Series RLC Circuit Analysis
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| Circuit Element |
Resistance, (R) |
Reactance, (X) |
Impedance, (Z) |
| Resistor |
R |
0 |
\(Z_R=R\\=R\angle0^\circ\) |
| Inductor |
0 |
\(\omega L\) |
\(Z_L=j\omega L\\=\omega L\angle+90^\circ\) |
| Capacitor |
0 |
\(\frac1{\omega C}\) |
\(Z_C=\frac1{j\omega C}\\=\frac1{\omega C}\angle-90^\circ\) |
\(i_{\left(t\right)}=I_{max}\sin\left(\omega t\right)\)
The instantaneous voltage across a pure resistor, \(V_R\) is “in-phase” with current
The instantaneous voltage across a pure inductor, \(V_L\) “leads” the current by \(90^o\)
The instantaneous voltage across a pure capacitor, \(V_C\) “lags” the current by \(90^o\)
Therefore, \(V_L\) and \(V_C\) are \(180^o\) “out-of-phase” and in opposition to each other.
Phasor Diagram for a Series RLC Circuit
Formula for RLC oscillation
\(Z=\sqrt{R^2+\left(X_L-X_C\right)^2}\)
\(X_L=\omega L\), \(X_C=\frac1{\omega C}\)
The maximum of \(Z\) is at the condition \(X_L-X_C = 0\).
\(\omega L=\frac1{\omega C}\)
\(\omega^2=\frac1{LC}\)
\(\mathrm\omega=\frac1{\sqrt{LC}}\)
\(2\mathrm{πf}=\frac1{\sqrt{LC}}\)
\(f_0=\frac1{2\mathrm\pi\sqrt{LC}}\)
where \(f_0\) is the resonant frequency of an RLC series circuit.
Series Resonance Frequency
Series RLC Circuit at Resonance
Series Circuit Current at Resonance
Bandwidth of a Series Resonance Circuit
If the series RLC circuit is driven by a variable frequency at a constant voltage, then the magnitude of the current, I is proportional to the impedance, \(Z\), therefore at resonance the power absorbed
by the circuit must be at its maximum value as \(P=I^2Z\).
If we now reduce or increase the frequency until the average power absorbed by the resistor in the series resonance circuit is half that of its maximum value at resonance, we produce two frequency points
called the half-power points which are \(-3dB\) down from maximum, taking \(0dB\) as the maximum current reference
These \(-3dB\) points give us a current value that is \(70.7%\) of its maximum resonant value which is defined as: \(0.5\left(I^2R\right)=\left(0.707\times I\right)^2R\). Then the point corresponding to
the lower frequency at half the power is called the “lower cut-off frequency”, labelled \(ƒ_L\) with the point corresponding to the upper frequency at half power being called the “upper cut-off
frequency”, labelled \(ƒ_H\).
The distance between these two points, i.e. \(( ƒ_H – ƒ_L )\) is called the Bandwidth, (BW) and is the range of frequencies over which at least half of the maximum power and current is provided as shown.
The frequency response of the circuits current magnitude above, relates to the “sharpness” of the resonance in a series resonance circuit. The sharpness of the peak is measured quantitatively and is
called the Quality factor, Q of the circuit.
The quality factor relates the maximum or peak energy stored in the circuit (the reactance) to the energy dissipated (the resistance) during each cycle of oscillation meaning that it is a ratio of
resonant frequency to bandwidth and the higher the circuit Q, the smaller the bandwidth, \(Q=\frac{f_r}{BW}\).
Q factor of an RLC circuit
The Q-factor or quality factor determines the quality of an RLC circuit. When you design an RLC circuit, you should aim for the highest possible Q-factor.
