Sinusoidal steady-state analysis

Phasor analysis is a technique to find the steady-state response when the system input is a sinusoid.

It is a powerful technique with which to find the steady-state portion of the complete response. Phasor analysis does not find the transient response.

Phasors, Phase Shift, and Phasor Algebra

\(V_A=5\;volts\;\angle0^o\)

\(V_B=5\;volts\;\angle60^o\)

\(V_{BA}=5\;volts\;\angle120^o\)

\(V_{BA}=V_B-V_A=(5\;volts\;\angle60^o)-(5\;volts\;\angle0^o)=5\;volts\;\angle120^o\)

\(V_o(\omega)=V_s(\omega)\times T(\omega)\)

vector = vector * vector

\(a+bj=\sqrt{a^2+b^2}(\cos\theta+j\sin\theta)=\sqrt{a^2+b^2}e^{j\theta}\;unit\;vector\)

\(\theta=\tan^{-1}\frac ba\)

[ex] \(2=2e^{j0^\circ},\;j=1e^{j90^\circ},\;-1=1e^{j180^\circ}\)

\(1+1j=\sqrt2e^{j\tan^{-1}\frac11}\;=\sqrt2e^{j45^\circ}\)

Define \(e^{j\theta_1}\cdot e^{j\theta_2}=e^{j(\theta_1+\theta_2)}\)

\(\frac{e^{j\theta_1}}{e^{j\theta_2}}=e^{j(\theta_1-\theta_2)}\)

\(j\cdot j=e^{j90^\circ}\cdot e^{j90^\circ}=e^{j180^\circ}=-1\)

Prove: \(left=(\cos\theta_1+j\sin\theta_1)(\cos\theta_2+j\sin\theta_2)\)

\(=\cos\theta_1\cos\theta_2+(j\cdot j)\sin\theta_1\sin\theta_2+j(\sin\theta_1\cos\theta_2+\sin\theta_2\cos\theta_1)\)

by trigonometry

\(right=\cos(\theta_1+\theta_2)+j\sin(\theta_1+\theta_2)\)

§ \(\frac d{dt}(e^{j\omega t})=\frac d{dt}(\cos\omega t+j\sin\omega t)=\omega(-1)\sin\omega t+j\omega\cos\omega t\)

\(=\omega(j\cdot j)\sin\omega t+j\omega\cos\omega t\)

\(=j\omega(\cos\omega t+j\sin\omega t)=j\omega e^{j\omega t}\)

§ RC ckt

Phasors are essential tool in circuit analysis.

The generator we originally had in the circuit is now just the real part of the phasor expression shown in Equation below,

\(v_s(t)=V\cos(\omega t+{\mathrm\Theta}_V)=\mathfrak R\{Vcos(\omega t+\mathrm\Theta)+jVsin(\omega t+\mathrm\Theta)\}\)

\(=\mathfrak R\{Ve^{j(\omega t+\Theta)}\}=\mathfrak R\{Ve^{j\Theta}e^{j\omega t}\}\)

We can now use the analysis from the previous section to replace the time-domain quantities in equation

\(v_s(t)=v_R(t)+v_C(t)\)

\(\mathfrak R\{V_se^{j\Theta_V}e^{j\omega t}\}=\mathfrak R\{R\,Ie^{j\Theta_I}e^{j\omega t}\}+\mathfrak R\{\frac1{j\omega C}Ie^{j\Theta_I}e^{j\omega t}\}\)

A common term in the previous equation is \(e^{j\omega t}\), and we can now drop \(\mathfrak R\),

as long as we later remember to take only the real part of the expression for the voltage and current phasors to get the time domain expression.

\(V_se^{j\Theta_V}=RIe^{j\Theta_I}+\frac1{j\omega C}Ie^{j\Theta_I}\)

\({\widetilde V}_s=R\widetilde I+\frac{\widetilde I}{j\omega C}\)

Since this is a linear equation, we can easily solve it:

\(\widetilde I=\frac{{\widetilde V}_s}{R+\frac1{j\omega C}}\)

Converting the phasor back to the time domain

In general, if the phasor is \(\widetilde V=\vert V\vert e^{j\Theta_V}\), to find the time-domain signal, we first multiply the phasor with \(e^{j \omega t}\) term, and then take the real part of it.

\(v(t)=\mathfrak R\{\widetilde Ve^{j\omega t}\}\)

\(v(t)=\mathfrak R\{\vert\widetilde V\vert e^{j(\omega t+\Theta_V)}\}\)

\(v(t)=\mathfrak R\{\vert\widetilde V\vert cos(\omega t+\Theta_V)+j\vert\widetilde V\vert sin(\omega t+\Theta_V)\}\)

\(v(t)=Vcos(\omega t+\Theta_V)\)

\(v_s=RC\frac{dv_o}{dt}+v_o\)

\(\Rightarrow V_s\cancel{e^{j\omega t}}=RCV_o\cdot j\omega\cdot\cancel{e^{j\omega t}}+V_o\cancel{e^{j\omega t}}\)

\(V_s=V_o(1+j\omega RC)\)

\(\frac{V_o}{V_s}=T(\omega)=\frac1{1+j\omega RC}\)

\(\tau=RC\Rightarrow\frac1\tau=\omega_o=\frac1{RC}\)

\(T(\omega)=\frac1{1+j\omega RC}=\frac1{1+{\frac{j\omega}{\omega_o}}}\)

\(=\frac{1e^{j0^\circ}}{\sqrt{1+{({\frac\omega{\omega_o}})}^2}e^{j\tan^{-1}(\frac\omega{\omega_o})}}\)

\(\frac{V_o}{V_s}=\frac1{\sqrt{1+{(\frac\omega{\omega_o})}^2}}e^{j\lbrack-\tan^{-1}(\frac\omega{\omega_o})\rbrack}\)

Define \(dB\)

\(20\log A\)

\(20\log(a\cdot b)=20\log(a)+20\log(b)\)

\(20\log(\frac ab)=20\log(a)-20\log(b)\)

\(\begin{array}{cc}A&dB\\10\times10\times10&20+20+20\\10\times10&20+20\\10&20\\1&0\\\frac1{10}&0-20\\\frac1{10}\times\frac1{10}&0-(20+20)\\\frac1{10}\times\frac1{10}\times\frac1{10}&0-(20+20+20)\end{array}\)

Bode plot

\(\frac{V_o}{V_s}=\left|\frac{V_o}{V_s}\right|e^{j\angle\frac{V_o}{V_s}}\)

\(\left|\frac{V_o}{V_s}\right|=\frac{1\rightarrow0dB}{\sqrt{1+{(\frac\omega{\omega_o})}^2}}\)

\(20\log(\sqrt{1+{(\frac\omega{\omega_o})}^2})=\frac{20}2\log\lbrack1+{(\frac\omega{\omega_o})}^2\rbrack\)

\(\simeq20\log(\frac\omega{\omega_o})\)

\(\simeq20\log(\frac\omega{\omega_o})\;when\;\omega\ll\omega_o\;or\;\omega\gg\omega_o\)

\(\pm20dB/decade\)

\(0.1\omega_o,\;\omega_o,\;10\omega_o\)

\(-3dB\;at\;\omega_o,\;20\log\frac1{\sqrt2}=-3dB\\0.1\omega_o,\;\omega_o,\;10\omega_o\)

\(\angle\frac{V_o}{V_s}=-\tan^{-1}(\frac\omega{\omega_o})\)

\(i=C\frac{dv}{dt}\Rightarrow I\cancel{e^{j\omega t}}=CVj\omega\cancel{e^{j\omega t}}\)

\(\frac VI=\frac1{j\omega C}=\frac1{\omega C}e^{j(-90^\circ)}\)

equivalent circuit

\(V_oe^{j\omega t}=V_se^{j\omega t}\cdot\frac{\frac1{j\omega C}}{R+\frac1{j\omega C}}\)

\(\frac{V_o}{V_s}=\frac1{1+j(\frac\omega{\omega_o})}=\frac1{\sqrt{1+{(\frac\omega{\omega_o})}^2}}e^{j(-\tan^{-1}\frac\omega{\omega_o})}\)

\(V_o=\frac1{\sqrt{1+{(\frac\omega{\omega_o})}^2}}\sin\lbrack\omega t-\tan^{-1}(\frac\omega{\omega_o})\rbrack\)

[Ex] allow the voice to pass \(20\sim20k\Rightarrow\omega=10^2\sim10^5\) - low pass filter

\(require\;10^5\leq0.1\omega_o\Rightarrow10^6\leq\omega_o\Rightarrow10^6\leq\frac1{RC}\)

\(10^6\leq\frac1{10^4C}\Rightarrow C\leq10^{-10}\)

\(\omega_o=\frac1{RC}=\frac1{10^4\cdot10^{-10}}=10^6\)

\(\left\{\begin{array}{l}for\;v_s=(1V)\sin10^6t\;\Rightarrow\;v_o=1\cdot\frac1{\sqrt2}\sin(10^6t-45^\circ)\\v_s=(1V)\sin10^7t\;\Rightarrow\;v_o=1\cdot\frac1{10}\sin(10^7t-90^\circ)\\v_s=(1V)\sin10^5t\;\Rightarrow\;v_o=1\cdot\sin(10^5t)\end{array}\right.\)

[Ex] allow the voice to pass \(20\sim20k\Rightarrow\omega=10^2\sim10^5\) - high pass filter

\(require\;10^\cancel2\geq\cancel{10}\omega_o\;\Rightarrow\;10\geq\frac1{10^4C}\;\Rightarrow\;C\geq10^{-5}F\)

\(let\;C=10^{-5}\;\Rightarrow\;\omega_o=\frac1{10^4\cdot10^{-5}}=10\)

\(\begin{array}{l}for\;v_s=1V\cdot\sin10t\;\Rightarrow\;v_o=1\cdot\frac1{\sqrt2}\sin(10t+45^\circ)\\v_s=1V\cdot\sin1t\;\Rightarrow\;v_o=1\cdot\frac1{10}\sin(1t+90^\circ)\\v_s=1V\cdot\sin10^2t\;\Rightarrow\;v_o=1\cdot\sin(10^2t)\end{array}\)

\(\left\{\begin{array}{l}C\;allow\;ac\;(voice)\;to\;pass\\C\;can\;block\;DC\;voltage\end{array}\right.\)

Laser point

\(\frac1{R_LC}\equiv\omega_o\)

\(\frac{I_c}{I_N}=\frac{R_L}{R_L+{\frac1{j\omega C}}}=\frac{j\omega R_LC}{1+j\omega R_LC}=\frac{\frac{j\omega}{\omega_o}}{1+{\frac{j\omega}{\omega_o}}}\)

\(=\frac{(\frac\omega{\omega_o})e^{j90^\circ}}{\sqrt{1+{({\frac\omega{\omega_o}})}^2}e^{j(\tan^{-1}{\frac\omega{\omega_o}})}}\)

\(\left|\frac{I_c}{I_N}\right|=\frac{(\frac\omega{\omega_o})}{\sqrt{1+{(\frac\omega{\omega_o})}^2}}\)

\(require\;\omega\geq10\omega_o\)

transfer \(v_I\) to \(i_o\)

\(i_o=k\;(mA/V^2){(V_I-V_{th})}^2\)

\(Q\;slope={\left.\frac{\partial i_o}{\partial V_I}\right|}_Q=2k\;(mA/V)(V_I-V_{th})\)

§ separate DC & ac

\(v=f(DC)+g(v_s)\)

Operational Amplifiers

OP is usually connected into a negative feedback fashion

voltage follower

\(A_d(v_+-v_-)=v_-\)

\(\Rightarrow v_-(1+A_d)=A_dv_+\)

\(\Rightarrow v_-=\frac{A_d}{1+A_d}v_+\simeq v_+\)

\(A_d\rightarrow\infty,\;virtual\;short\;ckt\)

\((v_+-v_-)=\frac{v_-}{A_d}\simeq0\;\Rightarrow\;v_-=v_+\)

§\(find\;R_s={\left.\frac{v_x}{i_x}\right|}_{v_i=0}\)

\(i_x=\frac{v_x-0}0\Rightarrow\frac{v_x}{i_x}=0\)

\(R_o={\left.\frac{v_x}{i_x}\right|}_{v_i=0}=0\)

\(v_o=v_{open}\frac{R_L}{R_o+R_L}=v_{open}\frac{\cancel{R_L}}{0+\cancel{R_L}}=v_i,\;indep.\;of\;R_L\)

\(R_i=\frac{v_i}0=\infty\)

\(v_i=v_s\frac{R_i}{R_s+R_i}=v_s\frac\infty{R_s+\infty}=v_s\)

\(indep.\;of\;R_s,\;R_L\)

Nonideal voltage source becomes ideal voltage source \(ideal\;v-v\;amp\)

\(v_-=v_{open}(\frac{R_1}{R_1+R_2})=v_{open}\cdot\beta\)

\(v_-=v_{open}\cdot\beta\Rightarrow v_{open}=\frac1\beta v_-\)

\(A_d(v_+-v_-)=\frac1\beta v_-\)

\(\beta A_d(v_+-v_-)=v_-\)

\(v_-(1+\beta A_d)=\beta A_dv_+\)

\(\Rightarrow v_-=\frac{\beta A_d}{1+\beta A_d}v_+\simeq v_+,\;if\;\beta A_d\gg1,\;(virtual\;short\;ckt)\)

\(v_{open}=v_i+\frac{v_i}{R_1}R_2=(1+\frac{R_2}{R_1})v_i\)

\(find\;R_o={\left.\frac{v_x}{i_x}\right|}_{v_i=0}\)

\(i_x=\frac{v_x-0}0\Rightarrow\frac{v_x}{i_x}=0\Rightarrow R_o=0\)

\(ideal\;v-v\;amp,\;v_i=v_s\)

\(v_{open}=(1+\frac{R_2}{R_1})v_i=(1+\frac{R_2}{R_1})v_s\)

\(v_{o=}(1+\frac{R_2}{R_1})v_s,\;indep.\;of\;R_s,\;R_L\)

