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| - | ====== Introduction ====== | ||
| - | |||
| - | |||
| - | ---- | ||
| - | |||
| - | ===== Inductance ===== | ||
| - | |||
| - | You may have heard in lecture that whenever the magnetic flux through a loop of wire changes, there is an induced EMF. There are many ways this can happen – moving a bar magnet near a stationary loop of wire, turning a loop of wire through a static magnetic field, etc. – but we will focus on a specific type of inductance in this lab: //the self-inductance in a loop of wire caused by varying the current through the wire.// | ||
| - | |||
| - | Mathematically, | ||
| - | | {FIXME $\mathcal{E} = -N\dfrac{d\varphi}{dt} = -L\dfrac{dI}{dt}$ {FIXME $\mathcal{E} = -N\dfrac{d\varphi}{dt} = -L\dfrac{dI}{dt}$ , | (1) | | ||
| - | |||
| - | where | ||
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| - | * {FIXME $N$ {FIXME $N$ is the total number of loops (also called //turns//) on the coil; * {FIXME $\varphi$ {FIXME $\varphi$ is the flux through the coil; * {FIXME $L$ {FIXME $L$ is the coefficient of self-inductance; | ||
| - | In this experiment, we will be using a long hollow coil of wire – called a // | ||
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| - | Have I heard of a different kind of solenoid? | ||
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| - | Possibly. The term is used for a long, coiled cylinder of wire in the context of Physics. Such coils are often parts of electromagnets that move objects linearly, and people have subsequently started calling such devices solenoids as well. In particular, what is called a ' | ||
| - | |||
| - | The flux at the center of a long solenoid is given by | ||
| - | |||
| - | | {FIXME $\varphi = BA = \mu_0nAI$ {FIXME $\varphi = BA = \mu_0nAI$ | ||
| - | |||
| - | where | ||
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| - | * {FIXME $B$ {FIXME $B$ is the magnitude of the magnetic field; | ||
| - | Differentiating Eq. (2) and substituting into Eq. (1), we have | ||
| - | |||
| - | | {FIXME $\mathcal{E} = -\mu_0nNA\dfrac{dI}{dt}$ {FIXME $\mathcal{E} = -\mu_0nNA\dfrac{dI}{dt}$ , | (3) | | ||
| - | |||
| - | and therefore | ||
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| - | | {FIXME $L_{\infty} = \mu_0nNA$ {FIXME $L_{\infty} = \mu_0nNA$ . | (4) | | ||
| - | |||
| - | Note that this self-inductance holds for an // | ||
| - | | {FIXME $L = \mu_0nNAK.$ {FIXME $L = \mu_0nNAK.$ | ||
| - | |||
| - | This factor depends on the ratio of the solenoid’s length({FIXME $\ell$ {FIXME $\ell$ ) to diameter({FIXME $D$ {FIXME $D$ ), {FIXME $\ell/D$ {FIXME $\ell/D$ . Calculation of {FIXME $K$ {FIXME $K$ is somewhat complex, so we will simply tabulate and plot the results. (See Table 1 and Fig. 1.) | ||
| - | | {FIXME $\ell/D$ {FIXME $\ell/ | ||
| - | **// | ||
| - | </ | ||
| - | | {FIXME $\infty$ {FIXME $\infty$ | ||
| - | | 100 | 0.996 | | ||
| - | | 50 | 0.992 | | ||
| - | | 20 | 0.979 | | ||
| - | | 10 | 0.959 | | ||
| - | | 5 | 0.920 | | ||
| - | | 3 | 0.873 | | ||
| - | | 2 | 0.818 | | ||
| - | | 1 | 0.688 | | ||
| - | |< | ||
| - | **Table 1**: End effect corrections for finite length solenoids | ||
| - | </ | ||
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| - | |||
| - | {FIXME ${/ | ||
| - | **Figure 1**: End effect corrections for finite length solenoids | ||
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| - | From the graph it should be clear how long a " | ||
| - | |||
| - | ===== Transient behavior of RC circuits ===== | ||
