Serial RLC Circuit Analysis

Series RLC circuits consist of a serially connected resistance, a capacitance and an inductor (coil) throughout an alternative feed.

So far, we have found that its three main passive components: Resistance , Inductor and Capacitance, have very different phase relationships with each other when connected to a sinusoidal alternative source.

Voltage waveforms in a pure omic resistance are "in the same phase" as the current.In a pure inducta, the voltage waveform current 90 o "progresses" and gives us the following statement: ELI.In a pure capacitance, the voltage waveform "outperforms" the current by 90 o and gives us the following statement: ICE.

This Phase Difference, Φ, depends on the reactive value of the components used. We know that reaccountance is zero if the circuit element (X) is resistant, positive if the circuit element is inductive and negative if it is capacitive. Here's how to give the resulting impedances:

Impedance of Circuit Elements

serial rlc circuit

Instead of analyzing each passive element individually, we can combine all three in a series of RLC circuits. The analysis of a series RLC circuit is the same as the double series RL and RC circuits that we have looked at before, but this time we need to take into account the sizes of both XL and Xcto find the overall circuit reassurance. Serial RLC circuits are classified as circumstrified circuits because they contain two energy storage elements, an inductans L and a capacitance C.

Serial RLC Circuit

serial rlc circuit

The series RLC circuit above has a single loop in which the instantaneous current flowing from the loop is the same for each circuit element. Since XL and XCof inductive and capacitive reactant are a function of the feeding frequency, the sinusoidal response of a series of RLC circuits will therefore vary according to the frequency. Next, the individual voltage drops of elements R, L and C along each circuit element will be "out of phase" with each other, as defined:

  • i (t) = I max sin(ωt)
  • The instant voltage on a pure resistance, VR, is "intra-phase" with current.
  • Instant voltage in a pure inductor, VL current 90o "advances"
  • Instant voltage in a pure capacitor, VC current 90it "outperforms"
  • Therefore, V L and V C 180are "out of phase" and opposite each other.

For the series RLC circuit above, this can be shown as follows:

serial rlc circuit

The amplitude of the welding voltage in all three components in a series RLC circuit consists of three separate component voltages, V R, V L and V C,and the current is common to the three components.Therefore, vector diagrams will have a current vector as a reference with three voltage vectors drawn according to this reference, as shown below.

Individual Voltage Vectors

serial rlc circuit

This means that we cannot combine VR, VL and VCto find the supply voltage, since all three voltage vectors point in different directions according to the current vector. Therefore, we must find the supply voltage as the Phaser Sum of the three component voltages that are vectorly combined.

Kirchhoff's voltage law (KVL) for both loop and node circuits states that around any closed loop, the sum of voltage drops around the loop is equal to the sum of emfs. Then applying this law to these three voltages will give us the amplitude of the source voltage.

Instant Voltages for Serial RLC Circuit

serial rlc circuit

The phaser diagram for a series of RLC circuits is produced by combining the above three separate phasers and adding these voltages vectorly.Since the current flowing from the circuit is common to all three circuit elements, we can use it as a reference vector with three voltage vectors drawn accordingly at the corresponding angles.

The resulting vector VSis obtained by combining two of the vectors, VL and VC,and then adding this sum to the remaining vector VR. The resulting angle between VS and I will be the phase angle of the circuits, as shown below.

Phaser Diagram for Serial RLC Circuit

serial rlc circuit

From the phaser diagram on the right side above, we can see that voltage vectors form a triangle from hypotenuse V S, horizontal axis V R and vertical axis V L – V C. We can use pythagorean theorem in this voltage triangle to obtain the VS mathematically.

Voltage Triangle for Serial RLC Circuit

serial rlc circuit

When using the equation above, please note that the last reactive voltage should always be positive, that is, the smallest voltage must always be taken from the largest voltage. A negative voltage cannot be added to VR,so it is right to have VL – VCor VC – VL. The smallest value must be subtracted from the largest, otherwise the calculation of VSwill be incorrect.

