E-9: A.C. Circuits

OBJECTIVE:

To study voltage and phase relationships in A.C. circuits

APPARATUS:

Dual trace oscilloscope & manual; differential amplifiers, circuit plug board & component kit; signal generator & frequency counter; digital multimeter (DMM).

Part A: SERIES A.C. CIRCUITS

INTRODUCTION:

Overview: Remember that ``rms'' value of the AC voltage or current means the ``root-mean-square'' and this value is useful even when the voltage or current is changing but not as perfect sine waves. For example, the voltage used to provide power for most lights and appliances is roughly $V_{rms}=110$ Volts. Some higher power appliances such as stoves use $V_{rms}=220$ Volts. Although the voltage and current may often be negative, the rms values are always positive. Similarly, the impedance $Z$ is always positive. Sometimes the word ``effective'' is used instead of ``rms'' but it means the same thing. For a pure sine wave voltage, the peak voltage and rms voltage are related by $V_{rms}=\frac{V_{\mbox{\tiny peak}}}{\sqrt{2}}$. Thus the peak to peak voltage swing of a 110 volt AC wire is $V_{\mbox{\tiny peak}}=110\times \sqrt{2} \mbox{ volts}\approx155 \mbox{ volts}$!

The rms concept allows us to describe any AC voltage as having a particular rms voltage and a particular phase.

In this experiment we will use the subscript ``rms'' for some measurements and will use ``$V$'' or ``$v$'' or ``$I$'' or ``$i$'' without this subscript when referring to instantaneous values of the voltage or current. Thus, even if two rms voltages are equal, ( $V_{rms}=v_{rms}$), $V$ may not be equal to $v$ since $V$ and $v$ may have different phases.

The impedance $Z$ (in ohms) of any part of a circuit is the ratio of the rms voltage across that part and the rms current though that part: Because impedance is defined as a ratio of voltage/current, impedance is measured in ohms. If $\omega=2\pi f$ and we measure $L$ and $C$ in henries and farads, respectively, then it can be shown that the impedance $X_{R}$ of an resistance $R$ is $R$ and the impedance $X_{L}$ (in ohms) of an inductance $L$ is $X_{L} =\omega L= 2\pi fL$. and the impedance $X_{C}$ (in ohms) of a capacitance $C$ is $X_{C} = \frac{1}{\omega C}= \frac{1}{2\pi fC}$. It can also be shown that the impedance Z (in ohms) of an RLC series circuit (shown in Fig. 1) is

$$Z = \sqrt{R^2 + (X_{L} - X _{C})^2}$$

where, $X_{L}$ and $X_{C}$ are the impedances of the individual inductor and capacitor.

The impedance $Z$ of the RLC series circuit is a minimum for $X_{L}=X_{C}$. The frequency for which this occurs is the resonant frequency. At this frequency, $f_{r}$, the current thru R is maximum, but the voltage $V_{rms\ LC}$ across the LC series combination is a minimum and in fact would be zero if the inductor had no resistance. Hence one can search for $f_{r}$ by varying the frequency and looking either
    1)  for a maximum $V_{rms R}$ signal,     or      2)  a minimum $V_{rms\ LC}$ signal.
A search for the minimum has a practical advantage that near the resonant frequency, $f_{r}$, one can increase enormously the detection sensitivity by going to maximum signal generator amplitude and also by going to higher scope gain.

Figure 1: A series LRC circuit.

Our dual trace scope allows the simultaneous observation of two voltages. If we use differential amplifiers (E-8) to avoid ground problems, then we can compare the signal across the resistor (displayed on one trace) with the signal across other circuit elements (displayed on the other trace). Since across a resistor the voltage and current are in phase, and since the current throughout a series circuit is the same, we can observe the relative phase relationship between the current and any other measured voltage.

SUGGESTIONED PROCEDURE:

1. Hook up the circuit plug board as in Fig. 1. (The signal generator supplies $V_{rms}$ (or $V_{AC}$.)
2. Use the digital multimeter (DMM) to measure $V_{rms}$, V$_{rms R}$, V$_{rms L}$, and V$_{rms C}$ at frequencies near 400, 700, and 1000 Hz. Is $V_{rms}$ = V$_{rms R}$ + V$_{rms L}$ + V$_{rms C}$?  Explain.
3. Calculate the frequency for which resonance should occur.
4. Use the scope to search for the resonant frequency f$_{r}$ by the following techniques and also estimate the uncertainty $\Delta$f$_{r}$ in each measurement. Use the trigger output of the signal generator to trigger the scope. (For the Krohn-Hite signal generator, use the ``TTL'' output. This output gives a ``rectangular'' signal, snapping between zero and +3 volts, which is suitable for digital circuitry such as computers and counters.)

