2009

From Mathsoc wiki

1 Thevenin & Norton Equivalent Circuits
2 Capacitors
3 RC Circuits
4 Magnetism
5 Inductors
6 RL Circuits
- 6.1 Series RL Circuits
- 6.2 Parallel RL Circuits
7 RLC Circuits
- 7.1 Series RLC Circuits
- 7.2 Parallel RLC Circuits
8 Transformers

Thevenin & Norton Equivalent Circuits

The Thevenin equivalent circuit consists of a voltage source in series with a resistor.

The Norton equivalent circuit consists of a current source in parallel with a resistor.

Capacitors

A capacitor consists of 2 parallel plates separated by an insulating material called the dielectric. When connected to a voltage source, current flows from one plate to another until the capacitor is fully charged. The amount of charge the capacitor can store per volt across the plates is its capacitance:

$C = \frac{Q}{V}$

Capacitors in series add as

$\frac{1}{C_{tot}} = \frac{1}{C_1} + \dots + \frac{1}{C_n}$

Capacitors in parallel add as

$C_{tot} = C_1 + \dots + C_n$

DC Circuits

Capacitors charge as

$q = C \varepsilon \left[ 1 - e^{-\frac{t}{RC}}\right]$

and discharge as

$q = C \varepsilon e^{-\frac{t}{RC}}$

The time constant is

$\tau = RC$

and the time it takes for the voltage to change by 99% is the transient time $= 5 \tau$

AC Circuits

Capacitors pass AC but with an opposition known as capacitive reactance

$X_c = \frac{1}{2\pi f C} \,\,\, (\Omega)$

The current leads the voltage by $\frac{\pi}{2}$

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RC Circuits

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Series RC Circuits

The total impedance is the phasor sum of the resistance and the capacitive reactance, with magnitude

$| Z | = \sqrt{R^2 + X_c^2}$

The phase angle with which the current leads the voltage is

$\theta = \tan^{-1} \left(\frac{X_C}{R}\right)$

The voltage through the resistor is in phase with the current, while the voltage in the capacitor lags the current by $\frac{\pi}{2}$ . Hence

$V_S = \sqrt{V_R^2 + V_C^2}$

In an RC lag circuit, the output voltage is taken across the capacitor, and this passes low frequencies, as we have

$V_{out} = \frac{X_C V_S}{\sqrt{R^2 + X_C^2}} = \frac{V_S}{\sqrt{\frac{R^2}{X_C^2} + 1}}$

so as $f \rightarrow 0$ , $\frac{R^2}{X_C^2} = R^2 2 \pi f C\rightarrow 0$ and $V_{out} \rightarrow V_s$

Similarly, in a RC lead circuit, the output voltage is taken across the resistor, and this passes high frequencies.

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Parallel RC Circuits

The total impedance is now

$\frac{1}{Z} = \sqrt{\frac{1}{R^2} + \frac{1}{X_C^2}}$

and the phase angle is

$\theta = \tan^{-1} \left(\frac{R}{X_C}\right)$

The current through the resistor is in phase with the voltage, while the current through the capacitor leads the voltage by $\frac{\pi}{2}$

$I_{tot} = \sqrt{I_R^2 + I_C^2}$

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Power in RC Circuits

Some energy is dissipated by the resistor, while some is alternately stored and returned by the capacitor. The power dissipated by the resistor is the true power

$P_{true} = I_{tot}^2 R \,\,\, (W)$

The power in the capacitor is the reactive power

$P_r = I_{tot}^2 X_C$

which is measured in VAR - volt ampere reactive.

The apparent power is the power that appears to be transferred,

$P_a = I_{tot}^2 Z = V_S I_{tot}$

measured in VA - volt ampere.

We have

$P_{true} = P_a \cos \theta$

$P_r = P_a \sin \theta$

where $\theta$ is the phase angle. The term $\cos \theta$ is known as the power factor. A higher power factor means more power is transferred to the resistor, and so the circuit is more efficient.

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Magnetism

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Electromagnetic Induction

A voltage is induced by a conductor moving through a magnetic field, inducing a current.

Faraday's Law: the voltage inducted across a coil of wire by a changing magnetic field through the coil is equal to the number of turns in the coil multiplied by the rate of change of magnetic flux.

Lenz's Law: The polarity of induced voltage is always such as to oppose the change in current creating it.

Electromagnetism!

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Hall Effect

Consider a flat strip conductor with a magnetic field $\vec{B}$ perpendicular to the plane of the conductor. The direction of current flow is along the strip with drift velocity $\vec{v}_D$ . We then have a force on the charge carriers

$\vec{F} = q \vec{v}_D \times \vec{B}$

which leads to a build-up of opposing charge along the edges of the strip.

The current density is given by $J = n q v_D$ , and the current then $I = n q v_D A$ where $A$ is the cross-sectional area of the strip, $A = l d$ with $l$ being the width and $d$ the thickness. So we have

$q v_D = \frac{I}{nA} \Rightarrow F = \frac{IB}{nA}$

and at equilibrium, the electric and magnetic forces on each charge cancel out, so

$|F_m| = |F_e| = \frac{q V_H}{l}$

using $V_H = El$ and $F_e = q E$ . Hence, we have the Hall voltage

$V_H = \frac{IB}{nqd}$

The polarity of this voltage is used to determine the sign of the charge carriers.

From $F_e +F_m = q v_D B + qE = 0$ we also have the Hall field

$E = - v_D B$

giving

$nq = - \frac{JB}{E}$

hence we can measure the the density charge carriers.

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Ampere's Law

Ampere's law can be stated as

$\oint \vec{B} \cdot \vec{dl} = \mu_0 I_{encl}$

where $I_{encl}$ is the current enclosed by the path of integration used in the line integral.

