog meters will deflect depending on the polarity and magnitude of the source voltage vS or series current i being measured. The instantaneous equations for the waveforms are shown on the lower right.

The circuits are arranged in Circuit dropdown from most lagging, L, to most leading C, with RLC being variable.

The instantaneous waveforms shown depend on type of circuit selected. Instantantaneous waveforms are:

• i - series current in red.
• vS - source voltage in blue.
• vR - voltage drop across resistor in teal (if the circuit has a resistor).
• vL - voltage drop across inductor in brown (if the circuit has an inductor).
• vC - voltage drop across capacitor in purple (if the circuit has an capacitor).
• p - power in lime.

Check boxes on lower left, select which waveforms to show. In some circuits (R, L and C), vS hides the component voltage, so turn off vS to see the component voltage.

Select ALL to show all relevant voltage and current waveforms for a specific circuit. ALL must be switched off if you want to see specific waveforms.

Power can be selected independently. Depending on the type of circuit, power is either being consumed or being returned to the source. Power is a scalar. When it is positive, power is being consumed, and negative, returned. The applet indicates the flow of power.

As the magnetic field of the inductor or the electric field of the capacitor is formed, the devices consume power. But as the fields reverse due to the reversal of the AC current, the energy stored in the magnetic or electric fields is returned to the source. This is easily seen by selecting the L or C circuits and showing the power waveform. The power waveform is 50% above and 50% below the x-axis.

The resistor consumes power, converting electrical power into heat. Again, this is easily seen by selecting the R circuit and showing power. The power waveform is only positive.

In a RLC circuit, vL and vC are opposite in polarity. So the capacitor supplies leading capacitive reactive power to the lagging inductive reactive power of the inductor. All the source must provide to the capacitor and inductor is the net difference.

The applet indicates the phase angle between VS and I. If adjustable, a slider is available to adjust the phase angle θ.

Radio buttons select Reference for the instantaneous waveforms (which waveform starts at 0°):

• I - current.
• VS - source voltage.

In series circuits, it is customary to make current I the Reference (current is constant in a series circuit), which aligns the phasor diagram to the x-axis. But for parallel and series/parallel circuits, it is more customary to make the source voltage VS the Reference (voltage is constant in a parallel circuit). This means the phasor diagram will be skewed depending on the phasor angle with VS along the x-axis. Selecting the different types of circuits with VS or I selected will draw the phasor diagram (using maximum quantities) with the appropriate Reference.

Final Controls:

• Phasor - Turn on/off Phasor Diagram
• Waveforms - Turn on/off Waveforms (not working)
• # of Waves - 1, 2, 3 or 4 full waveforms can be shown.
• Speed - Animation can be slowed down.
• Run - Clicked for animation to repeat or animation will draw one set of waveforms.
• Manual - Left Mouse Button - Holding and dragging allows mouse to control shown waveform.
• Connectors - Lines between vector rotation and waveforms.

## Limitation:

The purpose of the animation is to show instantaneous waveforms. This is shown best if the magnitudes of the source voltage vS, the current i and the power p are constant. Within this limitation, all parameters are mathematically correct. The animation must fit within the container.

The phase angle is limited to 5° or 85° for applet constraints. A vector at 1° would be hard to draw.

## Theory:

A memory mnemonic will help understand how voltages and currents are related in an AC circuit.

ELI the ICE man.

or ELI the ICEman drinks RIE. or ELI drinks RIE with ICE.

where rye is spelt rhyming with lie or die.

The keywords are ELI, RIE and ICE.

• E is source voltage (EMF).
• I is current.
• R, L and C indicate a circuit, which is resistive, purely inductive or capacitive.

The relationship of E and I to the circuit letter (R, L or C) describes the relationship between voltage and current in that circuit.

ELI, where E comes before I, indicates that in a purely inductive circuit, L, voltage leads current by 90° or using current as the reference, current lags voltage by 90°. The physical letter L shows the relationship, also.

RIE, where E and I are on the same side, indicates that in a resistive circuit, R, voltage and current are in phase with each other.

ICE, where I comes before E, indicates that in a capacitive circuit, C, current leads voltage by 90°.

Now, when we put different combinations of R, L or C in series with each other, these relationships are maintained. Take a series RL circuit (phasor diagram - upper right). The current flowing through the resistor produces a voltage drop VR, which is in phase with the current. The current flowing through the inductor produces a voltage drop VL, which leads the current by 90°. The source voltage VS is the vector addition of these two components. I lags VS by a phase angle θ between 0° and 90° (0° < θ < 90°).

With RLC series circuits, the relationship depends on the vertical legs, XL and XC.

• If XL > XC (or VL > VC or QL > QC), then I lags VS by θ.
• If XC > XL (or VC > VL or QC > QL), then I leads VS by θ.
• Finally, when XL = XC, then I is in phase with VS and the triangles disappear (VL = VC, VS = VR, QL = QC and S = P).

To see this illustrated, select vS and i to be shown and I as the reference. Now select the circuits in the order shown (L, then RL, etc.). I will be on the x-axis and the source voltage will start at +90° (90° lagging) and rotate towards -90° (90° leading). The angle for the RLC circuit depends on which leg is larger.

• L - I lags VS by 90°.
• RL - I lags VS by a phase angle θ between 0° and 90°.
• R - I and VS are 'in phase'.
• RC - I leads VS by a phase angle θ between 0° and 90°.
• C - I leads VS by 90°.
• RLC - depends on components:
• XL > XC - I lags VS by a phase angle θ between 0° and 90°.
• XC > XL - I leads VS by a phase angle θ between 0° and 90°.
• XL = XC - I and VS are in phase.