Fundamental concept of Phaser Method
Consider a 3rd order RLC circuit as an example
\(y^{(3)}+a_2\ddot y+a_1\dot y+a_0y=b_2\ddot v+b_1\dot v+b_0v\)
Laplace transform
\(s^3\widehat y(s)-s^2y(0)-s\dot y(0)-\ddot y(0)+a_2\lbrack s^2\widehat y(s)-sy(0)-\dot y(0)\rbrack+a_1\lbrack s\widehat y(s)-y(0)\rbrack+a_0\widehat y(s)\)
\(=b_2\lbrack s^2\widehat v(s)-sv(0)-\dot v(0)\rbrack+b_1\lbrack s\widehat v(s)-v(0)\rbrack+b_0\widehat v(s)\)
\(\widehat y(s)=\frac{p_2s^2+p_1s+p_0}{s^3+a_2s^2+a_1s+a_0}+\frac{b_2s^2+b_1s+b_0}{s^3+a_2s^2+a_1s+a_0}\widehat v(s)\)
\(=\widehat p(s)+\widehat h(s)\widehat v(s)\)
\(\widehat v(s)=\frac{Vs}{s^2+\omega^2}\)
Since the characteristic equation \(\lambda^3+a_2\lambda^2+a_1\lambda+a_0=0\) has 3 roots with negative real part, \({\left.p(t)\right|}_{t\rightarrow\infty}={\left.s\widehat
p(s)\right|}_{t\rightarrow0}=0\)
\(\therefore{\left.y(t)\right|}_{t\rightarrow\infty}=ℒ^{-1}\lbrack\widehat h(s)\frac{Vs}{s^2+\omega^2}\rbrack=h(t)\ast v(t)\)
\(As\;t\rightarrow\infty,\;y(t)=h(t)\ast v(t)\)
\(=ℒ^{-1}\lbrack\widehat h(s)\frac{Vs}{s^2+\omega^2}\rbrack\)
\(\widehat h(s)\frac{Vs}{s^2+\omega^2}=\frac{b_2s^2+b_1s+b_0}{s^3+a_2s^2+a_1s+a_0}\cdot\frac{Vs}{s^2+\omega^2}\)
\(=\frac{q_2s^2+q_1s+q_0}{s^3+a_2s^2+a_1s+a_0}+\frac{As+B\omega}{s^2+\omega^2}\)
\(\frac{q_2s^2+q_1s+q_0}{s^3+a_2s^2+a_1s+a_0}=\widehat q(s)\)
\({\left.y(t)\right|}_{t\rightarrow\infty}=ℒ^{-1}\lbrack\widehat h(s)\frac{Vs}{s^2+\omega^2}\rbrack=ℒ^{-1}\lbrack\widehat q(s)\rbrack+A\cos\omega t+B\sin\omega t\)
\(ℒ^{-1}\lbrack\widehat q(s)\rbrack\rightarrow0,\;t\rightarrow\infty\)
\((q_2s^2+q_1s+q_0)(s^2+\omega^2)+(As+B\omega)(s^3+a_2s^2+a_1s+a_0)=(b_2s^2+b_1s+b_0)Vs\)
\(s=j\omega\)
\((Aj\omega+B\omega){\left.(s^3+a_2s^2+a_1s+a_0)\right|}_{s=j\omega}={\left.(b_2s^2+b_1s+b_0)\right|}_{s=j\omega}Vj\omega\)
\(\Rightarrow Aj+B=\widehat h(j\omega)Vj\)
\(\Rightarrow A-Bj=V\widehat h(j\omega)\)
\(=V\vert\widehat h(j\omega)\vert e^{\angle\widehat h(j\omega)}\)
\(\Rightarrow\sqrt{A^2+B^2}e^{-j\theta},\;(\theta=\tan^{-1}(\frac BA))\)
\(\widehat y(s)=\widehat h(s)\widehat v(s)=\widehat h(s)\frac{Vs}{s^2+\omega^2}\)
\(y(t)=V\vert\widehat h(j\omega)\vert+\cos(wt+\angle\widehat h(j\omega))\)
\(s\) is replaced by \(j\omega\)
Phasor of sinusoidal signal
\(v(t)=V\cos(\omega t+\theta)=V\cos\theta\cos\omega t-V\sin\theta\sin\omega t\)
\(=Re(Ve^{j(\omega t+\theta)})\)
\(e^{j\phi}=\cos\phi+j\sin\phi\;(Euler\;formula)\)
\(Re(Ve^{j(\omega t+\theta)})=Re(Ve^{j\theta}e^{j\omega t}),\;Ve^{j\theta}\leftarrow Phasor\)