\(Q=\frac1R\sqrt{\frac LC}\)
The Impedance Triangle for a Series RLC Circuit
\(\mathrm{Impedance},\;\mathrm Z=\sqrt{\mathrm R^2+\left(\mathrm{ωL}-\frac1{\mathrm{ωC}}\right)^2}\)
\(\cos\phi=\frac RZ\)
\(\sin\phi=\frac{X_L-X_C}Z\)
\(\tan\phi=\frac{X_L-X_C}R\)
\(\)
Dividing Complex Numbers Formula
Dividing Complex Numbers
\(\dfrac{z_1}{z_2}=\dfrac{a+ib}{c+id}\)
\(\begin{aligned}\dfrac{z_1}{z_2}&=\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bc-ad}{c^2+d^2}\right)\end{aligned}\)
\(
\begin{aligned}\dfrac{z_1}{z_2}&=\dfrac{a+ib}{c+id}\\&=\dfrac{a+ib}{c+id}\times\dfrac{c-id}{c-id}\\&=\dfrac{(a+ib)(c-id)}{c^2-(id)^2}\\&=\dfrac{ac-iad+ibc-i^2bd}{c^2-(-1)d^2}\\&=\dfrac{ac-iad+ibc+bd}{c^2+d^2}\\&=\dfrac{(ac+bd)+i(bc-ad)}{c^2+d^2}\\&=\dfrac{ac+bd}{c^2+d^2}+i\left(\dfrac{bc-ad}{c^2+d^2}\right)\end{aligned}
\)
Division of Complex Numbers in Polar Form
\(
\begin{aligned}\dfrac{z_1}{z_2}&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)}{r_2\left(\cos\theta_2+i\sin\theta_2\right)}\left(\dfrac{\cos\theta_2-i\sin\theta_2}{\cos\theta_2-i\sin\theta_2}\right)\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2-i\sin\theta_2\right)}{r_2\left(\cos^2\theta_2-(i)^2\sin^2\theta_2\right)}\\&=\dfrac{r_1\left(\cos\theta_1+i\sin\theta_1\right)\left(\cos\theta_2-i\sin\theta_2\right)}{r_2(\cos^2\theta_2+\sin^2\theta_2)}\\&=\frac{r_1}{r_2}\left[\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)\right]\\&=r\left(\cos\theta+i\sin\theta\right)\end{aligned}
\)
Whre \(\theta=\theta_1-\theta_2\) and \(r=\dfrac{r_1}{r_2}\).
Parallel RLC Circuit Analysis
Impedance and Admittance
Immittance (導抗) is a term used for both impedance (阻抗) and admittance (導納).
The impedance 阻抗, \(Z\), is composed of real and imaginary parts,
\(Z=R+jX\)
\(R\) is the resistance 電阻, measured in ohms.
\(X\) is the reactance 電抗, measured in ohms.
The Admittance 導納, \(Y\), is composed of real and imaginary parts,
\(Y=G+jB\)
\(G\) is the conductance 電導, measured in siemens.
\(B\) is the susceptance 電納, measured in siemens.
\(Y=Z^{-1}=\frac1{R+jX}=\left(\frac1{R^2+X^2}\right)\left(R-jX\right)\)
\(G=\mathfrak R\left(Y\right)=\frac R{R^2+X^2}\)
\(B=\mathfrak I\left(Y\right)=-\frac X{R^2+X^2}\)
\(Z=Y^{-1}=\frac1{G+jB}=\left(\frac1{G^2+B^2}\right)\left(G-jB\right)\)
\(R=\mathfrak R\left(Z\right)=\frac G{G^2+B^2}\)
\(X=\mathfrak I\left(Z\right)=-\frac B{G^2+B^2}\)
Admittance is defined as, \(\)
\(\mathrm{\mathit{V}=\mathit{IZ}=\frac{\mathit{I}}{\mathit{Y}}}\)
\(Z=\) Total impedance of the parallel circuit,
\(Y=\frac1Z=\) Admittance of the parallel circuit.
\(Y\) is the admittance, measured in siemens. \(℧\)
\(Z\) is the impedance, measured in ohms. \(Ω\)
The admittance of the parallel circuit is given by
\(\mathrm{\mathit{Y}=\frac{1}{\mathit{R}}+\frac{1}{\mathit{j\omega L}}+\mathit{j\omega C}=\frac{1}{\mathit{R}}+ {\mathit{j}}(\mathit{\omega C}-\frac{1}{\mathit{\omega L}})=\mathit{G}+\mathit{jB}}\)
\(G=\frac1R=\) Conductance of the circuit, \(siemens (S)\)
\(B=\frac1X=\) Susceptance of the circuit,
\(\mathrm{Magnitude\:of\:admittance,|\mathit{Y}|=\sqrt{(\frac{1}{\mathit{R}})^{2}+(\mathit{\omega C}-\frac{1}{\mathit{\omega L}})^{2}}}\)
\(\mathrm{Phase\:angle\:of\:admittance,\:\varphi=\tan^{-1}(\frac{\mathit{\omega C}-\frac{1}{\mathit{\omega L}}}{\frac{1}{\mathit{R}}})=\tan^{-1}(\mathit{R}(\mathit{\omega C}-\frac{1}{\mathit{\omega
L}}))}\)
\(Therefore, \mathrm{\mathit{I}=\mathit{VY}=\mathit{V}×\sqrt{(\frac{1}{\mathit{R}})^{2}+(\mathit{\omega C}-\frac{1}{\mathit{\omega L}})^{2}}\angle\tan^{-1}(\mathit{R}(\mathit{\omega
C}-\frac{1}{\mathit{\omega L}}))}\)
\(Thus,\)
\(\mathrm{Magnitude\:of\:supply\:current,| \mathit{I}|=\mathit{V}×\sqrt{(\frac{1}{\mathit{R}})^{2}+(\mathit{\omega C}-\frac{1}{\mathit{\omega L}})^{2}}}\)
\(\mathrm{Phase\:angle\:of\:admittance,\:\varphi=\tan^{-1}(\mathit{R}(\mathit{\omega C}-\frac{1}{\mathit{\omega L}}))}\)
The current \(I_R\) through resistance being in phase with the supply voltage.
