# AC Series and Parallel Circuits 4: Inductors and Capacitors

1. Series and Parallel Circuits 1: The Basics
2. Series and Parallel Circuits 2: Resistors
3. Series and Parallel Circuits 3: Capacitors
4. AC Series and Parallel Circuits 4: Inductors and Capacitors (this article)
5. Series and Parallel Circuits 5: More About Circuits

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Inductors and capacitors, when subjected to alternating current (AC) and other, non-direct current (non-DC) sources, and placed in series or parallel circuit configurations, have numerous uses to the designer.

When stimulated with a sinusoidal AC voltage, ideal capacitors transmit current that “leads” the applied voltage by 90o (one cycle being 3600), while ideal inductors transmit current that lags the applied voltage by 90o as shown below.[1]

The inductive, capacitive and resistive components of impedance may be represented by the impedance diagram below.  In this diagram, the capacitive component (capacitive reactance or XC) works in a manner that is opposite the inductance component (inductive reactance or XL), and both are orthogonal to the resistance component (R or XR).  These three vectors combine to result in the impedance (Z) using the Pythagorean theorem.

[1] Adapted From:  Electronics Hub (http://jshavaluk.blogspot.com/2016/12/why-is-power-factor-important-in.html)

Thus, capacitors, inductors and resistors in series AC circuits exhibit overall impedance properties according to the relation:

where:
Z is impedance (Ω)

R is resistance (Ω)

XL is inductive reactance = ωL = 2πfL (Ω)

XC is capacitive reactance = 1/ωC = 1/2πfC (Ω)

This results in a “V” shaped impedance vs. frequency curve, or trough, as shown below for a series RLC circuit having R=0.1mΩ, L=100pH and C=1μF.

And as shown above, the series impedance vs. frequency curve reaches a minimum impedance when the inductive reactance (XL) equals the capacitive reactance (XC).  This is called the resonance frequency (fr) and occurs at frequency value fr, where:

where:

fr is Hertzian resonance frequency (Hz)

L is inductance (H)

C is capacitance (F)

At the resonance frequency (fr), the impedance (Z) equals or nearly equals the resistance (R) of the circuit.  So, if the circuit resistance is 0Ω, the impedance of the series circuit is ~0Ω at the resonance frequency.  And at frequencies above and below resonance (fr) the impedance (Z) rapidly increases.  Thus, the series RLC circuit passes frequencies in the range of fr.  This is known as a band pass or notch filter and is valuable for passing a certain range of frequencies (within the band), and attenuating frequencies of higher and lower frequency.  The bandwidth (BW) of the series RLC notch filter is the frequency range where Z is below approximately double the impedance value at resonance (Zr), as shown above in red (i.e., a 3 dB change).

In the case of a parallel RLC circuit, as shown below,[2] the total impedance of the parallel circuit is determined from its admittance (Y):

where:

Z = Impedance (Ω)

ω = angular frequency

f = Hertzian frequency

R = resistance (Ω)

L = Inductance (H)

C = Capacitance (F)

In the parallel RLC circuit, the impedance (Z) curve is inverted compared to the series RLC circuit.  An example is given below, also for R=0.1mΩ, L= 100 pH, and C=1μF.  This type of parallel RLC circuit is also known as a band stop or band rejection filter, since it attenuates signals over a range of frequencies as indicated by the red region below for 3dB from peak impedance.

The resonance frequency (fr) for the parallel RLC is:

which is identical to the series RLC circuit having the same L and C values.  The bandwidth of the parallel RLC circuit for the case above is also indicated for a 3 dB change in red.

The extent of the peak or trough of the resonance is set by the resistance (R) in both the series and parallel RLC circuits as shown below.  In this case of the parallel RLC circuit, the peak of the resonance does not exceed about 500mΩ even though higher R values (0.1Ω and 1Ω) are modeled.  This indicates that there is a limit of the effect of R on the peak of the parallel RLC circuit.  In all cases, the resonance frequency (fr) is the same.  This is valuable in establishing the selectivity, or quality factor (Q) of the filter circuit.  A higher Q factor results in a narrower bandwidth, passing (series) or rejecting (parallel) more signal (depending upon the type of filter) over a narrower range of frequency than a lower Q factor.  Thus, it is important to understand the details behind the signal that the circuit designer intends to pass or block, so that no intended signal (frequency) is blocked, and no unintended signal (frequency) is passed.

Series and parallel RLC series come in many variants.  They are also known as resonant circuits.  Each has a resonant frequency fr, where the capacitor and inductor interact so as to resonate.  In the case that R of the circuit is 0, signals of resonant frequency (fr) resonate in the RLC circuit in a manner that is unattenuated (undamped), and will resonate indefinitely.  This can cause unwanted effects in the circuit, such as “ringing” as shown below.[3]

When R is greater than 0, signals of resonant frequency (fr) will dampen.  The extent of damping depends upon the relative values of the resistance (R), and may result in a resonant signal which may be undamped, underdamped, critically damped, or over damped as shown below.[4]

The damping factor (αs) for the simple series RLC circuit is: [5,6]

The damping factor (αp) for the simple parallel RLC circuit is:

In the case where the damping factor (α) exceeds the resonant frequency (angular basis, i.e., α > ωr), where:

the circuit is said to be overdamped.  In the case where α = ωr, the circuit is critically damped, and in the case where α < ωr, the circuit is underdamped.  Thus, it is important to understand the damping required for your resonant circuit designs.  In our next post, we will discuss circuit types.

[4] Derived from:  https://gab.wallawalla.edu/~curt.nelson/engr228/lecture/chap8.pdf

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