In our earlier tutorials we have learned that Resistance, Inductance, and Capacitance, the three basic passive components, react differently to a sinusoidal alternating voltage in terms of phase.
However, a series RLC circuit is formed by connecting passive elements, such as resistors, inductors, and capacitors, in series with a voltage supply.
Voltage Waveforms in Pure Components
1. Ohmic Resistor
The voltage and current waveforms have the same phase in a pure ohmic resistor.
2. Pure Inductance
For a pure inductor, the current lags behind the voltage by a phase angle of 90°, which can be expressed by the mnemonic: ELI.
3. Pure Capacitance
In a pure capacitance, the current is 90° ahead of the voltage, which means that the voltage is 90° behind the current.
This relationship can be remembered by the mnemonic: ICE, which stands for I (current) leads C (capacitance) in E (voltage).
Phase Difference and Reactance
The relationship between the voltage and current in a circuit is determined by the phase difference, Φ, which depends on the reactance, (X), of the circuit elements.
Reactance is a measure of how the circuit element resists the flow of current. It is zero for resistors, positive for inductors, and negative for capacitors. The impedances of these elements are given by:
| Circuit Element | Resistance (R) | Resistance (X) | Impedance (Z) |
| Resistor | R | 0 | ZR = R = R∠0o |
| Inductor | 0 | ωL | ZL = jωL =ωL∠+90o |
| Capacitor | 0 | 1 / ωC | ZC = 1 / jωL = (1 / ωC)∠-90o |
Integrating Passive Elements: Series RLC Circuit
A series RLC circuit combines all three passive elements: resistance, inductance, and capacitance.
This way, we can analyze them together instead of separately.
Analytical Approach to Series RLC Circuit
To analyze a series RLC circuit, we can use the same method as for the series RL and RC circuits we examined before.
However, we also need to consider the magnitudes of both XL and XC, which determine the total circuit reactance.
Characteristics of Second-Order Series RLC Circuits
A series RLC circuit has two components that store energy: an inductor L and a capacitor C. This makes it a second-order circuit.
The following diagram shows an example of a series RLC circuit and some of its features and behaviors.
In the series RLC circuit above, the same instantaneous current flows through each element of the single loop.
The supply frequency, ƒ, affects the inductive and capacitive reactance’s XL and XC.
As the frequency, ƒ, varies, the sinusoidal response of the series RLC circuit will also vary.
The voltage drops across each circuit element of R, L and C will not be in sync with each other.
We can describe this by saying that they are "out-of-phase.
i(t) = Imax sin(ωt)
A pure resistor VR has the same phase for the current and the voltage across it.
A pure inductor has a voltage VL that is 90o ahead of the current in phase.
This means that the voltage reaches its peak value before the current does.
A pure capacitor has a 90o phase difference between the current and the voltage across it, such that the voltage, VC, is behind the current in time.
As a result, VL and VC have a phase difference of 180o and they cancel each other out.
The following diagram represents the series RLC circuit in the above equation:
A series RLC circuit has three components: a resistor, an inductor and a capacitor.
The source voltage amplitude in this circuit equals the sum of the individual voltages across each component, VR, VL and VC.
The same current flows through all three components.
As shown below, the current vector is the reference for the vector diagrams, and the three voltage vectors are plotted relative to this reference.
Voltage Vectors for RLC shown Separately
We cannot find the supply voltage, VS, across all three components by simply adding VR, VL and VC together.
This is because the three voltage vectors point in different directions relative to the current vector.
To find the supply voltage, VS, we need to add the three component voltages vectorially.
This means we will use the Phasor Sum of these voltages.
According to Kirchhoff's voltage law (KVL), the algebraic sum of the voltage drops and the electromotive forces (EMFs) in any closed loop is zero, regardless of whether the loop is a nodal or a mesh circuit.
By applying this law to the three voltages, we can obtain the amplitude of the source voltage, VS, as shown in the following formulas.
Voltages across Series RLC Circuit at any Instant
- KVL: VS - VR - VL - VC = 0
- VS - IR - L(di/dt) - Q/C = 0
- ∴ VS = IR + L(di/dt) + Q/C
To produce the phasor diagram for a series RLC circuit, we add the three individual phasors above as vectors.
These phasors represent the voltages across the circuit components.
The same current flows through all three elements of the circuit, so we can use it as the reference vector.
We draw the three voltage vectors relative to the reference vector at their respective angles.
A possible way to rewrite the text is:
To get the resulting vector VS, we first add the vectors VL and VC. Then we add this sum to the vector VR.
The angle between VS and i, as shown below, is the phase angle of the circuit.
Series RLC Circuit Phasor Diagram
A right triangle is formed by the voltage vectors in the phasor diagram on the right, where VS is the hypotenuse, VR is the horizontal side, and VL – VC is the vertical side.
As you can see, this creates the familiar Voltage Triangle that we have seen before.
We can apply Pythagoras's theorem to this triangle to calculate the value of VS mathematically, as shown below.
Series RLC Circuit Voltage Triangle
- V2S = V2R + (VL - VC)2
- VS = √[(V2R + (VL - VC)2]
A possible way to rewrite the text is:
When you apply the equation above, make sure that the final reactive voltage is always positive.
This means that you have to subtract the smaller voltage from the larger one.
You cannot add a negative voltage to VR. So, you can write VL – VC or VC – VL, depending on which one is bigger.
To calculate VS correctly, use the largest value from the smallest.
