E72(a) Review, Solutions

Problem 1:

1. R3, R5 and R6 are in series (Rx=R3+R5+R6),
and this is in parallel with R4 (Ry=Rx||R4),
and this is in series with R2 (Rz=R2+Ry),
and this is in parallel with R1,
RAB=Rz||R1=636 Ohm 2. R1 is in series with R2 and in parallel with R4 (Rx=(R1+R2)||R4,
and this is in series with R5 and R6 and in parallel with R3
RCD=(Rx+R5+R6)||R3=727 Ohm 3. The series combination of R3, R5 and R6 is in parallel with R4, (Rx=(R3+R4+R5)||R4),
and this is in parallel with the series combination of R1 and R2,
RBC=(R1+R2)||Rx = 545 Ohm Problem 2:

1. There are only independent sources so the Thevenin Resistance is 3k Ohm (=1k+1k+2k||2k).  The Thevenin Voltage is 4.5 volts (=2.5+2).
2. Thevenin Resistance is 1k Ohm (=2k||2k), and the Thevenin voltage is 3.5 volts (=2.5+1).

Problem 3:  This one is hard, but relates directly to some transistor concepts.

1. Apply a test voltage across nodes A and B, and find resulting current  2. Apply a test voltage across nodes C and D, and find resulting current.  Problem 4:

1. As seen from Rx, the Thevenin equivalent is Vt=9.5 V(=5+4.5), Rt=100 Ohm.  Therefore, for max power, Rx=100 Ohm.
2. Power dissipated in Rx is equal to the power in the other three resistors.  Power=4.75V*47.5mA=.23 W.

Problem 5:

For both parts of this problem t=RC and v(t)=v(∞)+(v(0)-v(∞))e-t/t

1. (a) v(0)=0, v(∞)=1.     b) v(0)=1, v(∞)=0.
2. (a) v(0)=1, v(∞)=-1.    b) v(0)=-2, v(∞)=0.

Problem 6:

a)

1. Exact in blue, asymptotic in red
2. From Bode plot (asymptotic) magnitude is unity (0 dB) and phase is 0 (-6o on exact plot), so output is equal to input (10sin(100t).
3. From Bode plot, magnitude is 0.1 (-20 dB) and phase is -90, so output is sin(10000t-90°)=-cos(10000t).
4. This is a lowpass filter

b)

1. Exact in blue, asymptotic in red
2. From Bode plot, magnitude is 0.1(-20 dB) and phase is 90, so output is sin(100t+90°)=cos(100t).
3. From Bode plot, magnitude is unity and phase is 0, so output is 10sin(10000t).
4. This is a highpass filter

Problem 7:

For both parts of this problem t=RC and v(t)=v(∞)+(v(0)-v(∞))e-t/t

1. When the switch is opened
•  the resistance "seen" by the capacitor is R=R3||(R2+R1)=667 Ohm.
• v(0)=V1*R3/(R3+R1)
• v(∞)=V1*R3/(R3+R2+R1)
2. When the switch is closed
•  the resistance "seen" by the capacitor is R=R3||R1=500 Ohm.
• v(0)=V1*R3/(R3+R2+R1)
• v(∞)=V1*R3/(R3+R1)

Problem 8:

For both parts of this problem t=RC and v(t)=v(∞)+(v(0)-v(∞))e-t/t

1. When the switch is opened
•  the resistance "seen" by the capacitor is R=R1+R2
• v(0)=(V1)*1/3
• v(∞)=V1
We want to solve for t such that: 2. When the switch is closed
•  the resistance "seen" by the capacitor is R=R1
• v(0)=(V1)*2/3
• v(∞)=0
We want to solve for t such that Note that the time constant is different for the two parts, though the form of the solution is not.

Problem 9:

1. 2. Problem 10:

Recall that for a generic first order system with a step input that: so Problem 11:

1. At low frequencies, inductor is short, output is zero.  At high frequencies, capacitor is short, output is zero.  Output must be non-zero at intermediate frequencies, therefore this is a bandpass filter.
2. 3. At s=0 and s=∞, the magnitude of the transfer function is zero.
4. It decreases.

Problem 12:

1. At low frequencies, capacitor is open circuit, output=input.  At high frequencies, capacitor is short circuit, output=0.  Therefore this is a lowpass filter.
2. 3. At s=0, Xfer function=1.  At s=∞, the Xfer function=0.
4. It increases.

Problem 13:

1. The expression below shows Vo=0 if r=0 2. see (i).
3. if r<<R, then Problem 14:

1. 2.5V+2V=4.5V
2. 2.5V+1V=3.5V

Problem 15: We take the positive root since resistance must be positive.