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| - | Equations (1) and (3) show that an EMF is induced in inductive circuits only when fluxes are //changing with time//. It is thus clear that inductances are important in determining // | ||
| - | |||
| - | {FIXME ${/ | ||
| - | **Figure 2**: Charging and discharging an RC circuit | ||
| - | |||
| - | ==== Charging ==== | ||
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| - | Consider the RC circuit in Fig. 2. You may recall from your studies of the charging and discharging of capacitors, if the switch S is moved to position a (so that the battery is connected across the resistor-capacitor combination) the potential across the capacitor {FIXME $V_C$ {FIXME $V_C$ will increase according to the equation | ||
| - | | {FIXME $V_C = V_E\left[1-e^{\Large{\left(-\frac{t}{RC}\right)}}\right]$ {FIXME $V_C = V_E\left[1-e^{\Large{\left(-\frac{t}{RC}\right)}}\right]$ . | (8) | | ||
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| - | Since the sum of the voltage across the capacitor and that across the resistor must equal_{FIXME $V_E$ {FIXME $V_E$_ , it follows that the voltage across the resistor, {FIXME $V_R$ {FIXME $V_R$ , must be | ||
| - | | {FIXME $V_R = V_Ee^{\Large{\left(-\frac{t}{RC}\right)}}$ {FIXME $V_R = V_Ee^{\Large{\left(-\frac{t}{RC}\right)}}$ . | (9) | | ||
| - | |||
| - | Eq.(9) is of the form {FIXME $V_R = V_Ee^{\Large -t/\tau}$ {FIXME $V_R = V_Ee^{\Large -t/\tau}$ , where {FIXME $\tau$ {FIXME $\tau$ is called the //time constant//. The time required for {FIXME $V_R$ {FIXME $V_R$ to drop to ({FIXME $1/e \approx 0.37$ {FIXME $1/e \approx 0.37$ ) of {FIXME $V_E$ {FIXME $V_E$ is just | ||
| - | | {FIXME $\tau = RC.$ {FIXME $\tau = RC.$ | (10) | | ||
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| - | ==== Discharging ==== | ||
| - | |||
| - | If now the switch is moved to point b, the capacitor will discharge through the resistor and the potential across the capacitor as a function of time will be | ||
| - | |||
| - | | {FIXME $V_C = V_E e^{\Large-\frac{t}{RC}}$ {FIXME $V_C = V_E e^{\Large-\frac{t}{RC}}$ . | (11) | | ||
| - | |||
| - | This time, the total potential across the capacitor and the resistor must be zero. Thus, {FIXME $V_R$ {FIXME $V_R$ is | ||
| - | | {FIXME $V_R = -V_E e^{\Large-\frac{t}{RC}}.$ {FIXME $V_R = -V_E e^{\Large-\frac{t}{RC}}.$ | ||
| - | |||
| - | In all these cases, Eq. (10) is the characteristic decay time of the RC circuit. | ||
| - | |||
| - | ===== Transient behavior of RL circuits ===== | ||
| - | |||
| - | {FIXME ${/ | ||
| - | **Figure 3**: Charging and discharging an RL circuit | ||
| - | |||
| - | ==== Charging ==== | ||
| - | |||
| - | Consider now the LR circuit shown in Fig. 3. Suppose we move the switch S to position a. The potential drop across the resistor must then be {FIXME $IR$ {FIXME $IR$ and, by the definition of self-inductance, | ||
| - | | {FIXME $V_E=IR + L\dfrac{dI}{dt}$ {FIXME $V_E=IR + L\dfrac{dI}{dt}$ . | (13) | | ||
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| - | The solution of this equation is | ||
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| - | | {FIXME $I = \dfrac{V_E}{R}\left[1 - e^{\Large{-\frac{Rt}{L}}}\right]$ {FIXME $I = \dfrac{V_E}{R}\left[1 - e^{\Large{-\frac{Rt}{L}}}\right]$ . | (14) | | ||
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| - | You can easily prove that Eq. (14) is the solution of Eq. (13) by substituting both Eq. (14) and its derivative into Eq. (13). | ||
