\section{E-7: A.C. Circuits}\label{e09_ac}


OBJECTIVE: 
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}To study voltage and phase relationships in A.C. circuits
\end{enumerate}

%\subsection{Preliminary Questions}
%\begin{enumerate}
%\item
%?
%\end{enumerate}

\noindent APPARATUS:
\begin{enumerate}
\item[] \latex{\vspace*{-.08in}}Dual trace oscilloscope \& manual; differential amplifiers,
circuit plug board \& component kit; 
signal generator \& frequency counter; digital multimeter (DMM).
\end{enumerate}


\begin{center}
Part A: {\bf SERIES~A.C.~CIRCUITS}
\end{center}

\noindent INTRODUCTION:
\begin{enumerate}
\item[] \latex{\vspace*{-.12in}}
\hrulefill  \\   {\em 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
{\em 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 {\em rms} voltage and
a particular {\em 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
{\em instantaneous}  values of the voltage or current. Thus, even if two rms voltages
are equal, ($V_{rms}=v_{rms}$),  $V$ may {\em not} be equal to $v$ since $V$ and $v$
may have different phases.\latex{\vspace*{-.13in}}

\hrulefill 

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 {\em minimum} for $X_{L}=X_{C}$.  
The frequency for which this occurs is the {\em 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 {\em maximum} $V_{rms\ R}$
signal,~~~~~or ~~~~  2)~~a  {\em 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. \\ \latex{\vspace*{-.05in}} 
\begin{figure}[!h] \centering 
\latex{\includegraphics[height=2.2in]{figs/e9-01.eps}
}
\html{\includegraphics[width=4.0in]{figs/e9-01.eps}}
\caption {A series LRC circuit.} 
\end{figure} \\
%\latex{\vspace*{-.1in}}
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.
\end{enumerate}

\noindent SUGGESTIONED PROCEDURE:  
\begin{enumerate}
\item \latex{\vspace*{-.09in}}Hook up the circuit plug board as in Fig.~1. (The signal 
generator supplies $V_{rms}$ (or $V_{AC}$.)
\item \latex{\vspace*{-.09in}}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.
\item \latex{\vspace*{-.09in}}Calculate the frequency for which resonance should occur.
\item \latex{\vspace*{-.09in}}Use the scope to
search for the  {\em 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.)
\begin{enumerate}
\item[(i)] \latex{\vspace*{-.06in}}As one varies f, observe the {\em maximum} in the {\em 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$_{rms\ R}$).
\item[(ii)] \latex{\vspace*{-.06in}}Observe the {\em minimum} in the {\em 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). 
\item[(iii)] \latex{\vspace*{-.06in}}{\em 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.
\end{enumerate}

\item \latex{\vspace*{-.06in}}Compare the calculated f$_{r}$ with your observed values. 
Is there agreement?

\html{\item \begin{tabular}{p{2.5in}c} }
\latex{\vspace*{0.05in}\hspace*{.0in}\parbox[b]{0.35\linewidth}}
{\latex{\hspace*{-.2in}6.~~}{\em 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. 
}
\html{ & }
\latex{\parbox[t]{0.4\linewidth}}{
\latex{\hspace*{.2in}\includegraphics[height=1.9in]{figs/e9-02.eps}\vspace*{-.1in}
\begin{center} \hspace*{.5in} Figure~2 \end{center}}
\html{
\begin{center}  {\bf Figure~2} 
\includegraphics[width=2.4in]{figs/e9-02.eps}
\end{center}
}}
\html{  \end{tabular}}

\begin{enumerate}
\item[(i)] \latex{\vspace*{-.06in}}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$^\circ $ out of phase, the input polarity is wrong on one 
differential amplifier; hence interchange
the input leads on {\em one} of the differential amplifiers.)

\noindent {\em Find the resonant 
frequency by this method}.  Adjusting gain and position controls so the 
signals nearly overlap will help.
\item[(ii)] \latex{\vspace*{-.06in}}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?
\item[(iii)] \latex{\vspace*{-.06in}}Remove the shorts and try $f<f_{r}$ and $f>f_{r}$. 
Explain phase behavior.
\end{enumerate}
\end{enumerate} 

\noindent OPTIONAL:
\begin{enumerate}
\item[] \latex{\vspace*{-.07in}}
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).
\end{enumerate}

\begin{center}
Part B: {\bf PARALLEL~A.C.~Circuits}
\end{center}
\setcounter{figure}{2}
\begin{figure}[!h] \centering 
\latex{\includegraphics[height=1.9in]{figs/e9-03a.eps}
}
\html{\includegraphics[width=6.0in]{figs/e9-03a.eps}}
\latex{\vspace*{-.1in}}
\caption {A parallel LRC circuit.}
\latex{\vspace*{-.1in}}
\end{figure}

\noindent INTRODUCTION:
\begin{enumerate}
\item[] \latex{\vspace*{-.06in}}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 $90^\circ$, and $i_{C}$ leads $V$ by $90^\circ $, 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 {\bf 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^\circ $ 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.

\begin{enumerate}
\item[] 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.\latex{\vspace*{-.1in}}
\end{enumerate}
\end{enumerate}

\begin{figure}[!h] \centering 
\latex{\includegraphics[height=2.in]{figs/e9-04.eps}
}
\html{\includegraphics[width=6.0in]{figs/e9-04.eps}}
\caption {A parallel LRC test circuit.}
\latex{\vspace*{-.1in}}
\end{figure}


\noindent SUGGESTED PROCEDURE: 
\begin{enumerate}
\item \latex{\vspace*{-.05in}}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.) 
\item \latex{\vspace*{-.05in}}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.  
\item \latex{\vspace*{-.05in}}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 $f_{r}$.
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}{\em 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.
\end{enumerate}
\item \latex{\vspace*{-.05in}}Starting with $f=f_{r}$ observe how the {\em 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 {\em not} in phase but are $180^\circ $ out of phase, interchange 
the input leads to one of the differential amplifiers).
\item \latex{\vspace*{-.06in}}{\em 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.
%\end{enumerate}

%INSERT
%\begin{enumerate}
\item \latex{\vspace*{-.05in}}Move the scope trace for $V$ near to the top of the scope screen. 
\item \latex{\vspace*{-.05in}}Move the scope trace for $I$ near to the bottom of the scope screen.
\item \latex{\vspace*{-.05in}}Then the ratio of    
$\frac{\mbox{\small Magnitude of upper trace}}{\mbox{\small Magnitude of lower trace}}$ is proportional to the
  impedance $Z$.
\item \latex{\vspace*{-.05in}}Notice that, as you change the frequency, $Z$ given by this ratio of traces,
goes through a maximum at the resonant frequency.
\end{enumerate}