A series RLC circuit has the same amplitude and phase of the current in all its components, as we have seen before.
A mathematical description of the voltage across each component can be derived from the current flowing through it and the voltage across each element as follows.
- VR = iR sin(ωt+0o) = i.R
- VL = iXL sin(ωt+90o) = i.jωL
- VC = iXC sin(ωt-90o) = i.(1/jωC)
We can use the Pythagoras formula for the voltage triangle to find the values by substituting them as follows:
- VR = I.R, VL = I.XL, VC = I.XC
- VS = √[(I.R)2 + (I.XL - I.XC)2]
- VS = I.√[(R2 + (XL - XC)2]
- ∴ VS = I x Z, where Z = √[(R2 + (XL - XC)2]
This shows that the current in the circuit varies with the source voltage's amplitude.
The circuit's impedance, which is a proportionality constant, depends on the resistance and the inductive and capacitive reactances.
The series RLC circuit above has three components that oppose the current flow: XL, XC and R.
The reactance, XT, of any series RLC circuit is the difference between the larger and the smaller of XL and XC.
That is, XT = XL – XC or XT = XC – XL, depending on which one is greater.
A current flowing through a circuit can be seen as the result of a voltage source that overcomes the total impedance of the circuit.
Series RLC Circuit's Impedance
The three vector voltages have different phases, so XL, XC and R also have different phases.
The vector sum of R, XL and XC shows the relationship between them.
The overall impedance, Z, of the RLC circuits can be calculated and depicted by an Impedance Triangle as shown below.
The circuit impedances are the sides of the triangle.
A Series RLC Circuit and Its Impedance Triangle
Z2 = R2 + (XL - XC)2
A series RLC circuit has an impedance Z that varies with the angular frequency, ω, as well as the inductive reactance XL and the capacitive reactance XC.
When XC is larger than XL, the circuit has a capacitive reactance and a positive phase angle.
Similarly, when the inductive reactance is higher than the capacitive reactance, XL > XC, the overall circuit reactance is inductive and the series circuit has a lagging phase angle.
When the two reactance’s are equal and XL = XC, this happens at a specific angular frequency called the resonant frequency.
This causes the effect of resonance, which we will explore further in another tutorial.
A series RLC circuit's current magnitude varies with the applied frequency. The current and the impedance, Z, have an inverse relationship.
When Z reaches its highest value, the current drops to its lowest value.
Conversely, when Z falls to its lowest value, the current rises to its highest value.
Therefore, we can rewrite the above equation for impedance as:
Impedance, Z = √[R2 + (ωL - 1/ωC)2]
The current, i, and the source voltage, VS, have the same phase angle, θ, as the angle between Z and R in the impedance triangle.
The phase angle can have a positive or negative sign, depending on whether the current in the circuit is behind or ahead of the source voltage.
We can use the ohmic values of the impedance triangle to calculate the phase angle mathematically as follows:
cosΦ = R/Z, sinΦ = (XL - XC) / Z, tanΦ (XL - XC) / R
Solving a Series RLC Circuit
Connected in series across a 100V, 50Hz supply is a series RLC circuit that comprises a 12Ω resistance, 0.15H inductance, and a 100μF capacitor.
Determine the total circuit impedance, circuit current, power factor, and illustrate the voltage phasor diagram.
Inductive Reactance, XL
XL = 2πfL = 1 / (2π x 50 x 0.15) = 47.13Ω
Capacitive Reactance, XC.
XC = 1 / 2πfC = 1 / (2π x 50 x 100 x 10-6) = 31.83Ω
Circuit Impedance, Z.
- Z = √[R2 + (XL - XC)2]
- Z = √[122 + (47.13 - 31.83)2]
- Z = √144 + 234 = 19.4Ω
Circuits Current, I.
I = VS / Z = 100 / 19.4 = 5.14 Amps
The series RLC circuit has three voltages: VR across the resistor, VL across the inductor, and VC across the capacitor.
- VR = I x R = 5.14 x 12 = 61.7 Volts
- VL = I x XL = 5.14 x 47.13 = 242.2 Volts
- VC = I x XC = 5.14 x 31.8 = 163.5 Volts
Now let's determine Power factor and Phase Angle, θ using the following formula.
- cosΦ = R / Z = 12 / 19.4 = 0.619
- ∴ cos-1 0.619 = 51.8o lagging.
Check the phasor diagram as shown below.
A positive value of 51.8o for the phase angle θ indicates that the circuit has an overall inductive reactance.
In a series RLC circuit, we use the current vector as our reference vector.
Therefore, the current "lags" the source voltage by 51.8o.
This means that the phase angle is lagging, as our mnemonic expression "ELI" confirms.
Conclusion
A series RLC circuit consists of a resistor, an inductor, and a capacitor.
The current is the same for all three components. The source voltage (VS) is the result of adding the phasor voltages of the resistor (VR), the inductor (VL), and the capacitor (VC).
A way to represent the total resistance to the flow of current is the impedance triangle, which is obtained by dividing each side of the voltage triangle by the same current (I).
The resistive voltage drop is I*R, and the voltage across the reactive elements is I*X = I*XL – I*XC, while the source voltage is I x Z.
The angle between VS and I indicates the phase angle (θ).
A series RLC circuit with multiple components can be simplified by combining the pure or impure resistances, capacitances, or inductances.
A complex circuit can be simplified by adding together all the resistances (RT = R1 + R2 + R3) and all the inductances (LT = L1 + L2 + L3) to get a single impedance for each component.