EE 101 Homework 5 Due: 09/27/16
To receive full credit all homework must:
• use correct format (see example 1st homework),
• be legible and organized,
• show complete work (including method, math, redrawn circuits, etc.),
• have complete (including units) and boxed answers,
• only write on front side of each page and staple multiple pages together,
• be turned in at beginning of class.
Start homework early. Help sessions are Sun-Thurs 5-9pm in Workman 116.
A few notes for you:
• τ = RC
• Charging equation for RC circuit: v
C
( t ) = v s
(1 − e
− t
τ
• Discharging equation for RC circuit: v
C
( t ) = v s e
− t
τ
V
) V
• The 10% to 90% rise time, t r
, is given as t
90
− t
10
, or the time when the voltage is at 90% minus the time when the voltage is at 10%.
• Math for finding t
10
: 0 .
1 v s
= v s t
10
(1 − e
(and t
90
− t
10
τ
) with simple subs):
1 multiply
−−−−−−−→ 0 .
1 = 1 − e
− mult. by − 1
−−−−−−−→ 0 .
9 = e so: t
10
− t
10
τ
−−−−−−−→ ln 0 .
9 = − t
10
τ
= 0 .
1054 τ and t
90
= 2 .
303 τ t
10
τ
−−−−−−−→ − mult. by τ
−−−−−−→ t
10
= − τ
0 .
9 = ln 0 .
9
− e
− t
10
τ
• Using t r or t
10 and t
90 to find τ : τ ≈ t r
2 .
2
= t
90
−
2 .
2 t
10
1. Consider the following circuit, where the switch can move between positions A and B.
250kΩ
A i
+ v
R
−
B
+
9V
−
+
2
µ
F v
C
−
(a) Assume the switch has been at position B for a long time, and it moves to position A at t = 0s. Find the following:
• Is this RC circuit charging or discharging?
• Determine the time-constant, τ ,
• Using the time-constant, estimate the 10% to 90% rise time, t r
,
• Write the mathematical expression for the capacitor’s voltage, v
C
(t),
• Accurately sketch v
C
(t) verses time using values at multiples of the time-constant as a guide (remember that we consider the capacitor charged after 5 time constants),
• Label the rise time on the sketch of v
C
(t) and check that your estimate appears correct,
• Find the mathematical expression for the resistor’s voltage, v
R
(t),
• Draw a rough sketch v
R
(t) verses time to show shape,
• Find the mathematical expression for the current, i (t), and
• Draw a rough sketch i (t) verses time to show shape.
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EE 101 Homework 5 Due: 09/27/16
(b) Assume the switch has been at position A for a long time, and it moves to position B at t = 0s. Find the following:
• Is this RC circuit charging or discharging?
• Determine the time-constant, τ ,
• Write the mathematical expression for the capacitor’s voltage, v
C
(t),
• Draw a rough sketch v
C
(t) verses time to show shape,
• Find the mathematical expression for the resistor’s voltage, v
R
(t),
• Draw a rough sketch v
R
(t) verses time to show shape,
• Find the mathematical expression for the current, i (t), and
• Draw a rough sketch i (t) verses time to show shape.
2. Assume the switch has been at position B for a long time, and it moves to position A at t = 0s, i.e., this is a ”charging” circuit.
470kΩ
A t = 0 s
+
B v
R
− i
+
5V
−
+
C v
C
−
Given that v
C
= 0 .
5V at time t = 0 .
5ms and v
C
= 4 .
5V at time t = 11 .
5ms.
• Is this RC circuit charging or discharging?
• Calculate the 10% to 90% rise time, t r
,
• Determine time-constant, τ ,
• Determine the value of the unknown capacitor, C,
• Write the mathematical expression for the capacitor’s voltage, v
C
(t), and
• Draw a rough sketch v
C
(t) verses time to show shape.
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