\(with\;a\;gain=1+\frac{R_2}{R_1}\)

Adder

\(v_o=v_+(1+\frac{R_2}{R_1})\)

\(v_+=Av_1+Bv_2\)

\(=v_1\frac{R_4}{(R_3+R_4)}+v_2\frac{R_3}{(R_3+R_4)}\) superposition principle

\(v_o=\frac{(R_1+R_2)}{R_1}\frac1{(R_3+R_4)}(R_4v_1+R_3v_2)\)

\(let\;R_1+R_2=R_3+R_4,\)

\(v_o=\frac{R_4}{R_1}v_1+\frac{R_3}{R_1}v_2\)

Averager

\(v_+=\frac12(v_1+v_2)\)

\(v_o=\frac12(v_1+v_2)\)

Inverter

\(\omega\ll\omega_B\Rightarrow\;virtual\;short\;ckt\)

\(v_o=0-\frac{v_i}{R_1}R_2=-\frac{R_2}{R_1}v_i\)

\(v_o=-v_i\)

Subtractor

\(v_o=\underbrace{Av_1}_{let\;v_2=0}+\underbrace{Bv_2}_{let\;v_1=0}=v_1\frac{R_2}{R_1+R_2}(1+\frac{R_4}{R_3})-\frac{R_4}{R_3}v_2\)

\(superposition\Rightarrow(1)\;let\;v_2=0,\;(2)\;let\;v_1=0\)

\(let\;R_1+R_2=R_3+R_4\)

\(v_o=\frac{R_2}{R_3}v_1-\frac{R_4}{R_3}v_2\)

\(if\;R_2=R_4,\;(R_1=R_3)\)

\(\Rightarrow v_o=\frac{R_4}{R_3}(v_1-v_2)\)

differential amp

\(R_1=R_3=R_4=R,\;R_2=R+\triangle R\)

\(v_o=v_1\frac{R+\triangle R}{\cancel2R+\cancel{\triangle R}}\cdot\cancel2-v_2\)

\(=v_1(1+\frac{\triangle R}R)-v_2=(v_1-v_2)+\frac{\triangle R}Rv_1\)

SI units and Prefixes

Base units

MKSA

Length m(meter)

Mass kg

Time s(second)

Current a(ampere)

Derived units

Volt V

Ohm Ω

Henry H

Mathematical Modeling

\(\overset.{V_c}(t)+\frac1{RC}V_c(t)=\frac1{RC}V(t)\)

unkown \(\leftarrow\vert\rightarrow\) given

Prefixes

Electric Source and Fields

\(\overset\rightharpoonup F=m\overrightarrow g+q\overrightarrow E+q\overrightarrow V\times\overrightarrow B\)

\(e^-=1.6x10^{-19}\;C\)

\(I=\frac{\triangle q}{\triangle t},\;(C/s)(A)\)

Instantaneous current

\(i(t)=\lim_{\triangle t\rightarrow0}\frac{\triangle q}{\triangle t}=\frac{dq(t)}{dt}\)

\(q(t)=\int_{t_0}^ti(\tau)d\tau+q(t_0)=\int_{-\infty}^ti(\tau)d\tau\)

\(q(-\infty)=0\)

\(q(t_0)=\int_{-\infty}^{t_0}i(\tau)d\tau\) initial value at initial time

\(i(t)=\frac{dq(t)}{dt}=\frac d{dt}\int_{t_0}^ti(\tau)d\tau=i(t),\;running\;integral\)

average current

\(I_{avg}=\frac1{t_2-t_1}\int_{t_1}^{t_2}i(\tau)d\tau\)

Voltage and Power

electric Potential \(V\)

electric Potential Energy \(qV\)

electric Work \(w=q(V_1-V_2)=qv\)

Instantaneous Power \(p(t)=\frac{dw(t)}{dt}=\frac{dw(t)}{dq(t)}\frac{dq(t)}{di}=v(t)i(t),\;\lbrack W.\;J/s\rbrack\)

\(\Rightarrow w(t)=w(t_0)+\int_{t_0}^tp(\tau)d\tau\)

Average Power in \(t\in\lbrack t_2-t_1\rbrack\)

\(P_{avg}=\frac{w(t_2)-w(t_1)}{t_2-t_1}=\frac1{t_2-t_1}\int_{t_1}^{t_2}p(\tau)d\tau\)

Kirchholff's Laws

KVL: Kirchhoff's voltage law

\(Loop:\sum_{k=1}^nv_k=0\)

KCL: Kirchhoff's current law

\(Node :\sum_{k=1}^mi_k=0\)

Conservation of Energy

\[\sum_{k=1}^nw_k(t)=\sum_{k=1}^n\int_{-\infty}^tp_k(\tau)d\tau=\sum_{k=1}^n\int_{-\infty}^tv_k(\tau)i_k(\tau)d\tau=0\]

\[\sum_{k=1}^nw_k(t)=\int_{-\infty}^t\{\sum_{k=1}^nv_k(\tau)i_k(\tau)\}d\tau=\int_{-\infty}^t\{\sum_{k=1}^np_k(\tau)\}d\tau=0\]

Conservation of total Power

\[\sum_{k=1}^np_k(\tau)=\sum_{k=1}^nv_k(t)i_k(t)=0\]

Power Dissipated in Resistor

\(p(t)=i(t)v(t)=Ri^2(t)=\frac{v^2(t)}R\)

\(p(t)=i(t)v(t)=Gv^2(t)=\frac{i^2(t)}G\)

Conductance

\(G_k=\frac1{R_k}\)

\(Resistors\;in\;parallet\Rightarrow equivalent\;conductor\;G_p=\sum_{k=1}^nG_k\)

\(i=i_1+i_2+...+i_n=(\sum_{k=1}^nG_k)V=G_pV\)

\(i_k=G_kV=\frac{G_k}{G_p}i\)

Capacitance

\(q=CV_c\)

\(C=\varepsilon\frac Ad\)

\(g(t)=CV_c(t)\)

\(\frac{dg(t)}{dt}=C\frac{dV_c(t)}{dt}\)

\(i_c(t)=C\frac{dV_c(t)}{dt}\)

\(V_c(t)=\frac1C\int_{-\infty}^ti_c(\tau)d\tau=\frac1C\int_{t_0}^ti_c(\tau)d\tau+V_c(t_0)\)

continuity in voltage

\(V_c(t^-)=V_c(t)=V_c(t^+)\)

Power and energy stored in a capacitor

\(P_c(t)=i_c(t)v_c(t)\)

\(=C\frac{dv_c(t)}{dt}v_c(t)\)

\(=\frac12C\frac{dv_c^2(t)}{dt}\)

\(=\frac d{dt}\lbrack\frac12Cv_c^2(t)\rbrack\;related\;to\;energy\;w(t)\)

\(p(t)=\frac{\partial W(t)}{\partial t}\)

\(Power\;p(t)\;Watt,\;Energy\;w(t)\;Joule\)

Inductance

\(\phi=Li\)

\(\frac{d\phi(t)}{dt}=L\frac{di_L(t)}{dt}\)

\(V_L(t)=L\frac{di_L(t)}{dt}\)

Power and energy stored in an inductor

\(p(t)=i_L(t)v_L(t)\)

\(=i_L(t)L\frac{di_L(t)}{dt}\)

\(=\frac d{dt}\lbrack\frac12Li_L^2(t)\rbrack\)

\(W_L(t)=\frac12Li_L^2(t)\)

Oscillator

Node-Voltage Analysis

Independent current sources

KCL - Kirchhoff's Current Law

Total current outward the node is zero.

\(A_{3x3}V_{3x1}=G_{3x3}I_{3x1}\)

\(V=A^{-1}GI\)

with dependent current source

\(A_{3x3}V_{3x1}=G_{3x2}I_{2x1}\)

\(V=A^{-1}GI\;if\;\left|A\right|\neq0\)

voltage source

KVL - Kirchhoff's Voltage Law

super node

Mesh-Current Analysis (planar circuit)

Independent voltage sources

KVL - Kirchhoff's Voltage Law

Mesh Current Analysis - planar circuit

Independent Voltage Source

Current Source

Supernode and Supermesh

Principle of Superposition

Voltage source in series with \(R_s\). Current source in parallel with \(R_s\).

\(V_s(t)=V_{open}\)

\(R_s={\left.\frac{v_x}{i_x}\right|}_{V_s=0}\)

\(i_{sh}=i_s\)

\(R_s={\left.\frac{v_x}{i_x}\right|}_{i_s=0}\)

Thevenin equivalent circuit

Norton equivalent circuit

\(i_N=i+\frac V{R_N}\)

to determine the \(i_N=i_{sc}\)

without dependent sources,

\(V_{sc}\;short, I_{sc}\;open\)

with dependent sources,

\(V_{ext}\;to\;get\;I_{ext}\)

Maximum Power Transfer Theorem

\(V_L=\frac{R_L}{R_L+r_s}V_s\)

\(i_L=\frac{V_s}{R_L+r_s}\)

\(P_L=V_Li_L=\frac{R_LV_s^2}{{(R_L+r_s)}^2}\)

max power \(\frac{dP_L}{dR_L}=0,\;\frac{d^2P_L}{dR_L^2}＜0\)

\(\frac{dP_L}{dR_L}=\frac{V_s^2{(R_L+r_s)}^2-R_LV_s^2\cdot2(R_L+r_s)}{{(R_L+r_s)}^4}=0\)

\(\Rightarrow{(R_L+r_s)}^2-2R_L(R_L+r_s)=(R_L+r_s)(r_s-R_L)=0\)

\(\therefore R_L=r_s\) matched

Power provided by the source

\(P_s=V_si_L=V_s\cdot\frac{V_s}{R_L+r_s}=\frac{V_s^2}{R_L+r_s}\)

\(P_L=i_LV_L=\frac{V_s}{R_L+r_s}\cdot\frac{R_LV_s}{R_L+r_s}=\frac{R_LV_s^2}{{(R_L+r_s)}^2}\)

\({\left.{P_L}_{max}\right|}_{R_L=r_s}=\frac{r_sV_s^2}{{4r_s}^2}=\frac{V_s^2}{4r_s}\)

\({\left.P_s\right|}_{R_L=r_s}=\frac{V_s^2}{2r_s}=2{P_L}_{max}\)

\(R_{ab}=R_a+R_b=R_3\parallel(R_1+R_2)\)

\(\therefore R_a+R_b=\frac{R_3(R_1+R_2)}{R_1+R_2+R_3}\)

\(R_b+R_c=\frac{R_1(R_2+R_3)}{R_1+R_2+R_3}\)

\(R_a+R_c=\frac{R_2(R_1+R_3)}{R_1+R_2+R_3}\)

\(R_a+R_b+R_c=\frac{R_1R_2+R_2R_3+R_1R_3}{R_1+R_2+R_3}\)

\(R_a=\frac{R_2R_3}{R_1+R_2+R_3}\)

\(R_b=\frac{R_1R_3}{R_1+R_2+R_3}\)

\(R_c=\frac{R_1R_2}{R_1+R_2+R_3}\)

\(R_1R_a+R_2R_b+R_3R_c=\frac{R_1R_2R_3}{R_1+R_2+R_3}\)

\(R_aR_b=\frac{R_1R_2R_3\cdot R_3}{{(R_1+R_2+R_3)}^2}\)

\(R_bR_c=\frac{R_1R_2R_3\cdot R_1}{{(R_1+R_2+R_3)}^2}\)

\(R_aR_c=\frac{R_1R_2R_3\cdot R_2}{{(R_1+R_2+R_3)}^2}\)

\(R_aR_b+R_bR_c+R_aR_c=\frac{R_1R_2R_3\cdot(R_1+R_2+R_3)}{{(R_1+R_2+R_3)}^2}=\frac{R_1R_2R_3}{(R_1+R_2+R_3)}\)

\(R_1R_a+R_2R_b+R_3R_c=\frac{R_1R_2R_3}{R_1+R_2+R_3}=R_aR_b+R_bR_c+R_aR_c\)

RLC - Passive sign convention (PSC)

Active sign convention (ASC) Passive sign convention (PSC)

component equation

\(i_c(t)=C\frac{dv_c(t)}{dt}\;(Q=CV),\;v_c(t)=\frac1C\int_{t_0}^t\;i_c(\tau)d\tau+v_c(t_0)\)

\(v_L(t)=L\frac{di_L(t)}{dt}\;(\Phi=LI),\;i_L(t)=\frac1L\int_{t_0}^tv_L(\tau)d\tau+v_L(t_0)\)

Laplace Transform

\(f(t)\Rightarrow\widehat f(s)=\int_0^\infty f(t)e^{-st}dt\)

\(ℒ\{f(t)\}=\int_{0^-}^\infty f(t)e^{-st}st\) used for singularity function \(\delta(t)\)

Ideal impulse function

\(\delta(t)=0,\;t\neq0\)

\(\int_{-\infty}^\infty\delta(t)dt=1\)

Laplace tranform of \(f'(t),\;t>0\)

\(ℒ\{f'(t)\}=\int_0^\infty f'(t)e^{-st}dt=\left.\int e^{-st}df(t)\right|_{t=0}^\infty\)

\(=\left.\int e^{-st}f(t)\right|_{t=0}^\infty-\left.\int f(t)de^{-st}\right|_{t=0}^\infty\)

\(=e^{-s\infty}f(\infty)-f(0)+s\int_0^\infty f(t)e^{-st}dt\)

\(=s\widehat f(s)-f(0)\)

Laplace tranform of \(\int_0^tf(\tau)d\tau\)

\(let\;g(t)=\int_0^tf(\tau)d\tau\)

\(then\;g'(t)=f(t)\)

\(and\;g(0)=0\)

\(ℒ\{g'(t)\}=ℒ\{f(t)\}\)

\(ℒ\{g'(t)\}=sℒ\{g(t)\}-g(0)\)

\(=sℒ\{g(t)\}\)

\(\therefore sℒ\{g(t)\}=ℒ\{f(t)\}\)

