Impedance and AC Analysis I

In this experiment, we will introduce how "real" inductor, which is model as a series resistance to account for the resistance of the wires inside the inductor

Th impedance looking into a real inductor will be
Z=R_L+jwL

and the magnitude of the impedance is the magnitude of Z
|Z|=sqrt(R_L^2+(wL)^)

first, we measured the internal resistance of the inductor
R_L=3.4ohm

next we want to apply a known AC voltage and measure the resultant AC current

 Here R_ext is used to limit the current. The function generator has an internal resistance of about 50ohm. Also, we assume the ammeter has no voltage drop
We will use R_ext =68.5ohm

Next we will set the RMS voltage of the function generator with 20KHZ

Secondly, we will device the circuit and  record the reading from the multimeter and ammeter

V_in=4.53V
I_in=88mA

The reading is different from the FB display because there is some internal resistance due to the function generator.

Calculations:
Z=V/I = 4.52/88mA=51.36ohm

Z=R_ext+R_L+jwL
|Z|=sqrt((R_ext+R_L)^2+(wL)^2))
W=2pif=2pi*20000=125664rad/s
Since we know the value of Z, R_ext, R_L and w
we will need to find L, which is about 0.4mH

Next we will consider the circuit that will add a capacitor that is in series with the inductor to try to cancel the inductive par of the real inductor impedance
to know what is the value of capacitor
we set
wL=1/wC
C=1.58*10-7 F
since there is no capacitor box to use
we find the closest value of capacitor to use in the circuit, which is 1.51*10^-7F
Thus, we need to go back and  redo the math to find the critical frequency
f = 1/(2*pi*50.27*.151*10^-6)=20.97KHz
 Next, we take scope to measurements at 20.97KHZ
V_PP_CH1=23.08V
V_PP_CH2=19.49
delta_t=19.46 us

phase angle = w*t*180/pi=125664*19.46*10^-6*180/pi=146 degree

Next, we used DDM to measure the voltage and current at different frequency

Frequency(KHz)	V_in(V)	I_In(mA)	|Z_in|(ohm)
5	6.21	29.57	210
10	6.23	48.33	108
20.97	5.02	69.15	72.6
30	5.56	66.99	83
50	5.96	46.93	127

Follow-up question
1. Why is the input current that largest at 12.7K?
The input current is largest because the impedance of the element is the smallest

2. Calculate the theoretical voltage phasor across the real inductor at 20.97KHz (use the DMM measurement value as the source voltage phasor magnitude). Compare this with the scope measurements. Convert the scope measurement to RMS for comparison purposes.

(23.08/(2sqrt(2))*2pi*20.97*1000*0.4*10^-3/(68.5+3.4)=5.98V

19.49/(2sqrt(2))=6.89

(6.89-5.98)/5.98=15.2%

3. Does the circuit look more capacitive or inductive at frequencies below 20kHZ?
capacitive

4. Does the circuit look more capacitive or inductive at frequencies above 20kHZ?
inductive