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| - | While Eq. (14) gives the current, it does not give us the potentials so that we can compare with the transient behavior of capacitors. The transition to potentials is, however, very easy. The current, {FIXME $I$ {FIXME $I$ //,// will have to flow through the resistor, {FIXME $R$ {FIXME $R$ . Thus, the potential drop {FIXME $V_R$ {FIXME $V_R$ across {FIXME $R$ {FIXME $R$ will be | ||
| - | | {FIXME $V_R = IR = V_E\left[1 - e^{\Large{-\frac{Rt}{L}}}\right]$ {FIXME $V_R = IR = V_E\left[1 - e^{\Large{-\frac{Rt}{L}}}\right]$ . | (15) | | ||
| - | |||
| - | Again, since_{FIXME $V_R + V_L$ {FIXME $V_R + V_L$_ must equal {FIXME $V_E$ {FIXME $V_E$ , we have | ||
| - | | {FIXME $V_L = V_E\ e^{\Large{-\frac{Rt}{L}}}$ {FIXME $V_L = V_E\ e^{\Large{-\frac{Rt}{L}}}$ . | (16) | | ||
| - | |||
| - | Note that {FIXME $V_C$ {FIXME $V_C$ for the {FIXME $RC$ {FIXME $RC$ circuit has the same form as {FIXME $V_R$ {FIXME $V_R$ for the {FIXME $RL$ {FIXME $RL$ circuit, and that {FIXME $V_R$ {FIXME $V_R$ for the {FIXME $RC$ {FIXME $RC$ case has the same form as {FIXME $V_L$ {FIXME $V_L$ for the {FIXME $RL$ {FIXME $RL$ case. Stated in another way, while the potential across the capacitor // | ||
| - | ==== Discharging ==== | ||
| - | |||
| - | If now the switch is moved to point b, the left side of Eq. (13) goes to zero and the discharge solutions are | ||
| - | |||
| - | | {FIXME $V_R = -V_E\ e^{\Large{-\frac{Rt}{L}}}$ {FIXME $V_R = -V_E\ e^{\Large{-\frac{Rt}{L}}}$ | ||
| - | |||
| - | and | ||
| - | |||
| - | | {FIXME $V_L = -V_E\ e^{\Large{-\frac{Rt}{L}}}$ {FIXME $V_L = -V_E\ e^{\Large{-\frac{Rt}{L}}}$ . | (18) | | ||
| - | |||
| - | (The negative signs are due to the current changing direction in the circuit.) | ||
| - | |||
| - | Equation (18) is of the form {FIXME $V_L = V_Ee^{\Large -t/\tau}$ {FIXME $V_L = V_Ee^{\Large -t/\tau}$ The time for the potential to drop to {FIXME $1/e \approx 37\%$ {FIXME $1/e \approx 37\%$ of the maximum value is when | ||
| - | | {FIXME $\tau = L/R$ {FIXME $\tau = L/R$ | (19) | | ||
| - | |||
| - | Comparison of Eq. (19) with Eq. (10) shows that there is a difference between the functional forms for the time constants of RC and RL circuits. | ||
| - | |||
| - | More LRC Circuit Theory | ||
| - | |||
| - | ===== Steady-state behavior ===== | ||
| - | |||
| - | ==== LR Circuits ==== | ||
| - | |||
| - | {FIXME ${/ | ||
| - | **Figure 4:** RL circuit | ||
| - | |||
| - | Consider the RL circuit shown in Fig. 4. Suppose the EMF, // | ||
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| - | | {FIXME $V(t) = V_0\sin\omega t,$ {FIXME $V(t) = V_0\sin\omega t,$ | (20) | | ||
| - | |||
| - | where {FIXME $\omega$ {FIXME $\omega$ is the angular frequency (typically measured in radians/ | ||
| - | In this case, the equation of the circuit becomes | ||
| - | |||
| - | | {FIXME $V(t) = V_R + V_L$ {FIXME $V(t) = V_R + V_L$ , | (21) | | ||
| - | |||
| - | or | ||
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| - | | {FIXME $V_0\sin\omega t = IR + L\dfrac{dI}{dt}$ {FIXME $V_0\sin\omega t = IR + L\dfrac{dI}{dt}$ . | (22) | | ||
| - | |||
| - | The resulting // | ||
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| - | | {FIXME $I(t) = I_0\sin(\omega t - \delta)$ {FIXME $I(t) = I_0\sin(\omega t - \delta)$ , | (23) | | ||
| - | |||
| - | where {FIXME $\delta$ {FIXME $\delta$ is an arbitrary phase angle. If we substitute this and its derivative into Eq. (21), use trigonometric identities for {FIXME $\sin(a-b)$ {FIXME $\sin(a-b)$ and {FIXME $\cos(a-b)$ {FIXME $\cos(a-b)$ , and rearrange terms, we obtain | ||