\(s\widehat g(s)=\widehat f(s)\)

\(\widehat g(s)=\frac1s\widehat f(s)\)

\(ℒ\{\int_0^tf(\tau)d\tau\}=\frac1s\widehat f(s)\)

integral operator \(\frac1s\)

\(ℒ\{e^{-at}f(t)\}\)

\(ℒ\{e^{-at}f(t)\}=\int_0^\infty e^{-at}f(t)e^{-st}dt\)

\(=\int_0^\infty f(t)e^{-(s+a)t}dt\)

\(=\widehat f(s+a)\)

\(ℒ\{1\}=\int_0^\infty1\cdot e^{-st}dt\)

\(=-\frac1s\left.e^{-st}\right|_0^\infty\)

\(=-\frac1s(e^{-s\infty}-1),\;(R_e(s)>0)\)

\(=\frac1s\)

\(f(t)=1,\;t\geq0\)

\(ℒ\{t\}=\int_0^\infty te^{-st}dt\)

\(=-\frac1s\int_0^\infty tde^{-st}\)

\(=-\frac1s(\left.t\cdot e^{-st}\right|_0^\infty-\int_0^\infty e^{-st}dt)\;(R_e(s)>0)\)

\(=\frac1{s^2}\)

\(ℒ\{1\}=\frac1s\)

\(ℒ\{e^{-at}f(t)\}=\widehat f(s+a)\)

\(ℒ\{e^{-at}\}=ℒ\{1\cdot e^{-at}\}=\frac1{s+a}\)

\(ℒ\{e^{-j\omega t}\}=\frac1{s+j\omega}\)

\(e^{-j\omega t}=\cos\omega t-j\sin\omega t\)

\(ℒ\{\cos\omega t\}-jℒ\{\sin\omega t\}\)

\(=\frac{s-j\omega}{s^2+\omega^2}\)

\(\therefore ℒ\{\cos\omega t\}=\frac s{s^2+\omega^2}\)

\(ℒ\{\sin\omega t\}=\frac\omega{s^2+\omega^2}\)

\(ℒ\{te^{-at}\}=\frac1{{(s+a)}^2}\)

\(ℒ\{e^{-at}\cos\omega t\}=\frac{(s+a)}{{(s+a)}^2+\omega^2}\)

\(ℒ\{e^{-at}\sin\omega t\}=\frac\omega{{(s+a)}^2+\omega^2}\)

\(\frac d{ds}F(s)=\frac d{ds}\int_0^\infty e^{-st}f(t)dt\)

\(=\int_0^\infty\frac\partial{\partial s}\lbrack e^{-st}f(t)\rbrack dt\)

\(=-s\int_0^\infty e^{-st}f(t)dt\)

\(=-ℒ\{t\cdot f(t)\}\)

\(\therefore ℒ\{tf(t)\}=-\frac d{ds}ℒ\{f(t)\}\)

Similarly

\(ℒ\{t^2f(t)\}=ℒ\{t\cdot tf(t)\}=-\frac d{ds}ℒ\{tf(t)\}=-\frac d{ds}\left(-\frac d{ds}ℒ\{f(t)\}\right)=\frac{d^2}{ds^2}ℒ\{f(t)\}\)

Theorem: Derivatives of transforms

\(ℒ\{t^nf(t)\}={(-1)}^n\frac{d^n}{ds^n}F(s)\)

RC circuit

Capacitor model with \(v_c(0)=v_{c0}\)

\(v_c(t)=\frac1C\int_0^ti_c(\tau)d\tau+v_c(0)\)

\(\Rightarrow{\widehat v}_c(s)=\frac1C\cdot\frac1s{\widehat i}_c(s)+\frac{v_{c0}}s\)

\(\Rightarrow{\widehat v}_c(s)=\frac{v_{c0}}s+\frac1{sC}{\widehat i}_c(s)\)

\({\widehat v}_c(s)=Z_c(s){\widehat i}_c(s)+\frac{v_{c0}}s\)

\(Z_c(s)=\frac1{sC},\;impedance\)

\({\widehat v}_T(s)=R_T{\widehat i}_c(s)+\frac{v_{c0}}s+\frac1{sC}{\widehat i}_c(s)\)

\(\Rightarrow(R_T+\frac1{sC}){\widehat i}_c(s)={\widehat v}_T(s)-\frac{v_{c0}}s\)

\({\widehat i}_c(s)=\frac1{(R_T+\frac1{sC})}({\widehat v}_T(s)-\frac{v_{c0}}s)\)

\(=-\frac{Cv_{c0}}{(sCR_T+1)}+\frac1{(R_T+\frac1{sC})}{\widehat v}_T(s)\)

\({\widehat v}_c(s)=\frac{v_{c0}}s+\frac1{sC}{\widehat i}_c(s)\)

\(=\frac{v_{c0}}s+\frac1{sC}\lbrack-\frac{Cv_{c0}}{(sCR_T+1)}+\frac1{(R_T+\frac1{sC})}{\widehat v}_T(s)\rbrack\)

\(=\frac{{\widehat v}_T(s)}{sCR_T+1}+\frac{v_{c0}}s-\frac{v_{c0}}{(s^2CR_T+s)}\)

\(=\frac{{\widehat v}_T(s)}{sCR_T+1}+\frac{sCR_T}{(CR_T+1)}v_{c0}\)

\(=\frac{R_TC}{sR_TC+1}v_{c0}+\frac1{sR_TC+1}{\widehat v}_T(s)\)

\(=\frac1{s+a}v_{c0}+\frac a{s+a}{\widehat v}_T(s),\;a=\frac1{R_TC}\)

\(v_c(t)=e^{-at}v_{c0}+ℒ^{-1}\{\frac a{s+a}{\widehat v}_T(s)\}\)

\(v_c(t)=e^{-\frac1{R_TC}t}v_{c0}+ℒ^{-1}\{\frac a{s+a}{\widehat v}_T(s)\}\)

\(t\rightarrow\infty,\;v_c(t)=ℒ^{-1}\{\frac a{s+a}{\widehat v}_T(s)\}\)

If \(v_T(t)=v_T\), it is constant.

\({\widehat v}_T(s)=\frac{v_T}s\)

\(then\;v_c(t)=v_{c0}e^{-at}+ℒ^{-1}\{\frac a{s+a}\frac{v_T}s\}\)

\(=v_{c0}e^{-at}+ℒ^{-1}\{(\frac1s-\frac1{s+a})v_T\}\)

\(=v_{c0}e^{-at}+(1-e^{-at})v_T\)

\(v_c(0)=v_{c0}\)

\(v_c(\infty)=v_T\)

\(i_c(\infty)=0\)

If \(v_T(t)=\cos\omega t\), then

\(v_c(t)=\frac1sv_{c0}+ℒ^{-1}\{\frac a{s+a}{\widehat v}_T(s)\}\)

\({\widehat v}_T(s)=\frac s{s^2+\omega^2}\)

\(\because\frac a{s+a}\cdot\frac s{s^2+\omega^2}=\frac{as}{(s+a)(s^2+\omega^2)}\)

\(=\frac A{s+a}+\frac{Bs+C\omega}{s^2+\omega^2}\)

\(v_c(t)=Ae^{-at}+B\cos\omega t+C\sin\omega t\)

\(=\frac{A(s^2+\omega^2)+(Bs+C\omega)(s+a)}{(s+a)(s^2+\omega^2)}\)

\(A(s^2+\omega^2)+(Bs+C\omega)(s+a)=as\)

\(s=-a\Rightarrow A(a^2+\omega^2)=-a^2\Rightarrow A=\frac{-a^2}{a^2+\omega^2}\)

\(A+B=0\Rightarrow B=-A=\frac{a^2}{a^2+\omega^2}\)

\(C\omega+aB=a\Rightarrow C\omega=a(1-B)=a\cdot\frac{\omega^2}{a^2+\omega^2}\)

\(C=\frac{a\omega}{a^2+\omega^2}\)

\(t\rightarrow\infty,\;v_c(t)=B\cos\omega t+C\sin\omega t\)

\(v_c(t)=\sqrt{B^2+C^2}(\frac B{\sqrt{B^2+C^2}}\cos\omega t+\frac C{\sqrt{B^2+C^2}}\sin\omega t)\)

\(\cos\theta=\frac B{\sqrt{B^2+C^2}}\)

\(=\sqrt{B^2+C^2}\cos(\omega t-\theta)\)

\(\sqrt{B^2+C^2}=\sqrt{\frac{a^4+a^2\omega^2}{{(a^2+\omega^2)}^2}}=\frac{a(\sqrt{a^2+\omega^2})}{(a^2+\omega^2)}\)

\(=\frac a{\sqrt{a^2+\omega^2}}\)

\(v_c(t)=\frac a{\sqrt{a^2+\omega^2}}\cos(\omega t-\theta)\)

\({\widehat v}_T(s)=(R_T+\frac1{sC}){\widehat i}_c(s)\Rightarrow{\widehat i}_c(s)={(R_T+\frac1{sC})}^{-1}{\widehat v}_T(s)\)

\({\widehat v}_c(s)=\frac1{sC}{\widehat i}_c(s)={\lbrack sC(R_T+\frac1{sC})\rbrack}^{-1}{\widehat v}_T(s)\)

\({\widehat v}_c(s)=\frac{{\widehat v}_T(s)}{1+sR_TC}\)

\({\widehat v}_c(s)=H(s){\widehat v}_T(s)\)

\(H(s)=\frac1{1+sR_TC}\)

\(H(j\omega)=\frac1{1+j\omega R_TC}\)

\(=\frac a{a+j\omega}\)

\(\vert H(j\omega)\vert=\frac a{\sqrt{a^2+\omega^2}}\)

low pass filter

Norton circuit

\(i_N(t)=\frac{v_c(t)}{R_N}+i_c(t)=\frac{v_c(t)}{R_N}+C\frac{dv_c(t)}{dt}\)

\({\widehat i}_N(s)=\frac{{\widehat v}_c(s)}{R_N}+C(s{\widehat v}_c(s)-v_{c0})\)

\(\Rightarrow{\widehat i}_N(s)=(\frac1{R_N}+sC){\widehat v}_c(s)-Cv_{c0}\)

\({\widehat i}_N(s)+Cv_{c0}=(\frac{1+sR_NC}{R_N}){\widehat v}_c(s)\)

\({\widehat v}_c(s)=\frac{R_N}{1+sR_NC}({\widehat i}_N(s)+Cv_{c0})\)

\(=\frac{R_N}{1+sR_NC}{\widehat i}_N(s)+\frac{R_NC}{1+sR_NC}v_{c0}\)

\(a=\frac1{R_NC}\)

\(=\frac{\frac{{\widehat i}_N(s)}C}{s+a}+\frac1{s+a}v_{c0}\)

\(=\frac{aR_N{\widehat i}_N(s)}{s+a}+\frac1{s+a}v_{c0}\)

\(v_c(t)=v_{c0}e^{-at}+ℒ^{-1}\{\frac a{s+a}R_N{\widehat i}_N(s)\}\)

RC circuit with current sources

\(i_c(t)=C\frac{dv_c(t)}{dt}\)

\(v_c(t)=\frac1C\int_{t_0}^ti_{c(\tau)}d\tau+v_c(t_0)\)

\({\widehat i}_c(s)=C\lbrack s{\widehat v}_c(s)-v_c(0)\rbrack,\;t_0=0\)

\({\widehat v}_c(s)=\frac1{sC}{\widehat i}_c(s)+\frac{v_c(0)}s\)

\({\widehat Z}_c(s)=\frac1{sC}\)

KCL: \({\widehat i}_N(s)=\frac{{\widehat v}_c(s)}{R_N}+\frac{{\widehat v}_c(s)-{\frac{v_{c0}}s}}{\frac1{sC}}\)

\({\widehat i}_N(s)=\frac{{\widehat v}_c(s)}{R_N}+sC{\widehat v}_c(s)-Cv_{c0}\)

\(\Rightarrow\frac{{\widehat i}_N(s)+Cv_{c0}}{sC+{\displaystyle\frac1{R_N}}}={\widehat v}_c(s)\)

\(\Rightarrow v_c(t)=ℒ^{-1}\left({\widehat v}_c(s)\right)\)

RL circuit

\(v_L(t)=L\frac{di_L(t)}{dt}\)

\({\widehat v}_L(s)=L\lbrack s{\widehat i}_L(s)-i_L(0)\rbrack\)

\(=sL{\widehat i}_L(s)-Li_L(0)\)

\(={\widehat Z}_L(s){\widehat i}_L(s)-Li_L(0)\)

\(\Rightarrow{\widehat i}_L(s)=\frac1{sL}{\widehat v}_L(s)+\frac1si_L(0)\)

Second order RLC circuits without any sources

Parallel RLC circuit

\(\frac{v_c(t)}R+i_L(t)+i_C(t)=0\)

\(\frac{v_L(t)}R+i_L(t)+C\frac{dv_L(t)}{dt}=0\)

\(i_L(t)\)

\(\frac LR\frac{di_L(t)}{dt}+i_L(t)+LC\frac{d^2i_L(t)}{dt^2}=0\)

\(\frac{d^2i_L(t)}{dt^2}+\frac1{RC}\frac{di_L(t)}{dt}+\frac1{LC}i_L(t)=0\)

\({\ddot i}_L(t)+\frac1{RC}{\dot i}_L(t)+\frac1{LC}i_L(t)=0\)

\(v_c(t)\)

\(\frac{v_c(t)}R+\frac1L\int v_c(t)dt+C\frac{dv_c(t)}{dt}=0\)

\(\frac1R{\dot v}_c(t)+\frac1Lv_c(t)+C{\ddot v}_c(t)=0\)

\({\ddot v}_c(t)+\frac1{RC}{\dot v}_c(t)+\frac1{LC}v_c(t)=0\)

Ex

\({\ddot v}_c(t)+\frac1{RC}{\dot v}_c(t)+\frac1{LC}v_c(t)=0\)

initial condition I - \(v_c(0)=v_{c0}\)