   (i)

   As one varies f, observe the maximum in the current $I_{rms}$ thru R. (Since $I_{rms}$ $\propto$ V$_{rms R}$, use a differential amplifier with scope channel Y$_2$ to look at V$_{rms R}$).

   (ii)

   Observe the minimum in the voltage $V_{rms\ LC}$ across the LC combination. (Use the other differential amplifier to input V$_{rms LC}$ to scope channel Y$_1$. To improve sensitivity, turn up signal amplitude and/or scope gain).

   (iii)

   Produce a Lissajous figure using V$_{rms R}$ and V$_{rms LC}$ as the X and Y inputs. (Use the X-Y setting of the TIME/CM or TIME/DIV knob and set mode lever to X-Y). At resonance the usual elliptical pattern becomes a straight line as V$_{rms LC}$ goes to zero. Again turn up the gains to improve sensitivity.

5. Compare the calculated f$_{r}$ with your observed values. Is there agreement?

6. Phase relations: Use scope channel 1 (plus a differential amplifier) to observe the total voltage $V_{rms RLC}$ across the RLC combination. Use scope channel 2 (plus differential amplifier) to observe V$_{rms R}$. Fig. 2 shows the phasor relationships.

   Figure 2

   (i)

   Set the signal generator near $f_{r}$. At resonance the two signals should be in phase. Why? (If at $f =f_{r}$ the signals are 180$^o$ out of phase, the input polarity is wrong on one differential amplifier; hence interchange the input leads on one of the differential amplifiers.)

   Find the resonant frequency by this method. Adjusting gain and position controls so the signals nearly overlap will help.

   (ii)

   With $f =f_{r}$, short out the capacitor by a jumper, (e.g. with metal bridge connector). Note what happens. Unshort capacitor and then short the inductor. Explain results in terms of Fig. 2. Does current lead or lag the applied voltage in each case?

   (iii)

   Remove the shorts and try $f<f_{r}$ and $f>f_{r}$. Explain phase behavior.

OPTIONAL:

A quality factor $Q$ (so named because the energy loss $\Delta$U per cycle of energy U stored in oscillating LC elements is inverse to the circuit's $Q$) is

$$Q = \frac{1}{R} \sqrt{\frac{L}{C}}.$$

Also $Q$ measures the sharpness of a resonance. Rewrite $Z$ in terms of $Q$:

$$Z = R\left[ 1 + Q^2\left( \frac{f}{f_{r}} - \frac{f_r}{f}\right) ^2\right] ^{1/2}.$$

Note that large $Q$ means $Z$ will assume large values when $f$ differs slightly from $f_{r}$. Hence sharp resonant effects occur. Calculate Q for your circuit. What $\Delta Z/Z$ results for $\Delta f/f_{r}= 0.1$? (Large $Q$ occurs for R small, but the resulting resonant current may be excessive. To achieve sharp tuning, radio circuits may have $Q$'s $\sim$ 100).

Part B: PARALLEL A.C. Circuits

Figure 3: A parallel LRC circuit.

INTRODUCTION:

The parallel RLC circuit is very common in radio and TV circuits. The circuit analysis follows from Kirchhoff's laws, but is not in many beginning texts. The common terminal voltage $V$ is across each element (see Fig. 3a), and the instantaneous current $i$ is the algebraic sum of the instantaneous $i_{R}$, $i_{L}$, and $i_{C}$.

Since $i_{R}$ is in phase with $V$, but $i_{L}$ lags $V$ by $90^o$, and $i_{C}$ leads $V$ by $90^o$, we can describe the situation by the rotating vectors in Fig. 3b where $I$ is the vector sum of $I_{R}$, $I_{L}$, and $I_{C}$. Hence from Fig. 3b

$$I_{rms} = \sqrt{I^2_{rms R} + (I_{rms C} - I_{rms L})^2}$$

where $I_{rms R} =  \frac{V_{rms}}{R}$, $I_{rms C}  = \frac{V_{rms}}{X_{C}}$, and $I_{rms L} =  \frac{V_{rms}}{X_{L}}$. Substituting these gives

$$I_{rms} = V_{rms}. \sqrt{ \frac{1}{R^2} + \left( \frac{1}{X_C} - \frac{1}{X_L} \right)^2}.$$

Resonance still occurs for $X_{L}=X_{C}$ but the total current, $I_{rms}$, is then just the current thru $R$ and thus is a MINIMUM (in contrast to the series resonant case).