The phasor of a sinusoidal signal \(v(t)=V\cos(\omega t+\theta)\) is designed as \(Ve^{j\theta}\) (implicitly contains single frequency \(\omega\))
Components in Phasor Method
voltage source \(v_s(t)=V_s\cos(\omega t+\theta_s)\)
\(V_se^{j\theta_s}=V_s\angle\theta_s\)
current source \(i_s(t)=I_s\cos(\omega t+\theta_s)\)
\(I_se^{j\theta_s}=I_s\angle\theta_s\\i_s(t)=I_s\cos(\omega t+\theta_s)\)
Resistor \(v_R(t)=Ri_R(t),\;i_R(t)=\frac1Rv_R(t)=Gv_R(t)\)
\(i_R(t)=I_R\cos(\omega t+\theta_I)\)
\(v_R(t)=V_R\cos(\omega t+\theta_V)\)
\(\Rightarrow v_R=RI_R,\;\theta_I=\theta_V\)
impedance \(Z_R=\frac{V_R}{I_R}=\frac{V_{R\angle}\theta_V}{I_R\angle\theta_I}=R\)
admittance \(Y_Z=\frac1{Z_R}=\frac1R=G\;(s,\;siemens)\)
Capacitor \(i_c(t)=C\frac{dv_c(t)}{dt}\)
\(v_c(t)=V_c\cos(\omega t+\theta_V)\)
\(i_c(t)=I_c\cos(\omega t+\theta_I)\)
\({\widehat i}_c(s)=Cs{\widehat v}_c(s)\Rightarrow{\widehat i}_c(j\omega)=j\omega C{\widehat v}_c(j\omega)\)
\(V_c=\frac1{j\omega C}I_c\)
\(I_c\cos(\omega t+\theta_I)=-\omega CV_c\sin(\omega t+\theta_V)=-\omega CV_c\cos(\omega t+\theta_V-\frac{\mathrm\pi}2)\)
\(e^{-\frac{\mathrm\pi}2}=-j\)
\(I_c\angle\theta_I=-\omega CV_c\angle(\theta_V-\frac{\mathrm\pi}2)=-\omega CV_c(-j)\angle\theta_V\)
\(=j\omega CV_c\angle\theta_V\)
\(I_c=j\omega CV_c\)
\(Z_c=\frac{V_c}{I_c}=\frac1{j\omega C}\)
Inductor \(V_L(t)=L\frac{di_L(t)}{dt}\)
\(v_L(t)=V_L\cos(\omega t+\theta_V)\)
\(i_L(t)=I_L\cos(\omega t+\theta_I)\)
\(\Rightarrow\frac{di_L(t)}{dt}=-\omega I_L\sin(\omega t+\theta_I)=-\omega I_L\cos(\omega t+\theta_I-\frac{\mathrm\pi}2)\)
\(V_L\cos(\omega t+\theta_V)=-\omega LI_L\cos(\omega t+\theta_I-\frac{\mathrm\pi}2)\)
\(V_L\angle\theta_V=-\omega LI_L\angle(\theta_I-\frac{\mathrm\pi}2)=j\omega LI_L\angle\theta_I\)
\(V_L=j\omega LI_L=Z_LI_L\)
\(Z_L=j\omega L\)
Circuit Analysis
Equivalent impedance
resistive impedance
inductive impedance
capacitive impedance
Complex Power
\(Z=R+jX\)
Resistance \(R\)
reactance \(X\)
\(v(t)=V\cos\omega t\)
\(i(t)=I\cos(\omega t-\theta)\)
\(V=V\angle0^\circ,\;I=I\angle\theta^\circ\)
Instantaneous power
\(p(t)=v(t)i(t)=V\cos(\omega t)I\cos(\omega t-\theta)\)
\(=VI\cos(\omega t)\cos(\omega t-\theta)\)
\(\cos(\alpha\pm\beta)=\cos(\alpha)\cos(\beta)\mp\sin(\alpha)\sin(\beta)\)
\(\alpha=\omega t,\;\beta=\omega t-\theta\)
\(=\frac12VI\lbrack\cos\theta+\cos(2\omega t-\theta)\rbrack\)
\(=\frac12VI\cos\theta+\frac12VI\cos(2\omega t-\theta)\)