\(i_c(t)=C{\dot v}_c(t)\Rightarrow{\dot v}_c(t)=\frac1Ci_c(t)\)

\({\dot v}_c(t)=\frac1Ci_c(t)=\frac1C\lbrack-\frac{v_c(t)}R-i_L(t)\rbrack\)

initial condition II - \({\dot v}_c(0)=-\frac{v_{c0}}{RC}-\frac1Ci_{L0}\)

\(s^2{\widehat v}_c(s)-sv_c(0)-{\dot v}_c(0)+\frac1{RC}\lbrack s{\widehat v}_c(s)-v_c(0)\rbrack+\frac1{LC}{\widehat v}_c(s)=0\)

\((s^2+\frac1{RC}s+\frac1{LC}){\widehat v}_c(s)=sv_c(0)+{\dot v}_c(0)+\frac1{RC}v_c(0)\)

characteristic polynomial \((s^2+\frac1{RC}s+\frac1{LC})\)

\({\widehat v}_c(s)=\frac{sv_c(0)+{\dot v}_c(0)+\frac1{RC}v_c(0)}{(s^2+\frac1{RC}s+\frac1{LC})}\)

\(s^2+\frac1{RC}s+\frac1{LC}=(s-\lambda_1)(s-\lambda_2)\)

where \(\lambda_1,\;\lambda_2\) are characteristic roots

\(Re(\lambda_1)＜0,\;Re(\lambda_2)＜0\)

\(s^2+\frac1{RC}s+\frac1{LC}=(s-\lambda_1)(s-\lambda_2)=s^2+2\alpha\omega_0s+\omega_0^2\)

\(\omega_0^2=\frac1{LC}\)

\(2\alpha\omega_0=\frac1{RC}\Rightarrow\alpha=\frac1{2\omega_0RC}\)

\(\alpha=\frac1{2\omega_0RC}=\frac{\sqrt{LC}}{2RC}=\frac1{2R}\sqrt{\frac LC}\)

\(\triangle={(2\alpha\omega_0)}^2-4\omega_0^2=4(\alpha^2-1)\omega_0^2\)

\(\alpha^2>1,\;\triangle>0\Rightarrow\lambda_1,\lambda_2\in\mathbb{R},\;\lambda_1\neq\lambda_2\)

I \(\alpha>1\)

\(s^2+\frac1{RC}s+\frac1{LC}=(s-\lambda_1)(s-\lambda_2),\;\lambda_1,\lambda_2\in\mathbb{R},\;\lambda_1\neq\lambda_2\)

\({\widehat v}_c(s)=\frac{as+b}{(s-\lambda_1)(s-\lambda_2)}\)

\(a=v_c(0),\;b={\dot v}_c(0)+\frac1{RC}v_c(0)\)

\({\widehat v}_c(s)=\frac{as+b}{(s-\lambda_1)(s-\lambda_2)}=\frac{A_1}{s-\lambda_1}+\frac{A_2}{s-\lambda_2}\)

\(\Rightarrow v_c(t)=A_1e^{\lambda_1t}+A_2e^{\lambda_2t}\)

\(t\rightarrow\infty,\;v_c(\infty)\rightarrow0\)

II \(\alpha=1\)

\(s^2+\frac1{RC}s+\frac1{LC}={(s-\lambda)}^2,\;\lambda_1=\lambda_2=\lambda\in\mathbb{R},\;\lambda＜0\)

\({\widehat v}_c(s)=\frac{as+b}{{(s-\lambda)}^2}=\frac{a(s-\lambda)+b+a\lambda}{{(s-\lambda)}^2}=\frac a{(s-\lambda)}+\frac c{{(s-\lambda)}^2},\;c=b+a\lambda\)

\(v_c(t)=ae^{\lambda t}+cte^{\lambda t}\)

\(t\rightarrow\infty,\;v_c(\infty)\rightarrow0\)

III \(\alpha＜1\)

\(s^2+\frac1{RC}s+\frac1{LC}={(s-\lambda)(s-\lambda\ast)},\;\lambda_1=\lambda_2\ast\)

\(\lambda_1=\alpha+j\beta,\;\lambda_2=\alpha-j\beta,\;(\alpha＜0,\;\beta>0)\)

\({\widehat v}_c(s)=\frac{as+b}{(s-\lambda_1)(s-\lambda_2)}=\frac{as+b}{(s-\alpha-j\beta)(s-\alpha+j\beta)}\)

\(=\frac{as+b}{{(s-\alpha)}^2+\beta^2}=\frac{A(s-\alpha)+B\beta}{{(s-\alpha)}^2+\beta^2}\)

\(v_c(t)=e^{\alpha t}(A\cos\beta t+B\sin\beta t)\)

damped oscillation

\(t\rightarrow\infty,\;v_c(\infty)\rightarrow0\)

\({\ddot v}_c(t)+\frac1{RC}{\dot v}_c(t)+\frac1{LC}v_c(t)=0\)

same unit

\({\ddot v}_c(t):v/sec^2,\;{\dot v}_c(t):v/sec,\;v_c(t):v\)

\(RC:\;sec\)

\(LC:sec^2\)

\(\frac1{LC}:sec^{-2}\)

\(\frac1{\sqrt{LC}}:sec^{-1}\;(Hz)\)

\(that\;is\;why\;\frac1{LC}\;assigned\;to{\;\omega_0}^2\)

\(\omega_0=\frac1{\sqrt{LC}}\)

Series RLC circuit

KVL \(V_R(t)+V_L(t)+V_C(t)=0\)

\(Ri_R(t)+L\frac{di_L(t)}{dt}+\frac1C\int_0^ti_c(\tau)d\tau+v_c(0)=0\)

\(\Rightarrow Ri(t)+L\frac{di(t)}{dt}+\frac1C\int_0^ti(\tau)d\tau+v_c(0)=0\)

\(\Rightarrow R\widehat i(s)+L\lbrack s\widehat i(s)-i_l(0)\rbrack+\frac1{sC}\widehat i(s)+\frac{v_c(0)}s=0\)

\(\Rightarrow(R+sL+\frac1{sC})\widehat i(s)=Li_l(0)-\frac{v_c(0)}s\)

\(\Rightarrow(s^2LC+sRC+1)\widehat i(s)=sLCi_l(0)-Cv_c(0)\)

\(\widehat i(s)=\frac{sLCi_l(0)-Cv_c(0)}{s^2LC+sRC+1}\)

\(=\frac{si_l(0)-{\frac1L}v_c(0)}{s^2+s{\frac RL}+{\frac1{LC}}}\)

\(\widehat i(s)=\frac{i_l(0)s-\frac1Lv_c(0)}{s^2+2\zeta\omega_0s+\omega_0^2}\)

\(2\zeta\omega_0=\frac RL,\;\omega_0^2=\frac1{LC}\)

\(\omega_0=\sqrt{\frac1{LC}},\;\zeta_0=\frac R2\sqrt{\frac CL}\)

\(\triangle=4\zeta^2\omega_0^2-4\omega_0^2=4(\zeta^2-1)\omega_0^2\)

A. \(\triangle>0\;(\zeta>1)\)

\(s^2+\frac RLs+\frac1{LC}=(s-\lambda_1)(s-\lambda_2),\;\lambda_1,\lambda_2\in R^-,\;\lambda_1\neq\lambda_2\)

\(\widehat i(s)=\frac{as+b}{(s-\lambda_1)(s-\lambda_2)},\;a=i_l(0),\;b=-\frac{v_c(0)}L\)

\(=\frac{A_1}{s-\lambda_1}+\frac{A_2}{s-\lambda_2}\)

\(=\frac{A_1(s-\lambda_2)+A_2(s-\lambda_1)}{(s-\lambda_1)(s-\lambda_2)}\)

\(as+b=A_1(s-\lambda_2)+A_2(s-\lambda_1)\)

\(A_1=\frac{a\lambda_1+b}{\lambda_1-\lambda_2},\;A_2=\frac{a\lambda_2+b}{\lambda_2-\lambda_1}\)

\(i(t)=A_1e^{\lambda_1t}+A_2e^{\lambda_2t}\)

\(i(\infty)=A_1e^{\lambda_1\infty}+A_2e^{\lambda_2\infty}\rightarrow0,\;t\rightarrow\infty\)

B. \(\triangle=0\;(\zeta=1)\)

\(s^2+\frac RLs+\frac1{LC}={(s-\lambda)}^2,\;\lambda_1=\lambda_2=\lambda\in R^-\)

\(\widehat i(s)=\frac{as+b}{{(s-\lambda)}^2},\;a=i_l(0),\;b=-\frac{v_c(0)}L\)

\(=\frac{a(s-\lambda)+b+a\lambda}{{(s-\lambda)}^2}\)

\(=\frac a{(s-\lambda)}+\frac{b+a\lambda}{{(s-\lambda)}^2}\)

\(i(t)=ae^{\lambda t}+(b+a\lambda)te^{\lambda t},\;\lambda(\infty)\rightarrow0\)

C. \(\triangle＜0\;(\zeta＜1)\)

\(s^2+\frac RLs+\frac1{LC}={(s-\alpha+j\beta)(s-\alpha-j\beta)},\;\alpha\in R^-,\beta\in R^+\)

\(\widehat i(s)=\frac{as+b}{{(s-\alpha)}^2+\beta^2}=\frac{a(s-\alpha)+\beta{\frac{(b+\alpha a)}\beta}}{{(s-\alpha)}^2+\beta^2}\)

\(i(t)=ae^{\alpha t}\cos\beta t+\frac{(b+\alpha a)}\beta e^{\alpha t}\sin\beta t\)

Parallel RLC circuit with const source

KCL \(i_R(t)+i_L(t)+i_C(t)=I_S\)

\(\Rightarrow\frac{v(t)}R+\frac1L\int_0^tv(\tau)d\tau+i_L(0)+C\frac{dv(t)}{dt}=I_S\)

\(\frac{\widehat v(s)}R+\frac1{sL}\widehat v(s)+\frac{i_L(0)}s+C\lbrack s\widehat v(s)-v_c(0)\rbrack=\frac{I_S}s\)

\(\Rightarrow(\frac1R+\frac1{sL}+sC)\widehat v(s)=\frac{I_S}s-\frac{i_L(0)}s+Cv_c(0)\)

\((\frac{sL}R+1+s^2LC)\widehat v(s)=LI_S-Li_L(0)+sLCv_c(0)\)

\(\Rightarrow(1+\frac s{RC}+\frac1{LC})\widehat v(s)=\frac{I_S}C-\frac{i_L(0)}C+sv_c(0)\)

\(\widehat v(s)=\frac{\frac{I_S}C-\frac{i_L(0)}C+sv_c(0)}{1+\frac s{RC}+\frac1{LC}}=\frac{as+b}{s^2+2\zeta\omega_0s+\omega_0^2}\)

\(\omega_0=\sqrt{\frac1{LC}},\;\zeta=\frac1{2R}\sqrt{\frac LC}\)

\(a=v_c(0),\;b=\frac1C\lbrack I_S-i_L(0)\rbrack\)

\(\widehat v(s)=\frac{\frac{I_S}C-\frac{i_L(0)}C+sv_c(0)}{s^2+2\zeta\omega_0s+\omega_0^2}\)

\(=\frac{as+b}{(s-\lambda_1)(s-\lambda_2)}\)

Series RLC circuit with sinusoidal source

KVL \(v_R(t)+v_L(t)+v_C(t)=v_s(t)\)

\(\Rightarrow Ri(t)+L\frac{di(t)}{dt}+\frac1C\int_0^ti_c(\tau)d\tau+v_c(0)=v_s(t)\)

\(\Rightarrow R\widehat i(s)+L\lbrack s\widehat i(s)-i_l(0)\rbrack+\frac1{sC}\widehat i(s)+\frac{v_c(0)}s={\widehat v}_s(s)\)

\((R+Ls+\frac1{sC})\widehat i(s)={\widehat v}_s(s)+Li_l(0)-\frac{v_c(0)}s\)

\((LCs^2+RCs+1)\widehat i(s)=sC{\widehat v}_s(s)+sLCi_l(0)-Cv_c(0)\)

\(\widehat i(s)=\frac{sC}{(LCs^2+RCs+1)}{\widehat v}_s(s)+\frac{LCi_l(0)s-Cv_c(0)}{(LCs^2+RCs+1)}\)

\(=\frac{\frac sL}{(s^2+{\frac RL}s+{\frac1{LC}})}{\widehat v}_s(s)+\frac{\ i_l(0)s-\frac1Lv_c(0)}{(s^2+{\frac RL}s+{\frac1{LC}})}\)

The particular solution represents the zero-state solution (it does not depend on the system initial conditions) and the homogeneous solution represents the zero-input solution (it depends only on the system initial conditions).