Since $V$ is the same for all the parallel elements, the relevant phase differences are between the currents. To measure the total current $I_{rms}$ and the phase between it and the voltage $V_{rms}$, we will insert a 22 k$\Omega$ sampling resistor $R_{S}$ in series with the signal generator. See Figure 4.

Measure the voltage $V_{rms S}$ across the sampling resistor $R_{S}$ by connecting the resistor ends to scope channel 2 via a differential amplifier. Since the voltage and current are in phase across a resistor, this signal $V_{rms S}$ is proportional to the total current $I_{rms}$.

We use channel 1 and the other differential amplifier to view the common voltage V across all the parallel elements. At resonant frequency $f_{r}$ the currents from $L$ and $C$ will cancel since they are of equal magnitude but (always) $180^o$ out of phase. Hence at $f_{r}$ the total current $I$ will be just that thru R, i.e. $I = I_{R}$, and $V$ and $V_{S}$ (or $I$) will be in phase.

The smaller the sampling resistor $R_{S}$, the less disturbance its voltage drop $V_{rms S}$ will have on the net voltage $V_{rms}$ applied across the parallel circuit. However distortion in the generated sine wave may bother when $R_{S}$ is small and large currents flow. Our suggested $R_{S} = 22 \mbox{k}\Omega$ is a compromise. Even so you may see a factor 2 change in $V_{rms}$ as $f$ varies; however we can compensate by adjusting the signal generator amplitude to keep $V_{rms}$ constant as the frequency changes.

Figure 4: A parallel LRC test circuit.

SUGGESTED PROCEDURE:

1. Hook up the circuit plug board as in Fig. 4 and again trigger the scope externally from the signal generator. (We chose a large $R=100 \mbox{k}\Omega$ circuit resistance to limit the resonant current and thus enhance sensitivity.)
2. Set the frequency near $f_{r}$ found in Part A: SERIES RESONANCE. Adjust the signal generator amplitude to give a 1 volt scope signal (p to p) for $V$ in channel 1.
3. Vary the frequency f on both sides of $f_{r}$ by about 300 Hz. Adjust the signal generator amplitude to keep $V_{rms}$ (i.e. $Y_1$) constant as $f$ and $V_{rms S}$ change. Since $I_{rms} \propto V_{rms S}$, plot $V_{rms S}$ versus $f$.  Note the resonant minimum in $I_{rms}$ at $f_{r}$.

   OPTIONAL: Replace the $22 \mbox{k}\Omega$ $R_{S}$ with a $1 \mbox{k}\Omega$ resistor. Repeat step #3 and note the increased distortion of $V_{S}$ at $f_{r}$, but the relative constancy now of the voltage $V$, (channel 1), across the circuit elements.

4. Starting with $f =f_{r}$ observe how the phase relations between I (which is in phase with $V_{S}$) and $V$ change as one goes to $f>f_{r}$ and $f<f_{r}$. (Again if at $f =f_{r}$, the two signals are not in phase but are $180^o$ out of phase, interchange the input leads to one of the differential amplifiers).
5. Use the Lissajous figure technique to find $f_{r}$: namely, with the horizontal deflection (TIME/CM or TIME/DIV) knob at X-Y (and the mode lever at X-Y) use $V$ and $V_{S}$ as the X-Y inputs. As $V_{S}$ (or $I$) goes thru its minimum, the elliptical pattern collapses to a straight line. Again one can increase gains to improve sensitivity.

6. Move the scope trace for $V$ near to the top of the scope screen.
7. Move the scope trace for $I$ near to the bottom of the scope screen.
8. Then the ratio of $\frac{\mbox{\small Magnitude of upper trace}}{\mbox{\small Magnitude of lower trace}}$ is proportional to the impedance $Z$.
9. Notice that, as you change the frequency, $Z$ given by this ratio of traces, goes through a maximum at the resonant frequency.