| - | | {FIXME $\left(\omega LI_0\cos\delta - RI_0 \sin \delta\right)\cos\omega t + \left(\omega L I_0 \sin \delta + RI_0 \cos \delta - V_0\right)\sin\omega t = 0$ {FIXME $\left(\omega LI_0\cos\delta - RI_0 \sin \delta\right)\cos\omega t + \left(\omega L I_0 \sin \delta + RI_0 \cos \delta - V_0\right)\sin\omega t = 0$ . | (24) | | ||
| - | |||
| - | This equation will be satisfied for all values of {FIXME $t$ {FIXME $t$ only if the coefficients of {FIXME $\cos\omega t$ {FIXME $\cos\omega t$ and {FIXME $\sin\omega t$ {FIXME $\sin\omega t$ are each individually equal to zero. | ||
| - | Setting the first coefficient to zero, we obtain for the phase angle | ||
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| - | | {FIXME $\tan\delta = \dfrac{\omega L}{R}$ {FIXME $\tan\delta = \dfrac{\omega L}{R}$ . | (25) | | ||
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| - | Setting the second to zero, we obtain | ||
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| - | | {FIXME $I_0 = \dfrac{V_0}{\omega L \sin\delta + R \cos \delta}$ {FIXME $I_0 = \dfrac{V_0}{\omega L \sin\delta + R \cos \delta}$ . | (26) | | ||
| - | |||
| - | Putting Eq. (25) in Eq. (26), and the result back into Eq. (23) gives | ||
| - | |||
| - | | {FIXME $I(t) = \dfrac{V_0\sin(\omega t - \delta )}{\sqrt{R^2 + (\omega L)^2}}$ {FIXME $I(t) = \dfrac{V_0\sin(\omega t - \delta )}{\sqrt{R^2 + (\omega L)^2}}$ . | (27) | | ||
| - | |||
| - | Let us consider now the following three cases: | ||
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| - | * Case 1: {FIXME $L=0$ {FIXME $L=0$ , i.e., a purely resistive circuit; | ||
| - | |||
| - | === Case 1: L = 0 === | ||
| - | |||
| - | Equations (25) and (27) reduce to | ||
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| - | | {FIXME $\delta = 0$ {FIXME $\delta = 0$ | (28a) | | ||
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| - | and | ||
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| - | | {FIXME $IR = V(t)$ {FIXME $IR = V(t)$ | (28b) | | ||
| - | |||
| - | Respectively. Thus, in the limiting case of no inductance, the equations reduce to Ohm's law with the current and voltage //in phase.// | ||
| - | |||
| - | === Case 2, R = 0 === | ||
| - | |||
| - | Equation (25) becomes | ||
| - | |||
| - | | {FIXME $\delta = 90^\circ$ {FIXME $\delta = 90^\circ$ | ||
| - | |||
| - | and Eq. (27) becomes | ||
| - | |||
| - | | {FIXME $I\omega L = V_0 \sin(\omega t - 90^\circ).$ {FIXME $I\omega L = V_0 \sin(\omega t - 90^\circ).$ | ||
| - | |||
| - | Thus, the inductor acts like a resistor with effective resistance (called //inductive reactance// | ||
| - | |||
| - | | {FIXME $X_L = \omega L$ {FIXME $X_L = \omega L$ . | (30) | | ||
| - | |||
| - | === Case 3: R ≠ 0 and L ≠ 0 === | ||
| - | |||
| - | By analogy with Eq. (29b) above, we can identify the denominator of Eq. (27) as the general impedance of the circuit, | ||
| - | |||
| - | | {FIXME $Z_{RL} = \sqrt{R^2 + (\omega L)^2}$ {FIXME $Z_{RL} = \sqrt{R^2 + (\omega L)^2}$ . | (31) | | ||
| - | |||
| - | Likewise, we see that the righthand side of Eq. (25) will always be positive, yielding a positive phase angle (i.e. a current //lag//) between 0 and {FIXME $90^\circ$ {FIXME $90^\circ$ . | ||
| - | ==== RC circuits ==== | ||
| - | |||
| - | **{FIXME ${/ | ||
| - | **Figure 5:** RC circuit | ||
| - | |||
| - | Consider now the RC circuit shown in Fig. 5. Again, we will impress a simple sinusoidal EMF at // | ||
| - | | {FIXME $V(t) = V_R + V_C$ {FIXME $V(t) = V_R + V_C$ | (32) | | ||
| - | |||
| - | or | ||