\(i(t)=i_p(t)+i_h(t)\)

\(i_p(t)=ℒ^{-1}\{\frac{\frac sL}{(s^2+{\frac RL}s+{\frac1{LC}})}{\widehat v}_s(s)\}\)

\(i_h(t)=ℒ^{-1}\{\frac{ i_l(0)s-\frac1Lv_c(0)}{(s^2+{\frac RL}s+{\frac1{LC}})}\}\)

\({\left.i_h(t)\right|}_{t\rightarrow\infty}=\lim_{s\rightarrow0}s(\frac{ i_l(0)s-\frac1Lv_c(0)}{(s^2+{\frac RL}s+{\frac1{LC}})})=0\)

\(i(t)=i_p(t),\;t\rightarrow\infty\)

\(t\rightarrow\infty,\;i(t)=i_p(t)=ℒ^{-1}\{\frac{\frac sL}{(s^2+{\frac RL}s+{\frac1{LC}})}{\widehat v}_s(s)\}\)

\({\widehat v}_s(s)=\frac s{s^2+\omega^2}\)

\(\widehat i(s)=\frac{\frac sL}{s^2+{\frac RL}s+{\frac1{LC}}}\cdot\frac s{s^2+\omega^2}=\frac{as+b}{s^2+\frac RLs+\frac1{LC}}\cdot\frac{cs+d\omega}{s^2+\omega^2}\)

\(\frac{s^2}L=(as+b)(s^2+\omega^2)+(cs+d\omega)(s^2+\frac RLs+\frac1{LC})\)

\(s=\lambda_1\Rightarrow\frac{\lambda_1^2}L=(a\lambda_1+b)(\lambda_1^2+\omega^2)\)

\(s=\lambda_2\Rightarrow\frac{\lambda_2^2}L=(a\lambda_2+b)(\lambda_2^2+\omega^2)\)

\(t\rightarrow\infty,\;\widehat i(s)=\frac{cs+d\omega}{s^2+\omega^2}\)

\(s=j\omega\Rightarrow\frac{-\omega^2}L=(j\omega c'+d\omega)\lbrack-\omega^2-\frac RL(j\omega)+\frac1{LC}\rbrack\)

\(t\rightarrow\infty,\;s=j\omega,\;c's+d\omega={\left.\frac{\frac{s^2}L}{s^2+{\frac RL}s+{\frac1{LC}}}\right|}_{s=j\omega}\)

\(c'\omega=Im\lbrack\frac{\frac{{(j\omega)}^2}L}{{(j\omega)}^2+\frac RL(j\omega)+\frac1{LC}}\rbrack\)

\(d\omega=Re\lbrack\frac{\frac{{(j\omega)}^2}L}{{(j\omega)}^2+\frac RL(j\omega)+\frac1{LC}}\rbrack\)

\(t\rightarrow\infty,\;i(t)=c'\cos\omega t+d\sin\omega t\)

General Linear RLC circuit

If there are no initail conditions, we can use the Laplace transform directly into the circuit.

About initial conditions

\(\dot y(t)=ay(t)=b(t),\;y(0)=y_0\)

\(\Rightarrow s\widehat y(s)-y(0)+a\widehat y(s)=\widehat b(s)\)

\(\Rightarrow(s+a)\widehat y(s)=\widehat b(s)+y(0)\)

I. \(b(t)=0,\;y(0)=y_0\)

\((s+a)\widehat y(s)=y_0\\b(t)=0,\;y(0)=y_0\)

II. \(b(t)=\delta(t),\;y(0)=0\)

\(\int_{-\infty}^\infty\delta(t)dt=1,\;\delta(t)=\;0\;when\;t\neq0\)

\(\delta(s)=\int_{0^-}^\infty\delta(t)e^{-st}dt=\int_{0^-}^\infty\delta(t)dt=1\)

\((s+a)\widehat y(s)=1\)

\((eq\;1)\;\dot y(t)+ay(t)=\delta(t).\;y(0)=0.\;t>0^-\)

\((eq\;2)\;\dot y(t)+ay(t)=0.\;y(0)=1.\;t>0\)

\((eq\;1)\;s\widehat y(s)+a\widehat y(s)=1\Rightarrow y_1(t)\)

\((eq\;2)\;s\widehat y(s)-y(0)+a\widehat y(s)=0\Rightarrow s\widehat y(s)+a\widehat y(s)=1\Rightarrow y_2(t)\)

\(y_1(t)=y_2(t),\;(t>0)\)

RC circuit - First-order (ODE) Ordinary Differential Equation

\[v(t)=R\;i(t)+\frac1C\int i(t)dt\]

\(Let\;v\;=\;0,\;i\left(t\right)\;=\;\alpha e^{\beta t}\)

\(0\;=\;R\alpha e^{\beta t}+\frac1C\int\alpha e^{\beta t}dt\)

\(-R\alpha e^{\beta t}=\frac1C\int\alpha e^{\beta t}dt\)

\(-R\alpha e^{\beta t}=\frac1{\beta C}\alpha e^{\beta t}\)

\(\beta=-\frac1{RC}\)

At time 0, \(V_C= 0, V_R = V_0\).

\(V_0\;=\;R\alpha e^{-\frac1{RC}t_0},\;t_0=0\)

\(\alpha=\frac{V_0}R\)

\(i(t)=\frac{V_0}Re^{-\frac t{RC}}\)

RL circuit - First-order (ODE) Ordinary Differential Equation

\[V=iR+L\frac{di}{dt}\]

\(V-iR=L\frac{di}{dt}\)

\(\frac{dt}L=\frac{di}{V-iR}\)

\(\int_0^t\frac{dt}L=\int_0^i\frac{di}{V-iR}\)

\(\frac tL=\int_0^i\frac{di}{V-iR}\)

Now integrate right hand side by using substitution method,

\(Let\;z=V-iR,\;\frac{dz}{dt}=-R\frac{di}{dt}\)

\(\frac{di}{dt}=-\frac1R\frac{dz}{dt}\)

\(\frac tL=-\frac1R\int_0^i\frac{dz}z\)

\(\frac tL=-\frac1R\left.\ln(z)\right|_0^i\)

\(-\frac{Rt}L=\left.\ln(V-iR)\right|_0^i\)

\(-\frac{Rt}L=\ln(V-iR)-\ln(V)\)

\(-\frac{Rt}L=\ln(\frac{V-iR}V)\)

\(e^{-\frac{Rt}L}=\frac{V-iR}V\)

\(Ve^{-\frac{Rt}L}=V-iR\)

\(iR=V-Ve^{-\frac{Rt}L}\)

\(iR=V(1-e^{-\frac{Rt}L})\)

\(i(t)=\frac{V(1-e^{-\frac{Rt}L})}R\)

\(V=i(t)R+L\frac{di(t)}{dt}\)

\(\frac VR=i(t)+\frac LR\frac{di(t)}{dt}\)

\(\frac VR-i(t)=\frac LR\frac{di(t)}{dt}\)

\(\frac{V/R-i(t)}1=\frac{(L/R)}1\cdot\frac{di(t)}{dt}\)

\(\frac{dt}{L/R}=\frac{di(t)}{V/R-i(t)}\)

\(-\frac{dt}{L/R}=\frac{di(t)}{i(t)-V/R}\)

\(-\int_0^t\frac{dt}{L/R}=\int_0^t\frac{di(t)}{i(t)-V/R}\)

\(-\frac t{L/R}=\ln\left(\frac{i(t)-V/R}{-V/R}\right)\)

\(e^{-Rt/L}=1-\frac{i(t)R}V\)

\(i(t)=\frac VR(1-e^{-Rt/L})\)

The term L/R in the equation is called the Time Constant, \((τ)\) of the RL series circuit, and it is defined as time taken by the current to reach its maximum steady state value and the term V/R represents the final steady state value of current in the circuit.

The Source-Free Series RLC Circuit (Second-order ODE)

The energy is represented by the initial capacitor voltage \(V_0\) and initial inductor current \(I_0\). Thus, at \(t=0\),

\(v(0)=V_0,\;i(0)=I_0\)

Applying KVL around the loop and differentiating with respect to \(t\),

\(Ri(t)+L\frac{di(t)}{dt}+\frac1C\int i(t)dt=0\)

\(R\frac{di(t)}{dt}+L\frac{d^2i(t)}{dt^2}+\frac1Ci(t)=0\)

\(\frac{d^2i(t)}{dt^2}+\frac RL\frac{di(t)}{dt}+\frac1{LC}i(t)=0\)

This is a second-order differential equation. The solution is of the form \(i=Ae^{st}\) and substituting this to the DE, the characteristic equation is

\(S^2+\frac RLs+\frac1{LC}=0\)

where \(s_1=-\frac R{2L}+\sqrt{{(\frac R{2L})}^2-\frac1{LC}},\;s_2=-\frac R{2L}-\sqrt{{(\frac R{2L})}^2-\frac1{LC}}\)

are the two roots of the characteristic equation of the differential. A more compact way of expressing the roots is

\(s_1=-\alpha+\sqrt{\alpha^2-\omega_0^2},\;s_2=-\alpha-\sqrt{\alpha^2-\omega_0^2}\)

where \(\alpha=\frac R{2L},\;\omega_0=\frac1{\sqrt{LC}}\)

The Characteristic roots \(s_1\) and \(s_2\) are called natural frequencies, measured in nepers (奈培) per second (Np/s).

\(\omega_0\) is known as the resonant frequency or strictly as the undamped natural frequency, expressed in radians per second (rad/s).

\(\alpha\) is the neper (奈培) frequency expressed in Np/s.

Since there are two possible solutions from the two values of \(s\),

\(i_1=A_1e^{s_1t},\;i_2=A_1e^{s_2t}\)

A complete or total solution would therefore require a linear combination of \(i_1\) and \(i_2\). Thus the natural response of the series RLC circuit is \(i(t)=A_1e^{s_1t}+A_2e^{s_2t}\), where the constants \(A_1\) and \(A_2\) are determined from initial values.

There are three types of solutions:

1. If \(\alpha\;>\;\omega_0\), the overdamped case; roots are unequal and real.

2. If \(\alpha\;=\;\omega_0\), the critically damped case; roots are equal and real.

3. If \(\alpha\;＜\;\omega_0\), the underdamped case; roots are complex.

Overdamped Case \((\;\alpha\;>\;\omega_0\;)\)

\(\alpha\;\omega_0\;\) implies \(C\;>4L/R^2\). When this happens, both roots \(s_1\) and \(s_2\) are negative and real. The response is \(i(t)=A_1e^{s_1t}+A_2e^{s_2t}\), which decays and approaches zero as \(t\) increases.

Critically Damped Case \((\;\alpha\;=\;\omega_0\;)\)

When \(\alpha\;=\;\omega_0\), \(C\;=\;4L/R^2\) and \(s_1=s_2=-\alpha=-R/2L\). The second-order differential equation becomes \(\frac{d^2i}{dt^2}+2\alpha\frac{di}{dt}+\alpha^2i=0\)

Solving the DE gives the natural response of the critically damped circuit: a sum of a negative exponential and a negative exponential multiplied by a linear term, \(i(t)=(A_2+A_1t)e^{-\alpha t}\)

The figure is a sketch of \(i(t)=te^{-\alpha t}\), which reaches a maximum value of \(e^{-1}/\alpha\) at \(t=1/\alpha\), one time constant, and then decays all the way to zero.

Underdamped Case \((\;\alpha\;＜\;\omega_0\;)\)

When \(\alpha\;＜\;\omega_0,\;C\;＜\;4L/R^2\). The roots may be written as \(s_1=-\alpha\;+\;j\omega_d,\;s_2=-\alpha\;-\;j\omega_d\), and \(\omega_d\;=\;\sqrt{\omega_0^2-\alpha^2}\) which is called the damped frequency.

Both \(\omega_0\) and \(\omega_d\) are natural frequencies because they help determine the natural response.

Using Euler's identities, \(e^{j\theta}=\cos\theta+j\sin\theta,\;e^{-j\theta}=\cos\theta-j\sin\theta\)

Replacing constants \((A_1+A_2),\;j(A_1-A_2)\) with constants \(B_1,\;B_2\), the natural response is \(i(t)=e^{-\alpha t}(B_1\cos\omega_dt+B_2sin\omega_dt)\)

The natural response for this case is exponentially damped and oscillatory in nature. It has a time constant of \(1/\alpha\) and a period of \(T\;=\;2\mathrm\pi/{\mathrm\omega}_{\mathrm d}\)

The Source-Free Parallel RLC Circuit (Second-order ODE)

Assume initial inductor current \(I_0\) and initial capacitor voltage |(V_0\) , \(i(0)=I_0\) and \(v(0)=V_0\) .

The three elements in parallel have the same voltage across. According to the passive sign convention, the current through each element is leaving the top node. Applying KCL at the top node, taking the derivative with respect to t and dividing by C results in

\(i(0)=I_0=\frac1L\int_{-\infty}^0v(t)dt\)

\(v(0)=V_0\)

\(i(0^+)=I_0,\;\;v(0^+)=V_0\)

The integro-differential equation:

\(\frac vR+\frac1L\int_0^tvdt-i(t_0)+C\frac{dv}{dt}=0\)

\(C\frac{d^2v}{dt^2}+\frac1R\frac{dv}{dt}+\frac1Lv=0\)

\(\frac{d^2v}{dt^2}+\frac1{RC}\frac{dv}{dt}+\frac1{LC}v=0\)

The characteristic equation is obtained as

\(S^2+\frac1{RC}s+\frac1{LC}=0\)

The roots of the characteristic equation are

\(s_1,s_2=-\alpha\pm\sqrt{\alpha^2-\omega_0^2}\)

\(\alpha=\frac1{2RC},\;\omega_0=\frac1{\sqrt{LC}}\)

Superposition:

\(v(t)=A_1e^{s_1t}+A_2e^{s_2t}\)

Overdamped Case \((\;\alpha\;>\;\omega_0\;)\)

\(\alpha>\omega_0\;when\;L>4R^2C\)

The roots of the characteristic equation are real and negative. The response is

\(v(t)=A_1e^{s_1t}+A_2e^{s_2t}\)

Critically Damped Case \((\;\alpha\;=\;\omega_0\;)\)

\(\alpha=\omega_0\;when\;L=4R^2C\)

The roots are real and equal so that the response is

\(v(t)=(A_1+A_2t)e^{-\alpha t}\)

Underdamped Case \((\;\alpha\;＜\;\omega_0\;)\)

\(\alpha＜\omega_0\;when\;L＜4R^2C\)

In this case the roots are complex and may be expressed as

\(s_{1,2}=-\alpha\pm j\omega_d\)

\(\omega_d=\sqrt{\omega_0^2-\alpha^2}\)

\(v(t)=e^{-\alpha t}(A_1\cos\omega_dt+A_2\sin\omega_dt)\)

The constants \(A_1\) and \(A_2\) in each case can be determined from the initial conditions. We need \(V_0\) and \(\frac{dv(0)}{dt}\)

\(\frac{V_0}R+I_0+C\frac{dv(0)}{dt}=0\)

\(\frac{V_0}{RC}+\frac{I_0}C+\frac{dv(0)}{dt}=0\)

\(\frac{dv(0)}{dt}=-\frac{(V_0+RI_0)}{RC}\)

refered to https://www.circuitbread.com/study-guides/dc-circuits/second-order-circuits

\(Z_{RLC}\) is the RLC circuit impedance in ohms (Ω),
\(ω = 2πf\) is the angular frequency in rad/s,
\(f\) is the frequency in hertz (Hz),
\(R\) is the resistance in ohms (Ω),
\(L\) is the inductance in henries (H),
\(C\) is the capacitance in farads (F),
\(Q\) is the quality factor of a parallel RLC circuit (dimensionless),
\(ω_0\) is the resonant angular frequency in radian per second (rad/s),
\(f_0\) is the resonant frequency in hertz (H),
\(φ\) is the phase shift between the total voltage \(V_T\) and the total current \(I_T\) in degrees (°).