The current \(I_L\) through inductor lags the applied voltage by 90°.
The current \(I_C\) in the capacitor leads the applied voltage by 90°.
\(\mathrm{\mathit{I}=\mathit{I}_{\mathit{R}}+\mathit{I}_{\mathit{L}}+\mathit{I}_{\mathit{C}}}\)
\(\mathrm{Magnitude\:of\:supply,|\mathit{I}|=\sqrt{(\mathit{I}_{\mathit{R}})^2+(\mathit{I}_{\mathit{C}}-\mathit{I}_{\mathit{L}})^2}}\)
Parallel Resonance
The condition of resonance occurs in the parallel RLC circuit, when the susceptance part of admittance is zero. However, admittance is
\(\mathrm{\mathit{Y}=\mathit{G}+\mathit{jB}=\frac{1}{\mathit{R}}+\mathit{j}(\mathit{\omega C}-\frac{1}{\mathit{\omega L}})}\)
\(\mathrm{\mathit{\omega C}-\frac{1}{\mathit{\omega L}}=0}\)
The frequency at which resonance occurs is
\(\mathrm{\mathit{\omega_{0} C}-\frac{1}{\mathit{\omega_{0} L}}=0}\)
\(\mathrm{\Rightarrow\:\mathit{\omega_{0}}=\frac{1}{\sqrt{\mathit{LC}}}}\)
Admittance – Frequency Curve
\(\mathrm{\mathit{Y}=\frac{1}{\mathit{R}}+\mathit{j}(\mathit{\omega C}-\frac{1}{\mathit{\omega L}})}\)
Current – Frequency Curve
\(\mathrm{\mathit{I}=\mathit{VY}\:or\:\mathit{I}\:\alpha\:\mathit{Y}}\)
Thus, current versus frequency curve of the parallel RLC circuit is same as that of admittance versus frequency curve.
Refer the current-frequency curve,
\(\mathrm{Lower\:cut\:off\:frequency,\mathit{\omega}_{1}=-\frac{1}{2\mathit{RC}}+\sqrt{(\frac{1}{2\mathit{RC}})^2+\frac{1}{\mathit{LC}}}}\)
\(\mathrm{Upper\:cut\:off\:frequency,\omega_{2}=+\frac{1}{2\mathit{RC}}+\sqrt{(\frac{1}{2\mathit{RC}})^2+\frac{1}{\mathit{LC}}}}\)
The bandwidth of the circuit is
\(\mathrm{BW=\mathit{\omega}_{2}-\mathit{\omega}_{1}=\frac{1}{\mathit{RC}}=\frac{\mathit{\omega}_{0}}{\mathit{\omega}_{0}\mathit{RC}}=\frac{\mathit{\omega}_{0}}{Q_{0}}}\)
Also,
\(\mathrm{\mathit{\omega}_{1}\:\mathit{\omega}_{2}=\frac{1}{\mathit{LC}}=\mathit{\omega}_{0}^2}\)
\(\mathrm{\Rightarrow\:\mathit{\omega}_{0}=\sqrt{(\mathit{\omega}_{1}\:\mathit{\omega}_{2})}}\)
Hence, the resonant frequency is the geometric mean of half-power frequencies.
Quality Factor of Parallel RLC circuit
\(\mathrm{∵\mathit{Q}\:−factor=\frac{Reactive \:Power}{Active\:Power}=\frac{\mathit{R}}{\mathit{\omega L}}=\mathit{\omega RC}}\)
At resonance, \(\mathrm{\mathit{Q}_{0}-factor=\frac{\mathit{R}}{\omega_{0}\mathit{L}}=\omega_{0}\mathit{RC}}\)
Since, the resonant frequency is
\(\mathrm{\omega_{0}=\frac{1}{\sqrt{\mathit{LC}}}}\)
So, the quality factor of parallel resonant circuit is
\(\mathrm{\mathit{Q}_{0}-factor=\frac{\mathit{R}}{\omega_{0}\mathit{L}}=\mathit{R}\frac{\sqrt{\mathit{LC}}}{\mathit{L}}=\mathit{R}\sqrt{\frac{\mathit{C}}{\mathit{L}}}}\)