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| - | | {FIXME $V_0 \sin\omega t = IR + \dfrac{Q}{C}$ {FIXME $V_0 \sin\omega t = IR + \dfrac{Q}{C}$ . | (33) | | ||
| - | |||
| - | Since {FIXME $I = \dfrac{dQ}{dt}$ {FIXME $I = \dfrac{dQ}{dt}$ , this equation is analogous to Eq. (22) and can be similarly solved. The solution, neglecting transient terms, is | ||
| - | | {FIXME $I(t) = \dfrac{V_0\sin(\omega t - \delta)}{\sqrt{R^2 + (1/\omega C)^2}}$ {FIXME $I(t) = \dfrac{V_0\sin(\omega t - \delta)}{\sqrt{R^2 + (1/\omega C)^2}}$ , | (34) | | ||
| - | |||
| - | where | ||
| - | |||
| - | | {FIXME $\tan \delta = - \dfrac{1}{\omega C}$ {FIXME $\tan \delta = - \dfrac{1}{\omega C}$ . | (35) | | ||
| - | |||
| - | In analogy to the three cases we considered above, | ||
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| - | * When {FIXME $C = \infty$ {FIXME $C = \infty$ this equation reduces to Ohm's law for a resistor. | ||
| - | | {FIXME $X_C = \dfrac{1}{\omega C}$ {FIXME $X_C = \dfrac{1}{\omega C}$ | (36a) | | ||
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| - | and a phase angle of | ||
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| - | | {FIXME $\delta = -90^\circ$ {FIXME $\delta = -90^\circ$ . | (36b) | | ||
| - | |||
| - | Contrary to the case of inductance, the current //leads// the voltage by 90˚ rather than following it. | ||
| - | |||
| - | * With both R and C in the circuit, the general impedance is | ||
| - | |||
| - | | {FIXME $Z_{RC} = \sqrt{R^2 + (1/\omega C)^2}$ {FIXME $Z_{RC} = \sqrt{R^2 + (1/\omega C)^2}$ | ||
| - | |||
| - | and the phase angle given by Eq. (35) will be negative, implying that the current will always //lead// the voltage. | ||
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| - | [[https:// | ||
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| - | ====== Experimental procedure ====== | ||
| - | |||
| - | |||
| - | ---- | ||
| - | |||
| - | ===== Calculable inductor (prediction) ===== | ||
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| - | In this experiment, you will use an inductor whose inductance is calculable from its dimensions. Determine the dimensions of the solenoid supplied, and use this information to predict what its inductance should be. | ||
| - | |||
| - | {FIXME ${/ | ||
| - | Calculable inductor(solenoid), | ||
| - | |||
| - | ===== Calculable inductor (measurement) ===== | ||
| - | |||
| - | | {FIXME ${/ | ||
| - | |||
| - | Using the solenoid, construct the circuit as shown in Fig. 6. Since we're observing the time-dependent behavior of fast signals, we use an oscilloscope as the voltage-measuring device. | ||
| - | |||
| - | * Set the function generator to be a **square wave**. | ||
| - | * Vary the frequency until you can easily observe the build-up and decay of the voltage across the inductor, similarly to what you did in the capacitance lab. Keep {FIXME $R$ {FIXME $R$ at about {FIXME $500 \;\Omega$ {FIXME $500 \;\Omega$ . Adjust the frequency to allow the voltage to drop to its full, asymptotic value. | ||
| - | * Calculate //L// from Eq. (19) ({FIXME $\tau = L/R$ {FIXME $\tau = L/R$ ). Remember that in this instance, {FIXME $R$ {FIXME $R$ is equal to the value of the variable resistance plus {FIXME $50 \;\Omega$ {FIXME $50 \;\Omega$ due to the output resistance of the function generator. | ||
| - | //Inductors are among the least ideal circuit elements. An inductor has resistance. Also, the adjacent turns of insulated wire have capacitance. To see the mixed character of your inductor, vary the frequency of the square wave over a large range while viewing the waveform across the inductor.// | ||