Transient Analysis of First Order RC and RL circuits

In an RL circuit, voltage across the inductor decreases with time while in the RC circuit the voltage across the capacitor increased with time.

Analysis of RC circuits. Charging and discharging processes

RC Circuit: Charging the Capacitor

\(V_s=R\cdot i(t)+V_c(t)\)

\(C=\frac{Q(t)}{V_s(t)},\;V_c(t)=\frac{Q(t)}C\)

\(i(t)=\frac{dQ(t)}{dt}\)

\(V_s=R\frac{dQ(t)}{dt}+\frac{Q(t)}C\)

\(\frac{V_s}R=\frac{dQ(t)}{dt}+\frac{Q(t)}{RC}\)

\(\frac{dQ(t)}{dt}=\frac{V_s}R-\frac{Q(t)}{RC}\)

\(\frac{dQ(t)}{dt}=\frac{V_sC-Q(t)}{RC}\)

\(\frac{dQ(t)}{V_sC-Q(t)}=\frac{dt}{RC}\)

\(\int_0^q\frac{dQ(t)}{V_sC-Q(t)}=\frac1{RC}\int_0^tdt\)

\(let\;u(t)=V_sC-Q(t),\;du(t)=-dQ(t)\)

\(-\int_0^q\frac{du(t)}{u(t)}=\frac1{RC}\int_0^tdt\)

\(-\left.\ln(u(t))\right|_0^q=\frac t{RC}\)

\(\ln(\frac{u(q)}{u(0)})=\frac{-t}{RC}\)

\(\frac{V_sC-q}{V_sC}=e^\frac{-t}{RC}\)

\(V_sC-q=V_sCe^\frac{-t}{RC}\)

\(Q(t)=V_sC(1-e^\frac{-t}{RC})\)

Capacitive time constant \(\tau_C = RC\)

\(Q(t)=Q(1-e^{-\frac t\tau})\)

Current:

\(I(t)=\frac{dQ(t)}{dt}\)

\(=\frac d{dt}CV_s(1-e^\frac{-t}{RC})\)

\(=CV_s(\frac1{RC})e^{-\frac t{RC}}\)

\(=\frac{V_s}Re^{-\frac t{RC}}\)

\(=I_0e^{-\frac t{RC}}\)

\(at\;tim\;t=0,\;I_0=\frac{V_s}R\)

\(I(t)=I_0e^{-\frac t{RC}}\)

\(V_c(t)=V_s(1-e^{-\frac t{RC}})\)

4T point is known as the Transient Period.
5T time period is commonly known as the Steady State Period.

RC Charging Table

RC Circuit: Discharging the Capacitor

\(V_R+V_C=0\)

\(i(t)R+\frac{Q(t)}C=0\)

\(\frac{dQ(t)}{dt}R+\frac{Q(t)}C=0\)

\(\frac{dQ(t)}{dt}=-\frac{Q(t)}{RC}\)

\(\frac{dQ(t)}{Q(t)}=-\frac{dt}{RC}\)

\(when\;t=0,\;Q(t)=Q_0\)

\(\int_{Q_0}^Q\frac{dQ}Q=-\frac1{RC}\int_0^tdt\)

\(\frac{dQ}Q\left.\ln Q\right|_{Q_0}^Q=-\frac1{RC}\left.t\right|_0^t\)

\(\ln\frac Q{Q_0}=-\frac t{RC}\)

\(\frac Q{Q_0}=e^{-\frac t{RC}}\)

\(Q(t)=Q_0e^{-\frac t\tau}\)

Current:

\(I(t)=\frac{dQ(t)}{dt}=\frac d{dt}(Q_0e^{-\frac t\tau})\)

\(I_{dis}(t)=-\frac{Q_0}\tau e^{-\frac t\tau}=-I_0e^{-\frac t\tau}\)

\(t=\;0,\;I_{dis}=-I_0\)

Capacitor charging and discharging

Analysis of RL circuits. Charging and discharging processes

RL Circuit: Charging the Inductor

The inductor, always opposing any change in current, will produce a voltage drop opposite to the change's direction. With that in mind, how much voltage the inductor will produce depends on how rapidly the current through it is decreased. As described by Lenz’s Law, the induced voltage will be opposed to the change in current. With a decreasing current, the voltage polarity will be oriented to try to keep the current at its former magnitude.

\(V_s=I(t)R+L\frac{d\;I(t)}{dt}\)

\(\frac{V_s}L=I(t)\frac RL+\frac{d\;I(t)}{dt}\)

\(\frac{d\;I(t)}{dt}=\frac{V_s-I(t)R}L\)

\(\frac{d\;I(t)}{V_s-I(t)R}=\frac{dt}L\)

\(\frac{d\;I(t)}{{\frac{V_s}R}-I(t)}=\frac{R\;dt}L\)

\(let\;u(t)=\frac{V_s}R-I(t),\;du(t)=-dI(t)\)

\(\frac{-du(t)}{u(t)}=\frac RLdt\)

\(\int_0^i\frac{du(t)}{u(t)}=-\frac RL\int_0^tdt\)

\(\;\ln(\frac{u(i)}{u(0)})=-\frac RLt\)

\(\frac{{\frac{V_s}R}-I(t)}{\frac{V_s}R}=e^{-\frac RLt}\)

\(\frac{V_s}R-I(t)=\frac{V_s}Re^{-\frac RLt}\)

\(I(t)=\frac{V_s}R(1-e^{-\frac RLt})\)

Inductive time constant \(\tau_L=\frac LR\)

\(I(t)=\frac{V_s}R(1-e^{-\frac t\tau})\)

\(V_L(t)=L\frac{d\;I(t)}{dt}\)

\(=L\cdot\frac{V_s}R\cdot\frac1\tau e^{-\frac t\tau},\;\tau=\frac LR\)

\(V_L(t)=V_se^{-\frac t\tau}\)

RL Circuit: Discharging the Inductor

\(0=L\frac{dI(t)}{dt}+I(t)R\)

\(L\frac{dI(t)}{dt}=-I(t)R\)

\(\frac{dI(t)}{I(t)}=-\frac RLdt\)

\(\int_0^i\frac{dI(t)}{I(t)}=-\frac RL\int_0^tdt\)

\(\ln\lbrack i(t)\rbrack\vert_0^i=-\frac RLt\)

\(\ln(\frac{I(t)}{I_0})=-\frac RLt,\;I_0=\frac{V_s}R\)

\(I(t)=I_0e^{-\frac RLt}=\frac{V_s}Re^{-\frac RLt}\;\)

\(V_L(t)=L\frac{d\;I(t)}{dt}\)

\(V_L(t)=-V_se^{-\frac RLt}\;=-V_se^{-\frac t\tau}\;\)

Resistor Y-Δ Transform

The Y-Δ transform is known by a variety of other names, mostly based upon the two shapes involved, listed in either order. The Y, spelled out as wye, can also be called T or star; the Δ, spelled out as delta, can also be called triangle, Π (spelled out as pi), or mesh. Thus, common names for the transformation include wye-delta or delta-wye, star-delta, star-mesh, or T-Π.

In many circuit applications, we encounter components connected together in one of two ways to form a three-terminal network: the “Delta,” or Δ (also known as the “Pi,” or π) configuration, and the “Y” (also known as the “T”) configuration.

It is possible to calculate the proper values of resistors necessary to form one kind of network (Δ or Y) that behaves identically to the other kind, as analyzed from the terminal connections alone. That is, if we had two separate resistor networks, one Δ and one Y, each with its resistors hidden from view, with nothing but the three terminals (A, B, and C) exposed for testing, the resistors could be sized for the two networks so that there would be no way to electrically determine one network apart from the other. In other words, equivalent Δ and Y networks behave identically.

Δ and Y Conversion Equations

There are several equations used to convert one network to the other:

Δ and Y networks are seen frequently in 3-phase AC power systems (a topic covered in volume II of this book series), but even then they’re usually balanced networks (all resistors equal in value) and conversion from one to the other need not involve such complex calculations. When would the average technician ever need to use these equations?

Capacitor Y-Delta and Delta-Y Transforms

Resistors and Inductors follow the same rules for Y-Delta and Delta-Y transforms because they combine similarly in series and parallel. This does not hold true for capacitors.
Through no small feat of algebraic manipulation, it can be shown that the Y-Delta and Delta-Y transforms for capacitors are as follows:

Circuit Diagram

Y-Delta Transformation

Delta-Y Transformation

A Railgun is a gun that fires a projectile (bullets or plasma) using a magnetic field produced by electricity. Unlike regular guns, it does not use gunpowder or other explosive propellants, and rarely needs to be cleaned or unjammed. Currently, most railguns are experimental and are not used in actual combat, but the U.S. Navy is expecting to have one operating soon. In theory, a railgun could fire a bowling-ball-sized projectile fast enough to destroy a small building, even over long distances. Railguns are usually very large and not portable because of the need for a large power supply. There have been proposals to use the same technology for non-weapon purposes, such as aircraft and spacecraft launchers and specialized tools.

The electrical current running down the railgun creates a magnetic field that propels the projectile out of the railgun

The driving force of a railgun power is the magnetic field created when electricity runs through the rails from the positive to negative ends.The more electricity running through the rail the stronger the magnetic field.The railgun projectile is positively charged so it is repulsed by the positive end.In most railguns millions of amperes are ran through the positive end to the negative end.When the positive charge run through the railgun nears and leaves the projectile lorentz force acts on the projectile.This propels it out of the rail gun.

Lorentz force [Nobel Prize for Physics (1902)]

In physics, specifically in electromagnetism, the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields.

A particle of charge \(q\) moving with a velocity \(v\) in an electric field \(\overset\rightharpoonup E\) and a magnetic field \(\overset\rightharpoonup B\) experiences a force of

\(\overset\rightharpoonup F=q\left(\overset\rightharpoonup E+\overset\rightharpoonup v\times\overset\rightharpoonup B\right)\)

Alessandro Volta

The first electric circuit was invented by Alessandro Volta. Alessandro Volta was an Italian scientist and inventor who developed the first battery in 1800.

The Smith chart is a graphical calculator or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. It was independently proposed by Tōsaku Mizuhashi in 1937, and by Amiel R. Volpert [ru] and Phillip H. Smith in 1939.

Impedance

The impedance Z contains reactance X and resistance R denoted by the following formula:

\(Z = R + jX\)

Inductor reactance written as,

\(X_{L} = 2\pi f L\)

Capacitor reactance written as,

\(X_{C} = \frac{1}{2\pi f C}\)

Normalisation

In practice, an Electrical engineer may encounter very small or very large values of resistances and reactances. To include all of them on the same chart would be impractical. To solve this, engineers use a process called "normalisation" when plotting values on the chart.

\(Z_0=50\Omega\)

\(Z' = \frac{R+jX}{50} = \frac{R}{50}+j\frac{X}{50}\)

[Ex] Impedance matching example

Consider a load \(Z_{Load}\) that has impedance in ohms and that this load is connected to a signal generator of 400 MHz.

\(Z_{Load} = 50 - j40\)

\(Z_{Load}' = \frac{50}{50}-j\frac{40}{50} = 1 - j0.8\)

\(X_{L} = X_{L}' \times 50 = 0.8\times50\)

\(X_{L} = 40 Ω\)

\(L=\frac{X_L}{2\pi f}=\frac{40}{2\pi\times400,000,000}\)

\(L\approx16\times10^{-9}=16nH\)

Admittance

\(Z = R + jX\)

\(Y = \frac{1}{Z} = \frac{1}{R + jX} = G + jB\)

We can see that the equation produces a new expression where resistance R is replaced with conductance G and reactance X is replaced with susceptance B which both compute a new quantity called admittance Y.

The conductance G, susceptance B and admittance Y are expressed in siemens or "S".

The susceptance is the reciprocal of reactance which depend on signal frequency in Hertz (Hz) and the inductor/capacitor values, L and C in Henries and Farads respectively.

Inductor susceptance written as B_L

\(B_L=\frac1{X_L}=\frac1{2\pi fL}\)

Capacitor susceptance written as B_C

\(B_C=\frac1{X_C}=2\pi fC\)

Normalisation

Since our measurements can involve very small or very large values, normalisation is necessary in order to fit our data onto the chart. For impedance Z, we divide by a known reference. By convention, the characteristic impedance Zo is used.

\(Z'=\frac Z{Zo}=\frac{R+jX}{Zo}\)

For admittance, it is the same proceedure. However, we divide by the characteristic admittance Yo which is the reciprocal of Zo, i.e. 1 / Zo.

\(Y'=\frac{Y}{Yo}=\frac{G+jB}{Yo}=(G+jB)Zo\)

[Ex]

Y'=1.2-j1.0

Z'=0.5+j0.4

ZY chart

We can use the ZY chart to analyse a circuit with mixture of series and parallel components.

Reading VSWR from the chart

Consider a transmitter connected to a cable with characteristic impedance Z_o = 50 ohms. A load (such as an antenna) with impedance Z = 50 + j0 will present a VSWR of 1 to the transmitter when connected.

It would therefore be the job of the engineer to use the chart to design some sort of matching network so that the final impedance becomes close to as much as possible, 50 + j0.

