電路學 Electrical Circuits

Mathematics   Yoshio    Sep 8th, 2023 at 8:00 PM    8    0   

電路學

Electrical Circuits


Passive device

Active device

Impedance 阻抗

The real part of impedance is resistance(電阻), while the imaginary part is reactance(電抗). Ideal inductors and capacitors have a purely imaginary reactive impedance:

電感 Inductance

The impedance of inductors increases as frequency increases,
\(Z_L={\mathrm{jω}L}\)


The inductance of an electric circuit is one henry when an electric current that is changing at one ampere per second results in an electromotive force of one volt across the inductor:

\[v=L\frac{di}{dt}\]

\[i(t)=\frac1L\int_0^Tv(t)dt+i_0\]

where V(t) is the resulting voltage across the circuit, I(t) is the current through the circuit, and L is the inductance of the circuit.

\(H=\frac{Kg\cdot m^2}{s^2\cdot A^2}=\frac{N\cdot m}{A^2}=\frac{Kg\cdot m^2}{C^2}=\frac J{A^2}=\frac{T\cdot m^2}A=\frac{Wb}A=\frac{V\cdot s}A=\frac{s^2}F=\frac\Omega{Hz}=\Omega\cdot s\)

where: H = henry, kg = kilogram, m = metre, s = second, A = ampere, N = newton, C = coulomb, J = joule, T = tesla, Wb = weber, V = volt, F = farad, Ω = ohm, Hz = hertz


Lenz's law and Faraday's law are both related to the direction of an induced current and the magnitude of the induced electromotive force (EMF). Lenz's law states that the direction of the induced current is always opposite to the change in flux that produced it. Faraday's law states that the magnitude of the induced EMF is directly proportional to the rate of change in the inducing magnetic field.


電容 Capacitance

The impedance of capacitors decreases as frequency increases,
\(Z_C=\frac1{\mathrm{jω}C}\)


The capacitance of a capacitor is one farad when one coulomb of charge changes the potential between the plates by one volt. Equally, one farad can be described as the capacitance which stores a one-coulomb charge across a potential difference of one volt

\[C=\frac QV\]

\[i=C\frac{dv}{dt}\]

\[v(t)=\frac1C\int_0^Ti(t)dt+v_0\]

\(F=\frac{s^4\cdot A^2}{m^2\cdot kg}=\frac{s^4\cdot C^2}{m^2\cdot kg}=\frac CV=\frac{A\cdot s}V=\frac{W\cdot s}{V^2}=\frac J{V^2}=\frac{N\cdot m}{V^2}=\frac{C^2}J=\frac{C^2}{N\cdot m}=\frac s\Omega=\frac1{\Omega\cdot Hz}=\frac S{Hz}=\frac{s^2}H\)

where: F = farad, C = coulomb, V = volt, W = watt, J = joule, N = newton, Ω = ohm, Hz = Hertz, S = siemens, H = henry, A = ampere.



Electrical Impedance

Alternating current, magnitude and phase

When an alternating current is being used the ratio \(\frac VI\) is not necessarily constant. This is because the voltage and current can peak at different times if the circuit contains components like coils or capacitors.

Impedance Z measures the ratio of the peak voltage to the peak current:

\(Z=\frac{V_{peak}}{I_{peak}}\)

The unit of Z is the ohm (Ω).

Sometimes we break the impedance down into two components. One part has the voltage and current in phase (peaking at the same time), and is the resistance, \(R\). The other part has the current peaking one quarter of a cycle after the voltage, and this is called reactance, \(X\). The impedance is the vector sum of the two:

\(Z=\sqrt{R^2+X^2}\)

The reactance of an inductor is positive \(X_L=\omega L\) and depends on the angular frequency \(\omega=2\pi f\) of the alternating current. The reactance of a capacitor is negative \(X_C=-\frac1{\omega C}\), showing that for a capacitor the current peaks one quarter of a cycle before the voltage.

In more advanced work it is convenient to write the impedance as a complex number with the resistance as the real part and the reactance as the imaginary part \(Z=R+iX\).





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
Time
Constant
RC Value Percentage of Maximum
Voltage Current
0.5 time constant 0.5T = 0.5RC 39.3% 60.7%
0.7 time constant 0.7T = 0.7RC 50.3% 49.7%
1.0 time constant 1T = 1RC 63.2% 36.8%
2.0 time constants 2T = 2RC 86.5% 13.5%
3.0 time constants 3T = 3RC 95.0% 5.0%
4.0 time constants 4T = 4RC 98.2% 1.8%
5.0 time constants 5T = 5RC 99.3% 0.7%

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

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

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

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

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

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

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

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

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

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


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



Railgun

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.