| - | |||
| - | ===== LRC circuits: Resonance ===== | ||
| - | |||
| - | A series LRC circuit acts like a playground swing. That is, if the swing is given a single push, it will oscillate (with decreasing amplitude, due to friction) with a natural frequency, determined by its length and the acceleration due to gravity. Similarly, if a voltage kick is given to an LRC circuit, the charge (and voltage) in the circuit will oscillate (with decreasing amplitude, due to the resistance of the circuit) with a natural resonance frequency | ||
| - | |||
| - | | {FIXME $f_{res} = \dfrac{1}{2\pi\sqrt{\vphantom | LC}}.$ {FIXME $f_{res} = \dfrac{1}{2\pi\sqrt{\vphantom | LC}}.$ | ||
| - | |||
| - | We wish to explore how the circuit behaves if we drive the circuit in a range of frequencies around its natural frequency. | ||
| - | |||
| - | ==== Observing resonance ==== | ||
| - | |||
| - | Construct the circuit shown in Fig. 8. Set //R// to about {FIXME $200\; | ||
| - | | {FIXME ${/ | ||
| - | |||
| - | For this circuit, | ||
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| - | * When {FIXME $f \ll f_{res}$ {FIXME $f \ll f_{res}$ , the impedance of the inductor is very low and the impedance of the capacitor is very high. The capacitor' | ||
| - | Impedance? | ||
| - | |||
| - | Impedance is a generalization of resistance. For inductors and capacitors, it depends on the frequency of a sine wave in a circuit. | ||
| - | |||
| - | What happens at resonance? That's what you'll be observing. | ||
| - | |||
| - | Record the amplitude of the voltage across the capacitor {FIXME $V_C$ {FIXME $V_C$ as measured on the oscilloscope as you vary the frequency {FIXME $f$ {FIXME $f$ of the signal generator. **Plot these data and describe the shape of the curve.** | ||
| - | ==== Quality (Q) factor ==== | ||
| - | |||
| - | The Q factor of a resonator refers to how strongly the system responds to being driven at its resonant frequency. There are several ways to characterize this, but for this lab we will take {FIXME $Q$ {FIXME $Q$ to be the ratio the amplitude at resonance to the amplitude of the signal at low frequencies: | ||
| - | | {FIXME $Q = \dfrac{\textit{amplitude at resonance}}{\textit{amplitude well below resonance}}.$ {FIXME $Q = \dfrac{\textit{amplitude at resonance}}{\textit{amplitude well below resonance}}.$ | ||
| - | |||
| - | For these circuits, measuring the amplitude at 1 to 2 kHz will be far enough away from resonance to be used for calculating {FIXME $Q$ {FIXME $Q$ . | ||
| - | The resistance {FIXME $R$ {FIXME $R$ acts to dampen the oscillations in this system, so it seems reasonable that it would have an impact on {FIXME $Q$ {FIXME $Q$ . Determine {FIXME $Q$ {FIXME $Q$ for a few different values of {FIXME $R$ {FIXME $R$ for your circuit. **Plot your results as you go.** | ||
| - | Keep {FIXME $R$ {FIXME $R$ at 1 k{FIXME $\Omega$ {FIXME $\Omega$ or less, otherwise our oscillator won't oscillate properly. | ||
| - | ====== Report submission ====== | ||
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| - | ---- | ||
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| - | Take a look over your report and make sure it's complete. Download your report as a PDF and upload it to the form below.**Make sure to log out of your Google account when you are done!** | ||
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| - | [[https:// | ||
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| - | .submit {FIXME $ margin: auto; width:50%; display: block; font-size: 200%; text-align: center; align: | ||