\(=\frac12VI\cos\theta+\frac12VI\lbrack\cos\theta\cos(2\omega t)+\sin\theta\sin(2\omega t)\rbrack\)
\(=\frac12VI\cos\theta\lbrack1+\cos(2\omega t)\rbrack+\frac12VI\sin\theta\sin(2\omega t)\)
\(=P\lbrack1+\cos(2\omega t)\rbrack+Q\sin(2\omega t)\)
\(=2P\cos^2\omega t+Q\sin(2\omega t)\)
\(P_{av}=\frac1T\int_Tp(t)dt\)
\(=\frac12VI\cos\theta=P\)
\(=\frac1T\int_T2P\cos^2\omega tdt\)
\(=\frac1T\cdot2P\cdot\frac T2=P\)
\(\omega\;=\frac{2\pi}T\)
\(\cos2\omega t=\cos^2\omega t-\sin^2\omega t=\cos^2\omega t-(1-\cos^2\omega t)\)
\(2\cos^2\omega t=\cos2\omega t+1\)
\(\cos^2\omega t=\frac12(\cos2\omega t+1)\)
\(\int_T\cos^2\omega t=\frac12\int_T(\cos2\omega t+1)=\frac T2\)
Root-Mean-Square
rms (effective) value
\(\widetilde V=\sqrt{\frac1T\int_Tv^2(t)dt}\)
\(=\sqrt{\frac1T\int_Tv^2\cos^2(wt)dt}\)
\(=\frac V{\sqrt2}\) in one period
\(\widetilde I=\sqrt{\frac1T\int_Ti^2(t)dt}\)
\(=\frac I{\sqrt2}\)
Define the effective phasors of \(v(t)=V\cos\omega t\) and \(i(t)=I\cos\omega t\) as
\(\widetilde V=\frac1{\sqrt2}V=\frac V{\sqrt2}\angle0^\circ=\widetilde V\angle0^\circ\)
\(\widetilde I=\frac1{\sqrt2}I=\frac I{\sqrt2}\angle-\theta=\widetilde I\angle-\theta\)
Define the complex power as
\(S=\widetilde V\widetilde I^\ast\)
\(=\widetilde V\widetilde I\angle\theta\)
\(=\frac V{\sqrt2}\frac I{\sqrt2}\angle\theta=\frac12VI\angle\theta\)
\(=\frac12VI\cdot e^{j\theta}\)
\(S=\frac12VI(\cos\theta+j\sin\theta)\)
\(=\frac12VI\cos\theta+j\frac12VI\sin\theta\)
\(=P+jQ\)
units:
\(P=\frac12VI\cos\theta,\;(W:watt)\)
\(Q=\frac12VI\sin\theta,\;(VAR:volt-ampere\;reac\tan ce)\)
\(S=\frac12VI,\;(VA:\;volt-ampere)\)
\(S=\widetilde V\widetilde I^\ast,\;\widetilde V=\widetilde IZ\)
\(=\widetilde V\widetilde I\angle\theta\)
\(=\widetilde V\widetilde I\cos\theta+j\widetilde V\widetilde I\sin\theta\)
\(S=\widetilde V\widetilde I^\ast=\widetilde I\widetilde I^\ast Z\)
\(=\widetilde I^2Z\)
\(=\widetilde I^2R+j\widetilde I^2X,\;Z=R+jX\)
\(S=\widetilde V\widetilde I^\ast=\widetilde{V(}\frac{\widetilde V}Z)^\ast=\frac{\widetilde V^2}{R-jX}\)
Conservation of Energy
\(\frac d{dt}e(t)=p(t)=0\)
\(p(t)=\sum_{k=1}^np_k(t)=\sum_{k=1}^nv_k(t)v_i(t)=0\)
\(p_k(t)=2P_k\cos^2\omega t+Q_k\sin2\omega t\)
\(\sum_{k=1}^np_k(t)=2(\sum_{k=1}^nP_k)\cos^2\omega t+(\sum_{k=1}^nQ_k)\sin2\omega t=0\)
\(\sum_{k=1}^nP_k=0,\;\sum_{k=1}^nQ_k=0\)
\(p(t)=0\Rightarrow\sum_{k=1}^nP_k=0,\;\sum_{k=1}^nQ_k=0,\;\sum_{k=1}^nS_k=0\)
\(S=\sum_{k=1}^nS_k=\sum_{k=1}^nP_k+j\sum_{k=1}^nQ_k=0\)
Maximum Power Transfer Theorem
