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

## 電路學

Electrical CircuitsPassive 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.