Thévenin's theorem & Norton's theorem

Thévenin's theorem

Thévenin's theorem states that "Any linear electrical network containing only voltage sources, current sources and resistances can be replaced at terminals A–B by an equivalent combination of a voltage source \(V_{th}\) in a series connection with a resistance \(R_{th}\)."

Norton's theorem

In direct-current circuit theory, Norton's theorem, also called the Mayer–Norton theorem, is a simplification that can be applied to networks made of linear time-invariant resistances, voltage sources, and current sources. At a pair of terminals of the network, it can be replaced by a current source and a single resistor in parallel.



Nobel Laureates in Physics
Hendrik Lorentz (1902)
Marie Curie (1903)
Lawrence Bragg (1915)
Max Planck (1918)
Albert Einstein (1921)
Niels Bohr (1922)
Arthur Compton (1927)
C.T.R. Wilson (1927)
Owen Richardson (1928)
Louis de Broglie (1929)
Werner Heisenberg (1932)
Paul Dirac (1933)
Erwin Schrödinger (1933)
Wolfgang Pauli (1945)
Max Born (1954)

Nobel Laureates in Chemistry
Marie Curie (1911)
Irving Langmuir (1932)
Peter Debye (1936)


bandpass cutoff frequency, bandwidth of the entire filter, center frequency \(f_0\)



AM and FM filters


Fleming's hand rule


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

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

\(\mathrm{\mathit{Q}_{0}-factor=\frac{\mathit{R}}{\omega_{0}\mathit{L}}=\mathit{R}\frac{\sqrt{\mathit{LC}}}{\mathit{L}}=\mathit{R}\sqrt{\frac{\mathit{C}}{\mathit{L}}}}\)






Kirchhoff's Circuit Laws

克希荷夫電路定律

Kirchhoff's Current Law (KCL)

克希荷夫電流定律
Kirchhoff's first law, or Kirchhoff's junction rule

\(\overset n{\underset{k=1}{\sum i_k}}=0\)

The algebraic sum of currents in a network of conductors meeting at a point is zero.


Kirchhoff's voltage law (KVL)

克希荷夫電壓定律
Kirchhoff's second law, or Kirchhoff's loop rule

\(\overset m{\underset{k=1}{\sum v_k}}=0\)

The directed sum of the potential differences (voltages) around any closed loop is zero.



Why AC is three-phase offset by 120 degrees?

Being 120 degrees apart makes the phases balanced such that power transfer at any instant is a constant. If you had phases 'closer together' as you suggest, there wouldn't be any real advantage over single phase power.



Among the benefits that 3-phase power brings is the ability to deliver nearly twice the power of single-phase systems without requiring twice the number of wires. It’s not three times as much power, as one might expect, because in practice, you typically take one hot line and connect it to another hot line.





In physics, this sort of addition occurs when sinusoids interfere with each other, constructively or destructively. The static vector concept provides useful insight into questions like this: "What phase difference would be required between three identical sinusoids for perfect cancellation?" In this case, simply imagine taking three vectors of equal length and placing them head to tail such that the last head matches up with the first tail. Clearly, the shape which satisfies these conditions is an equilateral triangle, so the angle between each phasor to the next is 120°(\(2π⁄3\) radians), or one third of a wavelength \(λ⁄3\). So the phase difference between each wave must also be 120°, as is the case in three-phase power.

In other words, what this shows is that:

\(\cos(\omega t)+\cos(\omega t+\frac{2\mathrm\pi}3)+\cos(\omega t-\frac{2\mathrm\pi}3)=0\)

In the example of three waves, the phase difference between the first and the last wave was 240°, while for two waves destructive interference happens at 180°. In the limit of many waves, the phasors must form a circle for destructive interference, so that the first phasor is nearly parallel with the last. This means that for many sources, destructive interference happens when the first and last wave differ by 360 degrees, a full wavelength \(\lambda\). This is why in single slit diffraction, the minima occur when light from the far edge travels a full wavelength further than the light from the near edge.



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波德圖(Bode Chart)頻域分析

史密斯圖(Smith Chart)


Point identity Reflection coefficient (polar form) Normalised impedance (rectangular form)
\(P_1\) (Inductive) \(0.63\angle60^\circ\) \(0.80+j1.40\)
\(P_2\) (Inductive) \(0.73\angle125^\circ\) \(0.20+j0.50\)
\(P_3\) (Capacitive) \(0.44\angle-116^\circ\) \(0.50-j0.50\)

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




電晶體Π模型


電晶體H型模型


What is a dB, dBm, dBu, dBc


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