\(\widetilde{V_T}=\widetilde{V_T}\angle0^\circ\)
\({Z_T}=R_T+jX_T\)
\(Z=R+jX\)
\(\widetilde V=\frac Z{Z_T+Z}\widetilde{V_T},\;\widetilde I=\frac1{Z_T+Z}\widetilde{V_T}\)
\(S=\widetilde V\widetilde I^\ast=\frac Z{(Z_T+Z){(Z_T+Z)}^\ast}\widetilde{V_T}^2=P+jQ\)
\(=\frac{R+jX}{{(R_T+R)}^2+{(X_T+X)}^2}\widetilde{V_T}^2=P+jQ\)
\(P=\frac R{{(R_T+R)}^2+{(X_T+X)}^2}\widetilde{V_T}^2\)
\(P_{max}:\frac{\partial P}{\partial R}=0,\;\frac{\partial P}{\partial X}=0\)
\(\frac{\partial P}{\partial R}=\frac{{(R_T+R)}^2+{(X_T+X)}^2-R\lbrack2(R_T+R)\rbrack}{{\lbrack{(R_T+R)}^2+{(X_T+X)}^2\rbrack}^2}\widetilde{V_T}^2=0\)
\(\Rightarrow R_T^2-R^2+{(X_T+X)}^2=0\)
\(\frac{\partial P}{\partial X}=\frac{R(X_T+X)}{{\lbrack{(R_T+R)}^2+{(X_T+X)}^2\rbrack}^2}\widetilde{V_T}^2\)
\(\frac{\partial P}{\partial X}=0\Rightarrow{X=-X_T}\)
\(\Rightarrow{R=R_T}\)
\(\therefore Z=R_T-X_T\)
\(P_{max}=Re(\widetilde V\widetilde I^\ast)=\widetilde I\widetilde I^\ast R_T\)
\(=\widetilde I^2R_T={(\frac{\widetilde{V_T}}{2R_T})}^2R_T=\frac{\widetilde{V_T}^2}{4R_T}\)
\(P_T=Re(\widetilde{V_T}\widetilde I^\ast)=\frac{\widetilde{V_T}^2}{2R_T}\)
\(P_{max}=\frac12P_T\)
\(Z=R_T-X_T=Z_T^\ast,\;matched\;impedance\)
\(if\;Z=R,\;then\)
\(\frac{\partial P}{\partial R}=\frac{{R_T^2-R^2+}X_T^2}{{\lbrack{(R_T+R)}^2+X_T^2\rbrack}^2}\widetilde{V_T}^2=0\)
\(\Rightarrow R=\sqrt{R_T^2+X_T^2},\;R=\vert Z_T\vert\)
\(P_{max}=\frac R{R+R_T}P_T\)
\(P_L = I^2 R_L\)
\(I = \frac{V_{Th}}{R_{Th} + R_L}\)
\(P_L = \lgroup \frac{V_{Th}}{(R_{Th} + R_L)} \rgroup ^2 R_L\)
\(\Rightarrow P_L = {V_{Th}}^2 \lbrace \frac{R_L}{(R_{Th} + R_L)^2} \rbrace\)
\(\frac{dP_L}{dR_L} = {V_{Th}}^2 \lbrace \frac{(R_{Th} + R_L)^2 \times 1 - R_L \times 2(R_{Th} + R_L)}{(R_{Th} + R_L)^4} \rbrace = 0\)
\(\Rightarrow (R_{Th} + R_L)^2 -2R_L(R_{Th} + R_L) = 0\)
\(\Rightarrow (R_{Th} + R_L)(R_{Th} + R_L - 2R_L) = 0\)
\(\Rightarrow (R_{Th} - R_L) = 0\)
\(\Rightarrow R_{Th} = R_L\:or\:R_L = R_{Th}\)
The value of Maximum Power Transfer
\(P_{L, Max} = {V_{Th}}^2 \lbrace \frac{R_{Th}}{(R_{Th} + R_{Th})^2} \rbrace\)
\(P_{L, Max} = {V_{Th}}^2 \lbrace \frac{R_{Th}}{4 {R_{Th}}^2} \rbrace\)
\(\Rightarrow P_{L, Max} = \frac{{V_{Th}}^2}{4 R_{Th}}\)
\(\Rightarrow P_{L, Max} = \frac{{V_{Th}}^2}{4 R_{L}}, \: since \: R_{L} = R_{Th}\)
\(P_{L, Max} = \frac{{V_{Th}}^2}{4R_{L}} = \frac{{V_{Th}}^2}{4R_{Th}}\)
Efficiency of Maximum Power Transfer
\(\eta_{Max} = \frac{P_{L, Max}}{P_S}\)
\(P_{L, Max}\) is the maximum amount of power transferred to the load.
