Capacitors and Inductors

Characteristics	Capacitor	Inductor
Unit	Farad (F)	Henry (H)
Voltage \(V\)	\(V_C = \frac{1}{C} \int i\ dt\)	\(V_L = L\frac{di}{dt}\)
Current \(i\)	\(i_C = C\frac{dv}{dt}\)	\(i_L = \frac{1}{L} \int Vdt\)
Prevents jumps	Of voltage	Of current
Series	\(\sum \frac{1}{C_n}\)	\(\sum L_n\)
Parallel	\(\sum C_n\)	\(\sum \frac{1}{L_n}\)
Time constant \(\tau\)	\(\tau_C = RC\)	\(\tau_L = \frac{L}{R}\)
Impedance	\(\Z_C = \frac{1}{j\omega C}\)	\(\Z_L = j\omega L\)
DC (\(\omega = 0\))	\(\Z_C = \infty\) (open)	\(\Z_L = 0\) (short)
AC (\(\omega\) high)	\(\Z_C \approx 0\) (short)	\(\Z_C \approx \infty\) (open)
Phasor sine wave	Leads	Lags

General Forms for C and L:

\[V = V(\infty) + (V(0)^+ - V(\infty)) e^{\frac{-t}{\tau}}\] \[i = i(\infty) + (i(0)^+ - i(\infty)) e^{\frac{-t}{\tau}}\]

DC Example

w7

AC Example

w9

Phasors

Phasor sources are in cosines:

\[\sin \theta = \cos(\theta - \frac{\pi}{2}) = \cos(\theta - 90°)\] \[v(t) = V_m \cos(\omega t + \theta),\ \ i(t) = I_m \cos(\omega t + \theta);\] \[v(t) = V_m \angle \theta = V_m (\cos \theta + j \sin \theta)\]

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\[A + jB = C\cos(\omega t + \theta)\] \[C = \sqrt{A^2 + B^2}\] \[\theta = \arctan \frac{B}{A}\]

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Impedance

\[Z = R + jX\]

Where R is the resistance and X is the reactance.

Root Mean Square (RMS)

Also known as effective voltage and current, it represents the power an AC source delivers to a load (resistor).

\[V_{rms} = \frac{V_L}{\sqrt{2}}\] \[I_{rms} = \frac{I_L}{\sqrt{2}}\]

For example:

• The U.S. uses AC 120V 60Hz, providing the same power as a DC 120V source.
• China uses AC 220V 50Hz, providing the same power as a DC 220V source.

If a load R is connected to various current phases \(I_n\), then:

\[I_{RMS} = \sqrt{\sum I^2_n}\]

Average Power

Resistors:

\[P_{Ravg} = \frac{1}{2} V_R I_R \cos \theta\] \[P_{Ravg} = V_{rms} I_{rms} \cos \theta\]

Where \(V_R, I_R\) is the amplitude of the AC voltage/current.

Capacitors & Inductances: Average power \(P_{avg} = 0\)

Apparent Power: \(P_{app} = V_{rms} I_{rms}\)

Maximum Power Transfer

Assume a Thévenin Circuit with AC volt source \(V_{TH}\) in series with an impedance \(\Z_{TH}\) and a load \(\Z_{L}\).

If \(\Z_L = \Z^*_{TH},\ \Z_L\) receives maximum power.

This means the resistances are the same, and the reactances have opposite sign:

\[R_L = R_{TH}\] \[X_L = -X_{TH}\]

Complex Power

Average Power \(P = V_{rms} I_{rms} \cos(\theta-\phi)\)

Reactive Power \(Q = V_{rms} I_{rms} \sin(\theta-\phi)\)

\[S = P + jQ = V_{rms} I^*_{rms} = V_{rms} I_{rms} e^{j(\theta - \phi)}\]

Power Table

Power	Equation	Description
Average \(P\)	\(V_{rms} I_{rms} \cos(\theta)\)	Real (Watts)
Reactive \(Q\)	\(V_{rms} I_{rms} \sin(\theta)\)	Exchange with C, L
Complex \(S\)	\(P + jQ = V_{rms} I_{rms}^*\)	Real + Reactive
Apparent \(S\)	\(V_{rms} I_{rms}\)	Total magnitude

Phasor Example

Complex power for:

• \(R1: 26.6\) VA
• \(C1: -j1331\) VA
• \(R2 + L1: 532 + j1065\) VA
• \(V1: -559 + j266\) VA

pg 453

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S-domain

Laplace transform table

f(t)	F(s) = L(f(t))
Unit step function, \(u(t)\)	\(\frac{1}{s}\)
Dirac delta function, \(\delta(t)\)	\(1\)
Ramp function, \(t\ u(t)\)	\(\frac{1}{s^2}\)
\(e^{-at}\ u(t)\)	\(\frac{1}{s + a}\)
\(t\ e^{-at}\ u(t)\)	\(\frac{1}{(s + a)^2}\)
\(t^n\ u(t)\) (n = integer)	\(\frac{n!}{s^{n+1}}\)
\(\sin(\omega t)\ u(t)\)	\(\frac{\omega}{s^2 + \omega^2}\)
\(\cos(\omega t)\ u(t)\)	\(\frac{s}{s^2 + \omega^2}\)
\(e^{-at} \sin(\omega t)\ u(t)\)	\(\frac{\omega}{(s + a)^2 + \omega^2}\)
\(e^{-at} \cos(\omega t)\ u(t)\)	\(\frac{s + a}{(s + a)^2 + \omega^2}\)
\(t\ \sin(\omega t)\ u(t)\)	\(\frac{2\omega s}{(s^2 + \omega^2)^2}\)
\(t\ \cos(\omega t)\ u(t)\)	\(\frac{s^2 - \omega^2}{(s^2 + \omega^2)^2}\)
\(\frac{1}{\omega^2}(1 - \cos(\omega t))\ u(t)\)	\(\frac{1}{s(s^2 + \omega^2)}\)

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Laplace Transform Properties

Property	Operation	Laplace Transform
Linearity	\(a f(t) + b g(t)\)	\(a F(s) + b G(s)\)
Time Shifting	\(f(t - k)u(t - k)\)	\(e^{-ks}F(s)\)
Frequency Shifting	\(e^{at}f(t)\)	\(F(s - a)\)
Time Derivative	\(f'(t)\)	\(sF(s) - f(0^-)\)
Second Derivative	\(f''(t)\)	\(s^2F(s) - sf(0^-) - f'(0^-)\)
Integral	\(\int_0^t f(\tau) d\tau\)	\(\frac{1}{s}F(s)\)
Convolution	\(f(t) * g(t)\)	\(F(s) \cdot G(s)\)
Final Value Theorem	\(\lim_{t \to \infty} f(t)\)	\(\lim_{s \to 0} sF(s)\) (if poles in left half-plane)
Initial Value Theorem	\(\lim_{t \to 0^+} f(t)\)	\(\lim_{s \to \infty} sF(s)\)

For time shifting, \(k > 0\) or else \(f(t)\) diverges (no Laplace transform).

Common Laplace Transforms

Time Domain: \(f(t)\)	Frequency Domain: \(F(s)\)
\(u(t)\)	\(\frac{1}{s}\)
\(u(t - k)\)	\(\frac{e^{-ks}}{s}\)
\(\delta(t)\)	\(1\)
\(\delta(t - k)\)	\(e^{-ks}\)
\(t^n u(t)\) (\(n \geq 0\))	\(\frac{n!}{s^{n+1}}\)
\(e^{-at}u(t)\)	\(\frac{1}{s + a}\)
\(t^n e^{-at}u(t)\)	\(\frac{n!}{(s + a)^{n+1}}\)
\(\sin(\omega t)u(t)\)	\(\frac{\omega}{s^2 + \omega^2}\)
\(\cos(\omega t)u(t)\)	\(\frac{s}{s^2 + \omega^2}\)
\(e^{-at}\sin(\omega t)u(t)\)	\(\frac{\omega}{(s + a)^2 + \omega^2}\)
\(e^{-at}\cos(\omega t)u(t)\)	\(\frac{s + a}{(s + a)^2 + \omega^2}\)
\(\sinh(\alpha t)u(t)\)	\(\frac{\alpha}{s^2 - \alpha^2}\)
\(\cosh(\alpha t)u(t)\)	\(\frac{s}{s^2 - \alpha^2}\)

Reactive Element Transforms

In S-domain, we talk about impedances \(Z\) just like in phasors.

Inductor

Inductor	Time Domain	s-Domain
Voltage	\(v_L(t) = L \frac{di_L(t)}{dt}\)	\(V_L(s) = L[sI_L(s) - i_L(0^-)]\)
Current	\(i_L(t) = \frac{1}{L} \int_0^t v_L(\tau)d\tau + i_L(0^-)\)	\(I_L(s) = \frac{V_L(s)}{sL} + \frac{i_L(0^-)}{s}\)
Impedance	\(Z_L(s)\)	\(sL\)

Capacitor

Capacitor	Time Domain	s-Domain
Voltage	\(v_C(t) = \frac{1}{C} \int_0^t i_C(\tau)d\tau + v_C(0^-)\)	\(V_C(s) = \frac{I_C(s)}{sC} + \frac{v_C(0^-)}{s}\)
Current	\(i_C(t) = C \frac{dv_C(t)}{dt}\)	\(I_C(s) = C[sV_C(s) - v_C(0^-)]\)
Impedance	\(Z_C(s)\)	\(\frac{1}{sC}\)

Example

\[Zeq = \frac{R_E r_{\pi}}{R_E + r_{\pi} + sR_E r_{\pi} C_{\pi}}\]

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Pending:

• 3-Phase [c12]
• Mutual Inductance (transformers) [c13]
• Transfer functions
• Op amps
• Two-port networks