From above, we know that in all components of a series of RLC circuits, the current has the same amplitude and phase. Then, the voltage along each component can also be mathematically defined according to the current flowing through it and the voltage along each element.

serial rlc circuit

For the voltage triangle, in the Pythagorean equation above, it will replace these values and give us:

serial rlc circuit

Thus, we can see that the amplitude of the source voltage is proportional to the amplitude of the current that passes through.This proportional constant is called the impedance of the circuit, which ultimately depends on resistance and inductive and capacitive recess.

Then, in the series RLC circuit above, it can be seen that defying current flow consists of three components, XL , XC and R, and with reactance, XTis defined as follows : XT = XL – XC or XT = XC – XL whichever is larger.Thus, the total impedance of the circuit is considered as the voltage source required to pass a current through it.

Impedance of the Serial RLC Circuit

Because the three vector voltages are out of phase with each other, XL, XC, and R must be "out of phase" with the relationship between R, XL and XC, the vector sum of these three components. This will give us the general impedance of RLC circuits. These circuit impedances can be drawn and represented by an Empedans Triangle, as shown below.

Impedance Triangle for Serial RLC Circuit

serial rlc circuit

The impedance of a series of RLC circuits depends on the angular frequency such as Z, XL and XC, if capacitive reactanse is larger than inductive recess, XC > XL, then the general circuit reassurance is capacitive and gives a leading phase angle.

Similarly, if the inductive reactance is larger than capacitive reactance, X L > X C, then the general circuit reactance is inductive and gives a delayed phase angle to the series circuit.If the two reacques are the same and X L = X C, the angular frequency at which this occurs is called the resonance frequency and produces the resonance effect that we will examine in more detail in another tutorial.

Then the size of the current depends on the frequency applied to the serial RLC circuit.The impedance is at the maximum when Z is at maximum, the current is minimal, and in the same way the current is at the maximum when Z is at the minimum.Therefore, the above equation for impedance can be rewritten as follows:

serial rlc circuit

The phase angle between the source voltage and the current is the same as the angle between Z and R in the impedance triangle. This phase angle can be positive or negative depending on whether the welding voltage leads to the circuit current and can be mathematically calculated from the omic values of the impedance triangle:

serial rlc circuit

Serial RLC Circuit Question Sample 1

A series of RLC circuits with 12Ω resistance, 0.15H inducing and 100uF capacitors are connected serially via a 100V, 50Hz feed.Calculate the total circuit impedance, circuit current, power factor and draw the voltage phaser diagram.

serial rlc circuit

Inductive Reactance, X L.

serial rlc circuit

Capacitive Reactance, X C .

serial rlc circuit

Circuit Impedance, Z.

serial rlc circuit

Circuit Current, I

serial rlc circuit

Voltages in serial RLC Circuit, V R , V L , V C .

serial rlc circuit

Power factor and Phase Angle, ε .

serial rlc circuit

Phaser Diagram:

serial rlc circuit

Since the phase angle is calculated as ε 51.8 o positive value, the total reassurance of the circuit should be inductive.Since we take the current vector as our reference vector in a series of RLC circuits, the current is 51.8 o "behind" the source voltage, so that we can say that the phase angle is behind, as confirmed by our reminder expression "ELI".

Series RLC Circuit Summary

In a series of RLC circuitsthat include a resistance, an inductor and a capacitor, welding voltage VS is the sum of phasers consisting of three components, VR , VL and VC are common to all three currents.Since the current is common to three components, it is used as a horizontal reference when creating a voltage triangle.

The impedance of the circuit is the total opposition to the current flow. For a series of RLC circuits, and the impedance triangle can be drawn by dividing each side of the voltage triangle into its current, I. Voltage drop along the resistant element is equal to IR, the voltage along the two reactive elements is IX =I XL – IXC, while the welding voltage is equal to I *Z. The angle phase angle between VS and I will be ε.

When working with a series of RLC circuits with multiple resistances, capacitance or inductans are not pure, they can all be assembled together to form a single component.For example, all resistors are collected, R T = ( R 1 + R 2 + R 3 ) … etc. or all induced L T = ( L 1 + L 2 + L 3 ) … etc. can be a circuit that contains many elements in this way. can be easily reduced to a single circuit element.

serial rlc circuit

In the next tutorial on parallel RLC Circuits, we will look at the voltage-current relationship of the three components connected to each other in a parallel circuit configuration when a stable state sinusoidal AC waveform is applied together with the corresponding phaser diagram representation.