The Smith chart is a mathematical transformation of the two-dimensional Cartesian complex plane. Complex numbers with positive real parts map inside the circle. Those with negative real parts map outside the circle. If we are dealing only with impedances with non-negative resistive components, our interest is focused on the area inside the circle. The transformation, for an impedance Smith chart, is: \(\Gamma=\frac{Z-Z_0}{Z+Z_0}=\frac{z-1}{z+1},\;z=\frac Z{Z_0}\)

\(\Gamma=\frac{Z_L-Z_0}{Z_L+Z_0}\)

The variation of complex reflection coefficient with position along the line

\(V_F=Ae^{j\omega t}e^{+\gamma l}\)

\(V_R=Be^{j\omega t}e^{-\gamma l}\)

\(e^{j\omega t}\) is the temporal part of the wave

\(e^{\pm\gamma l}\) is the spatial part of the wave

\(l\) is the distance measure along the transmission lime from the load toward the gnerator in metres (m)

\(\gamma=\alpha+j\beta\)

where \(\alpha\) is the attenuation constant in nepers per metre (Np/m)

\(\beta\) is the phase constant in radians per metre (rad/m)

The variation of complex reflection coefficient with position along the line

\(\Gamma=\frac{V_R}{V_F}=\frac{Be^{-\gamma l}}{Ae^{+\gamma l}}=Ce^{-2\gamma l},\;where\;C\;is\;also\;a\;constant.\)

The blue circle, centered within the impedance Smith chart, is sometimes called an SWR circle (short for constant standing wave ratio).

The variation of normalised impedance with position along the line

\(V_F+V_R=V\)

\(V_F-V_R=Z_0I\)

\(\Gamma=\frac{V_R}{V_F}\)

the normalised impedance of the termination \(z=\frac V{Z_0I}\)

\(z=\frac{1+\Gamma}{1-\Gamma}\)

the reflection coefficient \(\Gamma=\frac{z-1}{z+1}\)

These are the equations which are used to construct the Z smith chart. Mathematically speaking \(\Gamma\;and\;z\) are related via a Möbius transformation.

To locate a normalized impedance of z = r + jx, we find the intersection of the corresponding constant-resistance circle and constant-reactance arc.

Normalized Impedance Formula,

\(Z_n=\frac{Z_a}{Z_0}\)

Impedance Angle Formula,

\(\theta=a\tan(\frac XR)\cdot\frac{180}{\mathrm\pi}\)

For example, the impedance z = 0.5 + j2 is marked as point A.

For a short circuit, we substitute z = 0 and obtain Γ = -1, marked as point B.

On the other hand, for an open circuit (z = ∞), Γ works out to +1, denoted as point D.

For an impedance with r = 0, z = j0.5 gives us point C.

With a resistor equal to the normalizing impedance z = 1, we’re at the center of the Smith chart (point E) where Γ = 0.

[Ex 1] Adding a Series Capacitor

With a normalized load impedance of \(z_1=0.5+j\), the reflection coefficient is \(\Gamma_1=0.62\angle82.87^\circ\). If we add to this impedance a 10pF series capacitor (\(C_1=10pF\)), Assume that the operating frequency is \(211.7MHz\) and the reference impedance is \(Z_0=50\Omega\).

At \(211.7MHz\), a \(10pF\) capacitor has a normalized reactance of,

\(jx=-\frac j{2\pi fCZ_0}\)

\(=-\frac j{2\pi\times211.7\times10^6\times10\times10^{-12}\times50}=-1.5j\)

The capacitor reduces the initial reactance by \(1.5j\), moving us from the \(1j\) to \(-0.5j\) constant-reactance arc.

the new impedance \(z_2=0.5-0.5j\). Measuring the length and phase angle of the vector from the origin to \(z_2\), we obtain \(\Gamma_2\;=0.45\angle-116.4^\circ\).

We add a second series \(10pF\) capacitor (\(C_2=10pF\)) to \(z_2\),

he new capacitor reduces the reactance by another \(1.5j\). Therefore, we end up at point \(z_3=0.5-2j\).

[Ex 2] Sweeping the Value of a Series Capacitor

If we add a series capacitor \(C_s\) to a normalized load impedance of \(z_1=0.5+j\) and sweep the capacitor value from \(+∞\) to \(5pF\)

by sweeping the value of the series capacitor from a large value to a smaller value, the overall impedance moves along the constant-resistance circle in a counterclockwise direction. Or we can say that increasing the series capacitor moves us along the constant-resistance circle in a clockwise direction

[Ex 3] Adding a Series Inductor

Assume that the operating frequency is \(211.7MHz\) and \(Z_0=50Ω\). If we add a \(37.58nH\) series inductor to \(z_1=0.5+j\),

\(jx=\frac{j2\pi fL}{Z_0}\)

\(=\frac{j2\pi\times211.7\times10^6\times37.58\times10^{-9}}{50}=1j\)

This moves us to \(z_2=0.5+2j\).

Adding an additional inductor of \(3×37.58nH=112.74nH\) produces an equivalent normalized impedance of \(0.5+5j\).

[Ex 4] A Series RC Network Frequency Sweep

Assume that \(R=25Ω\), \(C=10pF\), and \(Z_0=50Ω\). The impedance of a series RC circuit is:

\(Z=R+\frac1{j\omega C}=R-\frac1{\omega C}j\)

The reactive component of the normalized impedance is \(x=-\frac1{\omega CZ_0}\), which is always negative.

As the frequency is swept from DC to infinity, x changes from \(-∞\) to \(0\).

the blue arc

[Ex 5] A Series RC Network Frequency Sweep

The reactive component of the normalized impedance is \(x=\frac{L\omega}{Z_0}\), which is always positive.

As the frequency is swept from DC to infinity, x changes from \(0\) to \(+∞\).

the blue arc

[Ex 6] A Series RLC Network Frequency Sweep

A series RLC circuit with \(R=10Ω\) and \(L=20nH\), and \(C=2pF\)

With \(Z_0=50Ω\), the impedance of this series RLC network is on the constant-resistance circle of \(r = 0.2\)

At the resonant frequency \(f_r\), the reactance of the inductor cancels that of the capacitor.

the blue circle

When \(0＜f＜f_r\) the circuit is capacitive, the magnitude of the capacitive reactance is larger than the inductive reactance. We are at the bottom half of the constant-resistance circle.

When \(f_r＜f＜\infty\) the circuit is inductive, we are at the upper half of the constant-resistance circle.

When impedance mismatch exists between any load (e.g. antenna) and the rest of the system, power from the transmitter flows to the antenna and a fraction of the power is reflected back towards the transmitter. These forward and reflected waves interfere with each other to produce standing waves along the transmission line. By looking at the maximum and minimum voltage amplitude of the standing wave, we can calculate the VSWR.

Reflection coefficient

\(\Gamma=\frac{Z_L-Zo}{Z_L+Zo}\)

Once we know the reflection coefficient, Γ we can calculate VSWR through this equation. The |Γ| represents the magnitude only (in the above table, both"+1" and "-1" are equivalent to "1").

\(VSWR=\frac{1+\vert\Gamma\vert}{1-\vert\Gamma\vert}\)

The VSWR ranges from 1 or 1:1 (ideal) to infinity. When the load or antenna impedance, Z (whether real or complex) changes we get different values of VSWR that fall in this range.

Two-port network

In electronics, a two-port network (a kind of four-terminal network or quadripole) is an electrical network (i.e. a circuit) or device with two pairs of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them satisfy the essential requirement known as the port condition: the current entering one terminal must equal the current emerging from the other terminal on the same port.

Examples include small-signal models for transistors (such as the hybrid-pi model), filters and matching networks. The analysis of passive two-port networks is an outgrowth of reciprocity theorems first derived by Lorentz.

The parameters used to describe a two-port network are the following: z, y, h, g. Each choice corresponds to a different choice for which pair of variables from port 1 and port 2 are chosen to be independent, externally applied sources and which two will be the dependent resultant variables determine by the two-port parameters (see the figure). The port currents and voltages are denoted as follows:

V₁ = Port 1 voltage

I₁  = Port 1 current

V₂ = Port 2 voltage

I₂  = Port 2 current

The relations between inputs and outputs usually are expressed in matrix notation.

Impedance parameters (z-parameters)

The figure shows the two-port driven by two external current sources, making the input currents I₁ and I₂ the independent variables controlled from outside the two-port. By convention, dependent sources (controlled by the independent sources) are depicted using diamond shapes. The port voltages are determined in terms of these input currents by the z-parameters defined by:

\(\begin{bmatrix}V_1\\V_2\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}\\z_{21}&z_{22}\end{bmatrix}\begin{bmatrix}I_1\\I_2\end{bmatrix}\)

\(z_{11}={\left.\frac{V_1}{I_1}\right|}_{I_2=0}\qquad z_{12}={\left.\frac{V_1}{I_2}\right|}_{I_1=0}\)

\(z_{21}={\left.\frac{V_2}{I_1}\right|}_{I_2=0}\qquad z_{22}={\left.\frac{V_2}{I_2}\right|}_{I_1=0}\)

Notice that all the series connected elements represented by z-parameters have dimensions of ohms, as do the dependent source parameters.

Admittance parameters (y-parameters)

The figure shows the two-port driven by two external voltage sources, making the input voltages V₁ and V₂ the independent variables controlled from outside the two-port. The port currents are determined in terms of these input voltages by the y-parameters defined by:

\(\begin{bmatrix}I_1\\I_2\end{bmatrix}=\begin{bmatrix}y_{11}&y_{12}\\y_{21}&y_{22}\end{bmatrix}\begin{bmatrix}V_1\\V_2\end{bmatrix}\)

\(y_{11}={\left.\frac{I_1}{V_1}\right|}_{V_2=0}\qquad y_{12}={\left.\frac{I_1}{V_2}\right|}_{V_1=0}\)

\(y_{21}={\left.\frac{I_2}{V_1}\right|}_{V_2=0}\qquad y_{22}={\left.\frac{I_2}{V_2}\right|}_{V_1=0}\)

The network is said to be reciprocal if \(y_{12}=y_{21}\). Notice that all the shunt-connected elements are represented by y-parameters with dimensions of siemens, as are the dependent source parameters.

Hybrid parameters (h-parameters)

The figure shows the two-port driven by two external sources, a current source at port 1 and a voltage source at port 2, making the input current I₁ and input voltage V₂ the independent variables controlled from outside the two-port. The voltage at port 1, V₁, and the current at port 2, I₂, are determined in terms of these inputs by the h-parameters defined by:

\(\begin{bmatrix}V_1\\I_2\end{bmatrix}=\begin{bmatrix}h_{11}&h_{12}\\h_{21}&h_{22}\end{bmatrix}\begin{bmatrix}I_1\\V_2\end{bmatrix}\)

\(h_{11}={\left.\frac{V_1}{I_1}\right|}_{V_2=0}\qquad h_{12}={\left.\frac{V_1}{V_2}\right|}_{I_1=0}\)

\(h_{21}={\left.\frac{I_2}{I_1}\right|}_{V_2=0}\qquad h_{22}={\left.\frac{I_2}{V_2}\right|}_{I_1=0}\)

Often this circuit is selected when a current amplifier is described, because the port 1 input is the independent input current and port 2 output is the dependent current.

Notice that off-diagonal h-parameters are dimensionless, while the series-connected diagonal element has dimensions of ohms, while the shunt-connected diagonal element has dimensions of siemens.

\(V_1 = h_{11} I_1 + h_{12} V_2\)

\(I_2 = h_{21} I_1 + h_{22} V_2\)

\(h_{11} = \frac{V_1}{I_1},\: when\: V_2 = 0\)

\(h_{12} = \frac{V_1}{V_2},\: when\: I_1 = 0\)

\(h_{21} = \frac{I_2}{I_1},\: when\: V_2 = 0\)

\(h_{22} = \frac{I_2}{V_2},\: when\: I_1 = 0\)

Inverse hybrid parameters (g-parameters)

The figure shows the two-port driven by two external sources, a voltage source at port 1 and a current source at port 2, making the input voltage V₁ and input current I₂ the independent variables controlled from outside the two-port. The current at port 1, I₁, and the voltage at port 2, V₂, are determined in terms of these inputs by the g-parameters defined by:

\(\begin{bmatrix}I_1\\V_2\end{bmatrix}=\begin{bmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{bmatrix}\begin{bmatrix}V_1\\I_2\end{bmatrix}\)

\(g_{11}={\left.\frac{I_1}{V_1}\right|}_{I_2=0}\qquad g_{12}={\left.\frac{I_1}{I_2}\right|}_{V_1=0}\)

\(g_{21}={\left.\frac{V_2}{V_1}\right|}_{I_2=0}\qquad g_{22}={\left.\frac{V_2}{I_2}\right|}_{V_1=0}\)

Often this circuit is selected to describe a voltage amplifier, as the port 1 input is an independent voltage, and the port 2 output is a dependent voltage.

Notice that off-diagonal g-parameters are dimensionless, while the series-connected diagonal element has dimensions of ohms, while the shunt-connected diagonal element has dimensions of siemens.

Scattering parameters (S-parameters)

The previous parameters are all defined in terms of voltages and currents at ports. S-parameters are different, and are defined in terms of incident and reflected waves at ports. S-parameters are used primarily at UHF and microwave frequencies where it becomes difficult to measure voltages and currents directly. On the other hand, incident and reflected power are easy to measure using directional couplers. The definition is,

\(\begin{bmatrix}b_1\\b_2\end{bmatrix}=\begin{bmatrix}S_{11}&S_{12}\\S_{21}&S_{22}\end{bmatrix}\begin{bmatrix}a_1\\a_2\end{bmatrix}\)

where the a_k are the incident waves and the b_k are the reflected waves at port k. It is conventional to define the a_k and b_k in terms of the square root of power. Consequently, there is a relationship with the wave voltages.