\(P_S\) is the amount of power generated by the source.
\(P_S = I^2 R_{Th} + I^2 R_L\)
\(\Rightarrow P_S = 2 I^2 R_{Th},\:since\:R_{L} = R_{Th}\)
Substitute \(I = \frac{V_{Th}}{2 R_{Th}}\)
\(P_S = 2\lgroup \frac{V_{Th}}{2 R_{Th}} \rgroup ^2 R_{Th}\)
\(\Rightarrow P_S = 2\lgroup \frac{{V_{Th}}^2}{4 {R_{Th}}^2} \rgroup R_{Th}\)
\(\Rightarrow P_S = \frac{{V_{Th}}^2}{2 R_{Th}}\)
\(\eta_{Max} = \frac{\lgroup \frac{{V_{Th}}^2}{4R_{Th}} \rgroup}{\lgroup \frac{{V_{Th}}^2}{2R_{Th}}\rgroup}\)
\(\Rightarrow \eta_{Max} = \frac{1}{2}\)
\(\% \eta_{Max} = \eta_{Max} \times 100\%\)
\(\Rightarrow \% \eta_{Max} = \lgroup \frac{1}{2} \rgroup \times 100\%\)
\(\Rightarrow \% \eta_{Max} = 50\%\)
\(1.\;P_{max}:\;Z=Z^\ast=R_T-jX_T\)
\(2.\;P_{max}:R=\sqrt{R_T^2+X_T^2}\)
Power factor correction
\(\widetilde{V_S}=\widetilde{V_S}\angle0^\circ\)
\(\widetilde I=\frac{\widetilde{V_S}}{r+Z}=\frac{\widetilde{V_S}}{r+R+jX}\)
\(S_Z=\widetilde{V_Z}\widetilde I^\ast=\widetilde IZ\widetilde I^\ast=\frac{\widetilde{V_S}^2}{{(r+R)}^2+X^2}(R+jX)\)
\(=P+jQ,\;P=\frac{R\widetilde{V_S}^2}{{(r+R)}^2+X^2},\;Q=\frac{X\widetilde{V_S}^2}{{(r+R)}^2+X^2}\)
\(=S_Z\cos\theta+jS_Z\sin\theta\)
Power factor=pf=\(=\frac{S_Z\cos\theta}{S_Z}=\cos\theta\)
\(=\frac P{\vert S_Z\vert}=\frac P{\sqrt{P^2+Q^2}}\)
\(P_r=r\widetilde I\widetilde I^\ast=r\widetilde I^2\)
\(=\frac{r\widetilde V^2}{{(r+R)}^2+X^2}\)
\(Z_{new}=Z\parallel\frac1{j\omega C}\)
\(Y_{new}=\frac1{Z_{new}}=\frac R{R^2+X^2}-\frac{jX}{R^2+X^2}+j\omega C\)
\(\omega C=\frac X{R^2+X^2}\)
\(\therefore C=\frac X{\omega(R^2+X^2)}\)
\(Y_{new}=\frac R{R^2+X^2}\Rightarrow Z_{new}=R+\frac{X^2}R\)
\(P_r=\frac{r\widetilde{V_S}^2}{{(r+R)}^2+2(1+{\frac rR})X^2+{\frac{X^4}{R^2}}}<\frac{r\widetilde{V_S}^2}{{(r+R)}^2+X^2}\)
