Consider the case where the incoming system and the running system are running a
ID: 2250216 • Letter: C
Question
Consider the case where the incoming system and the running system are running at two different frequencies. Assume the running system is 60 Hz sinusoids of the form.
V_An=sin(260t) (Eq. 1a)
V_Bn=sin(260t-2/3) (Eq. 1b)
V_Cn=sin(260t+2/3) (Eq. 1c)
And the incoming voltages at f Hz,
V_an=sin(2ft) (Eq. 2a)
V_bn=sin(2ft-2/3) (Eq. 2b)
V_cn=sin(2ft+2/3) (Eq. 2c)
The voltage across the lamps is the difference between each phase of the 2 systems. Applying the trig identity:
sin()-sin()=2 sin(1/2 (-))cos(1/2 (+))
Then the voltage across each lamp is:
V_Aa=V_An-V_an=[2sin((60-f)t)] cos((60+f)t) (Eq. 3a)
V_Bb=V_Bn-V_bn=[2sin((60-f)t)] cos((60+f)t+2/3) (Eq. 3b)
V_Cc=V_Cn-V_cn=[2sin((60-f)t)] cos((60+f)t-2/3) (Eq. 3c)
Now we can assume the sine term is a changing amplitude. Let f be 59 Hz, then the voltages are:
V_Aa=[2sin(t)] cos(2 59.5 t) (Eq. 3a)
V_Bb=[2sin(t)] cos(2 59.5 t+2/3) (Eq. 3b)
V_Cc=[2sin(t)] cos(2 59.5 t-2/3) (Eq. 3c)
Therefore, the three lamps blink at a rate of 0.5 Hz (once per second) with the sequence of a, b, and then c (thus the term “rotating” lamps).
Use Matlab or your favorite plotting program to plot the voltages in (4) over 10 seconds and note the changing voltage.
For the following conditions, calculate the lamp voltages and qualitatively state the lighting conditions of the lamps.
The two systems are perfectly in phase and operating at the same frequency.
The running system is operating at 60 Hz and the incoming system is operating at 61 Hz.
The two systems are 90 degrees out of phase and operating at the same frequency.
The two systems are 180 degrees out of phase and operating at the same frequency.
Bonus - For each of these 4 conditions above, verify your hypothesis by plotting the voltages across each of the bulbs over a 2 second period. Use Matlab or your favorite plotting program to calculate and plot the voltages.
Explanation / Answer
MATLAB CODE:
clc;
clear all;
close all;
t=0:0.01:1;
V_An=sin(2*pi*60*t); %(Eq. 1a)
V_Bn=sin(2*pi*60*t-2*pi/3); %(Eq. 1b)
V_Cn=sin(2*pi*60*t+2*pi/3); %(Eq. 1c)
f= 59;
%-- Incoming signal
V_an=sin(2*pi*f*t); %(Eq. 2a)
V_bn=sin(2*pi*f*t-2*pi/3); %(Eq. 2b)
V_cn=sin(2*pi*f*t+2*pi/3); %(Eq. 2c)
V_Aa=V_An-V_an;
V_Bb=V_Bn-V_bn;
V_Cc=V_Cn-V_cn;
plot(t,V_Aa,t,V_Bb,t,V_Cc);
title('System f=59 Hz');
%-----------In phase & same frequency ------
V_An=sin(2*pi*60*t); %(Eq. 1a)
V_Bn=sin(2*pi*60*t-2*pi/3); %(Eq. 1b)
V_Cn=sin(2*pi*60*t+2*pi/3); %(Eq. 1c)
f= 60;
%-- Incoming signal
V_an=sin(2*pi*f*t); %(Eq. 2a)
V_bn=sin(2*pi*f*t-2*pi/3); %(Eq. 2b)
V_cn=sin(2*pi*f*t+2*pi/3); %(Eq. 2c)
V_Aa=V_An-V_an;
V_Bb=V_Bn-V_bn;
V_Cc=V_Cn-V_cn;
figure;
plot(t,V_Aa,t,V_Bb,t,V_Cc);
title('System with In phase & same frequency f=60 Hz');
%----------- In phase & f=61------------
V_An=sin(2*pi*60*t); %(Eq. 1a)
V_Bn=sin(2*pi*60*t-2*pi/3); %(Eq. 1b)
V_Cn=sin(2*pi*60*t+2*pi/3); %(Eq. 1c)
f= 61;
%-- Incoming signal
V_an=sin(2*pi*f*t); %(Eq. 2a)
V_bn=sin(2*pi*f*t-2*pi/3); %(Eq. 2b)
V_cn=sin(2*pi*f*t+2*pi/3); %(Eq. 2c)
V_Aa=V_An-V_an;
V_Bb=V_Bn-V_bn;
V_Cc=V_Cn-V_cn;
figure;
plot(t,V_Aa,t,V_Bb,t,V_Cc);
title('System with In phase & same frequency f=61 Hz');
%-------------- out phase = 90deg & same frequency =60 ---------
V_An=sin(2*pi*60*t); %(Eq. 1a)
V_Bn=sin(2*pi*60*t-2*pi/3); %(Eq. 1b)
V_Cn=sin(2*pi*60*t+2*pi/3); %(Eq. 1c)
f= 60;
%-- Incoming signal
V_an=sin(2*pi*f*t+pi/2); %(Eq. 2a)
V_bn=sin(2*pi*f*t-2*pi/3+pi/2); %(Eq. 2b)
V_cn=sin(2*pi*f*t+2*pi/3+pi/2); %(Eq. 2c)
V_Aa=V_An-V_an;
V_Bb=V_Bn-V_bn;
V_Cc=V_Cn-V_cn;
figure;
plot(t,V_Aa,t,V_Bb,t,V_Cc);
title('System with Out phase = 90deg & same frequency f=60 Hz');
%--------------- Out phase = 180deg & same frequency=60 --------------
V_An=sin(2*pi*60*t); %(Eq. 1a)
V_Bn=sin(2*pi*60*t-2*pi/3); %(Eq. 1b)
V_Cn=sin(2*pi*60*t+2*pi/3); %(Eq. 1c)
f= 60;
%-- Incoming signal
V_an=sin(2*pi*f*t+pi); %(Eq. 2a)
V_bn=sin(2*pi*f*t-2*pi/3+pi); %(Eq. 2b)
V_cn=sin(2*pi*f*t+2*pi/3+pi); %(Eq. 2c)
V_Aa=V_An-V_an;
V_Bb=V_Bn-V_bn;
V_Cc=V_Cn-V_cn;
figure;
plot(t,V_Aa,t,V_Bb,t,V_Cc);
title('System with Out phase=180deg & same frequency f=60 Hz');