For reciprocal networks S₁₂ = S₂₁. For symmetrical networks S₁₁ = S₂₂. For antimetrical networks S₁₁ = –S ₂₂. For lossless reciprocal networks \(|S_{11}|=|S_{22}|\) and \(|S_{11}|^{2}+|S_{12}|^{2}=1.\)

\(R_1(I_1-I_3)+R_2I_1+R_3(I_1-I_2)=0\;(V)\)

\(R_3(I_2-I_1)+R_4(I_2-I_3)+R_5I_2=0\;(V)\)

\(R_1(I_3-I_1)+R_4(I_3-I_2)=450\;(V)\)

\(150(I_1-I_3)+50I_1+100(I_1-I_2)=0\;\Rightarrow3(I_1-I_3)+I_1+2(I_1-I_2)=0\)

\(100(I_2-I_1)+300(I_2-I_3)+250I_2=0\;\Rightarrow2(I_2-I_1)+6(I_2-I_3)+5I_2=0\)

\(150(I_3-I_1)+300(I_3-I_2)=450\;\Rightarrow(I_3-I_1)+2(I_3-I_2)=3\)

\(6I_1-2I_2-3I_3=0\)

\(-2I_1+13I_2-6I_3=0\)

\(-I_1-2I_2+3I_3=3\)

\(\begin{bmatrix}6&-2&-3\\-2&13&-6\\-1&-2&3\end{bmatrix}\begin{bmatrix}I_1\\I_2\\I_3\end{bmatrix}=\begin{bmatrix}0\\0\\3\end{bmatrix}\)

\(\begin{bmatrix}I_1&I_2&I_3\end{bmatrix}=\begin{bmatrix}\frac{51}{29}&\frac{42}{29}&\frac{74}{29}\end{bmatrix}\)

Balanced Wheatstone bridge

In the figure, R_x is the fixed, yet unknown, resistance to be measured. R₁, R₂, and R₃ are resistors of known resistance and the resistance of R₂ is adjustable. The resistance R₂ is adjusted until the bridge is "balanced" and no current flows through the galvanometer V_g. At this point, the potential difference between the two midpoints (B and D) will be zero. Therefore the ratio of the two resistances in the known leg (R₂ / R₁) is equal to the ratio of the two resistances in the unknown leg (R_x / R₃). If the bridge is unbalanced, the direction of the current indicates whether R ₂ is too high or too low.

At the point of balance,

\(\frac{R_2}{R_1}=\frac{R_x}{R_3}\)

\(\Rightarrow R_x=\frac{R_2}{R_1}\cdot R_3\)

Buck Converter

A buck converter or step-down converter is a DC-to-DC converter which decreases voltage, while increasing current, from its input (supply) to its output (load).

The basic concept of a buck converter is:

1. Use the higher-than-needed voltage of the source to quickly induce a current into an inductor.

2. Disconnect the source and use the inertia of the current in the inductor to provide more current than the source delivers. To close the circuit with the source disconnected, a second switch, usually a diode, is needed.

Continuous mode

Buck converters operate in continuous mode if the current through the inductor \(({\displaystyle I_{\text{L}}})\) never falls to zero during the commutation cycle.

(1) When the switch is closed, the voltage across the inductor is \({\displaystyle V_{\text{L}}=V_{\text{i}}-V_{\text{o}}}\). The current through the inductor rises linearly. During this time, the inductor stores energy in the form of a magnetic field.

(2) When the switch is opened, the diode is forward biased. The voltage across the inductor is \({\displaystyle V_{\text{L}}=-V_{\text{o}}}\) (neglecting diode drop). Current \({\displaystyle I_{\text{L}}}\) decreases.

Evolution with time of the voltages \({\displaystyle V_{o},V_{D},V_{L}}\) and the current \({\displaystyle I_{L}}\) in an ideal buck converter operating in continuous mode.

he energy stored in inductor L is, \(E=\frac12LI_L^2\)

\(V_L=L\frac{dI_L}{dt}\)

With \({\displaystyle V_{\text{L}}}\) equal to \({\displaystyle V_{\text{i}}-V_{\text{o}}}\) during the on-state and to \({\displaystyle -V_{\text{o}}}\) during the off-state.

\({\displaystyle {\begin{aligned}\Delta I_{L_{\text{on}}}&=\int _{0}^{t_{\text{on}}}{\frac {V_{\text{L}}}{L}}\,\mathrm {d} t={\frac {V_{\text{i}}-V_{\text{o}}}{L}}t_{\text{on}},&t_{\text{on}}&=DT\end{aligned}}}\)

\({\displaystyle {\begin{aligned}\Delta I_{L_{\text{off}}}&=\int _{t_{\text{on}}}^{T=t_{\text{on}}+t_{\text{off}}}{\frac {V_{\text{L}}}{L}}\,\mathrm {d} t=-{\frac {V_{\text{o}}}{L}}t_{\text{off}},&t_{\text{off}}&=(1-D)T\end{aligned}}}\)

where \(D\) is a scalar called the duty cycle with a value between 0 and 1.

Assuming that the converter operates in the steady state, the energy stored in each component at the end of a commutation cycle T is equal to that at the beginning of the cycle. That means that the current \({\displaystyle I_{\text{L}}}\) is the same at \({\displaystyle t=0}\) and at \({\displaystyle t=T}\),

\({\displaystyle {\begin{aligned}\Delta I_{L_{\text{on}}}+\Delta I_{L_{\text{off}}}&=0\\{\frac {V_{\text{i}}-V_{\text{o}}}{L}}t_{\text{on}}-{\frac {V_{\text{o}}}{L}}t_{\text{off}}&=0\end{aligned}}}\)

\({\displaystyle t_{\text{on}}=DT}\) and \({\displaystyle t_{\text{off}}=(1-D)T}\)

\({\displaystyle {\begin{aligned}\left(V_{\text{i}}-V_{\text{o}}\right)DT-V_{\text{o}}(1-D)T&=0\\DV_{\text{i}}-V_{\text{o}}&=0\\{}\Rightarrow D&={\frac {V_{\text{o}}}{V_{\text{i}}}}\end{aligned}}}\)

From this equation, it can be seen that the output voltage of the converter varies linearly with the duty cycle for a given input voltage. As the duty cycle \(D\) is equal to the ratio between \({\displaystyle t_{\text{on}}}\) and the period \(T\), it cannot be more than 1. Therefore, \({\displaystyle V_{\text{o}}\leq V_{\text{i}}}\). This is why this converter is referred to as step-down converter.

Discontinuous mode

In some cases, the amount of energy required by the load is too small. In this case, the current through the inductor falls to zero during part of the period. The only difference in the principle described above is that the inductor is completely discharged at the end of the commutation cycle

We still consider that the converter operates in steady state. Therefore, the energy in the inductor is the same at the beginning and at the end of the cycle (in the case of discontinuous mode, it is zero). This means that the average value of the inductor voltage (\(V_L\)) is zero

\({\displaystyle \left(V_{\text{i}}-V_{\text{o}}\right)DT-V_{\text{o}}\delta T=0}\)

\({\displaystyle \delta ={\frac {V_{\text{i}}-V_{\text{o}}}{V_{\text{o}}}}D}\)

The output current delivered to the load (\(I_o\)) is constant, as we consider that the output capacitor is large enough to maintain a constant voltage across its terminals during a commutation cycle. This implies that the current flowing through the capacitor has a zero average value.

\({\displaystyle {\overline {I_{\text{L}}}}=I_{\text{o}}}\)

Where \({\displaystyle {\overline {I_{\text{L}}}}}\) is the average value of the inductor current.

\({\displaystyle {\begin{aligned}{\overline {I_{\text{L}}}}&=\left({\frac {1}{2}}I_{L_{\text{max}}}DT+{\frac {1}{2}}I_{L_{\text{max}}}\delta T\right){\frac {1}{T}}\\&={\frac {1}{2}}I_{L_{\text{max}}}\left(D+\delta \right)\\&=I_{\text{o}}\end{aligned}}}\)

The inductor current is zero at the beginning and rises during \(t_{on}\) up to \(IL_{max}\). That means that \(IL_{max}\) is equal to:

\({\displaystyle I_{L_{\text{max}}}={\frac {V_{\text{i}}-V_{\text{o}}}{L}}DT}\)

Substituting the value of \(IL_{max}\) in the previous equation leads to:

\({\displaystyle I_{\text{o}}={\frac {\left(V_{\text{i}}-V_{\text{o}}\right)DT\left(D+\delta \right)}{2L}}}\)

And substituting δ by the expression given above yields:

\({\displaystyle I_{\text{o}}={\frac {\left(V_{\text{i}}-V_{\text{o}}\right)DT\left(D+{\frac {V_{\text{i}}-V_{\text{o}}}{V_{\text{o}}}}D\right)}{2L}}}\)

This expression can be rewritten as:

\({\displaystyle V_{\text{o}}=V_{\text{i}}{\frac {1}{{\frac {2LI_{\text{o}}}{D^{2}V_{\text{i}}T}}+1}}}\)

It can be seen that the output voltage of a buck converter operating in discontinuous mode is much more complicated than its counterpart of the continuous mode.

Boost Converter

Boost converters are a type of DC-DC switching converter that efficiently increase (step-up) the input voltage to a higher output voltage. By storing energy in an inductor during the switch-on phase and releasing it to the load during the switch-off phase, this voltage conversion is made possible. Power electronics applications requiring a greater output voltage than the input source, in particular, depend on boost converters.

It is a class of switched-mode power supply (SMPS) containing at least two semiconductors, a diode and a transistor, and at least one energy storage element: a capacitor, inductor, or the two in combination.

Continuous mode

When a boost converter operates in continuous mode, the current through the inductor (\({\displaystyle I_{L}}\)) never falls to zero.

The left-hand side of L is at \(V_i\), and the right-hand side of L sees the \(V_s\). The direction of the inductor's voltage reverses when switch turns on and off.

Switch-on period

During this stage, the input voltage (\(V_i\)) is applied across the inductor (\(L\)), causing the current through the inductor to increase linearly. The inductor current can be expressed as:

\(\Delta I_L=\frac{V_i}L\cdot t_{on}\)

Switch-off period

When the switch opens, the inductor current must continue to flow. This forces the diode \(D\) to become forward-biased, and the inductor releases its stored energy to the load (\(R\)) and the output capacitor (\(C\)).

During this period, the voltage across the inductor (\(V_L\)) is equal to the difference between the output voltage (\(V_o\)) and the input voltage (\(V_i\)). The inductor current decreases linearly as the energy is transferred to the load, and the equation for the inductor current becomes:

\(\Delta I_L=\frac{(V_o-V_i)}L\cdot t_{off}\)

By equating the inductor current equations for both stages and rearranging the terms, we can derive the voltage conversion relationship for the boost converter:

\(\frac{V_i}{L}\cdot t_{on}=\frac{(V_o-V_i)}{L}\cdot t_{off}\)

\(V_i\cdot D=(V_o-V_i)\cdot(1-D)\)

\(V_i=(1-D)V_o\)

\(V_o=\frac{V_o}{(1-D)}\)

Rearranging the equation reveals the duty cycle to be:

\(D=\frac{t_{on}}{(t_{on}+T_{off})}=1-\frac{V_i}{V_o}\)

Buck-Boost Converter

The buck–boost converter is a type of DC-to-DC converter that has an output voltage magnitude that is either greater than or less than the input voltage magnitude. It is equivalent to a flyback converter using a single inductor instead of a transformer.

\((S_+=E\cdot T_{ON})=-(S_-=U\cdot T_{OFF})\Rightarrow U=-E\cdot\frac{\displaystyle T_{ON}}{\displaystyle T_{OFF}}\)

\(\frac{\operatorname dI_L}{\operatorname dt}=\frac{V_i}L\)

\(\Delta I_{L_{ON}}=\frac{V_i\,}LDT\)

\(\frac{\operatorname dI_L}{\operatorname dt}=\frac{V_o}L\)

\(\Delta I_{L_{OFF}}=\frac{V_o}L\left(1-D\right)T\)

\(\Delta I_{L_{ON}}+\Delta I_{L_{OFF}}=\frac{V_i\,D\,T}L+\frac{V_o\left(1-D\right)T}L=0\)

\(\frac{V_o}{V_i}=-\frac D{1-D}\)

\(D=\frac{V_o}{V_o-V_i}\)

Ćuk converter

A Ćuk converter can be seen as a combination of boost converter and buck converter, having one switching device and a mutual capacitor, to couple the energy.

Similar to the buck-boost converter with inverting topology, the output voltage of non-isolated Ćuk converter is typically inverted, with lower or higher values with respect to the input voltage. While DC-to-DC converters usually use the inductor as a main energy-storage component, the Ćuk converter instead uses the capacitor as the main energy-storage component. It is named after Slobodan Ćuk of the California Institute of Technology, who first presented the design.

\(V_L=L\frac{dI}{dt},\)

\({\overline V}_{L1}=D\cdot V_s+\left(1-D\right)\cdot\left(V_s-V_C\right)=\left(V_s-(1-D)\cdot V_C\right)\)

\({\overline V}_{L2}=D\left(V_o+V_C\right)+\left(1-D\right)\cdot V_o=\left(V_o+D\cdot V_C\right)\)

As both average voltage have to be zero to satisfy the steady-state conditions, using the last equation we can write:

\(V_C=-\frac{V_o}D\)

\({\overline V}_{L1}=\left(V_s+(1-D)\cdot\frac{V_o}D\right)=0\)

\(\therefore\frac{V_o}{V_s}=-\frac D{1-D}\)

\(U=\frac D{1-D}\cdot E=E\cdot\frac{T_{ON}}{T_{OFF}}\)

A voltage doubler is an electronic circuit which charges capacitors from the input voltage and switches these charges in such a way that, in the ideal case, exactly twice the voltage is produced at the output as at its input.

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What is meant by an inverting and non-inverting "input" for op amplifiers?

Inverting and Non Inverting input relates to the action of the input pin on the output of the device. If you apply a +ve signal to the non-inverting input, the output will go positive. If you apply a +ve signal to the inverting input, the output will go negative. This applies to both Logic and Analog circuits. In Analog Operation Amplifiers there are both Inverting and non-inverting inputs; these operate differentially with negative feedback to create the very valuable and accurate operation of these incredible devices.

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∮ ∯ ∰ ∫∬ ∭⨌ ＞ ＜ ⋉

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