Examples

Current inside wire of radius $R$ : the current density is

$J = \frac{I_0}{\pi R^2}$

We perform our line integral around a circle of radius $r < R$ , hence

$B 2 \pi r = \mu_0 J \pi r^2$

$\Rightarrow B = \frac{\mu_0 I_0 r}{2 \pi R^2}$

Current outside wire: we again perform our line integral around a circle of radius $r$ , this time outside the wire, obtaining

$B = \frac{\mu_0 I}{2 \pi r}$

Solenoid: Let $n$ denote the number of turns per unit length of the solenoid. We perform our line integral around a rectangular path, with one edge of length $l$ just under the surface of the solenoid (below the turns of wire), and the edge parallel to that above the surface. We find

$Bl = \mu_0 nl I$

$\Rightarrow B = \mu_0 n I$

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Inductors

When a length of wire is made into a coil it becomes an inductor. Inductance is given by

$L= \frac{BA}{I}$

and its unit is the Henry. Faraday's law can be written as

$V_L = N \frac{d \Phi}{dt} = L \frac{di}{dt}$

Inductors in series and parallel add as do resistors.

The time constant for an inductor in a DC circuit is $\tau = \frac{L}{R}$

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RL Circuits

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Series RL Circuits

The total impedance is

$Z = \sqrt{R^2 + X_L^2}$

and the phase angle is

$\theta =\tan^{-1} \left(\frac{X_L}{R}\right)$

The resistor voltage is in phase with the current, while the inductor voltage leads current by $\frac{\pi}{2}$ . We have

$V_S^2 = \sqrt{V_R^2 + V_L^2}$

In an RL lag circuit, the output voltage is taken across the resistance, and this passes low frequencies.

In an RL lead circuit the output voltage is taken across the inductor, and this passes high frequencies.

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Parallel RL Circuits

The total impedance is

$\frac{1}{Z} = \sqrt{\frac{1}{R^2} + \frac{1}{X_L^2}}$

and the phase angle is

$\theta = \tan^{-1} \left(\frac{X_L}{R}\right)$

The source voltage, voltage across resistor and voltage across capacitor are all equal and in phase. The current through the inductor lags the voltage by $\frac{\pi}{2}$ , while the current through the resistor is in phase with the voltage.

$I_{tot} = \sqrt{I_R^2 + I_L^2}$

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RLC Circuits

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Series RLC Circuits

We can write the impedance as

$Z = R + j 2 \pi f L + \frac{1}{j 2\pi f C}$

using $X_L = 2 \pi f L$ and $X_C = \frac{1}{2\pi f C}$ , with $j^2 = -1$ . Thus,

$|Z| = \sqrt{R^2 + \left(2 \pi f L - \frac{1}{2\pi f C}\right)^2} = \sqrt{R^2 + \left(X_L - X_C\right)^2}$

and the phase angle is

$\theta = \tan^{-1} \left( \frac{X_{tot}}{R}\right)$

where

$X_{tot} = | X_L - X_C|$

If $X_L > X_C$ , the circuit is primarily inductive. If $X_C > X_L$ , the circuit is primarily capacitive. When $X_C = X_L$ then the circuit is resonant, and $Z= R$ . At resonance the voltages through the inductor and capacitor are a maximum, but are $\pi$ out of phase and so cancel. The resonant frequency is given by

$f_R = \frac{1}{2 \pi \sqrt{LC}}$

We can use a series RLC circuit as a band-pass filter by taking the output voltage across the resistor. This only allows signals at the resonant frequency and within a certain range above and below to pass. This range is known as the band-width, and is defined as the range of frequencies for which

$I > \frac{I_{max}}{\sqrt{2}}$ .

The frequencies at which the current equals this value are known as the cut-off frequencies.

The bandwidth may be found using the formula

$BW = \frac{f_r}{Q}$

where $Q$ is the quality factor, $Q = \frac{X_L}{R}$ , with the resonant value of $X_L$ being used.

The true power delivered at the cut-off frequencies is half the power delivered at resonance.

By taking the output across the RC combination we would instead have a stop-band filter, which would not pass frequencies within a certain range of the resonant frequency.

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Parallel RLC Circuits

The total impedance is

$|Z| = \sqrt{\frac{1}{R^2} + \left(\frac{1}{X_L} - \frac{1}{X_C}\right)^2}$

and the phase angle is

$\theta = \tan^{-1} \frac{X_{tot}}{R}$

where

$X_{tot} = \left| \frac{1}{X_L} - \frac{1}{X_C}\right|$

We have the voltage across each element is the same, but the current through the inductor and capacitor are $\pi$ out of phase. If $X_L > X_C$ , the circuit is primarily capacitive. If $X_C > X_L$ , the circuit is primarily inductive. When $X_C = X_L$ then the circuit is resonant, the currents through the inductor and capacitor cancel, and $Z= R$ .

We can use a parallel RLC circuit as a band-pass filter by taking the output voltage across the inductor and capacitor combination.

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Transformers

When two coils of wire are placed side by side, a changing electromagnetic field produced by the first coil induces a voltage in the second. This is known as mutual inductance.

A transformer consists of two coils placed side by side. The first, known as the primary, is connected to a voltage source, while the second, known as the secondary, is connected to a load. We have the turns ratio

$n = \frac{N_{sec}}{N_{pri}}$

where $N_{sec}$ is the number of turns in the secondary and $N_{pri}$ is the number of turns in the primary.

We have that

$\frac{V_{sec}}{V_{pri}} = n$

thus for $n>1$ the voltage in the secondary is greater than the voltage in the primary - this is known as a step-up transformer. Similary, if $n<1$ then the voltage in the secondary is less than the voltage in the primary - this is known as a step-down transformer.