First-Order Circuits

EENG223: Circuit Theory I
DC Circuits:
First-Order Circuits
Hasan Demirel
EENG223: Circuit Theory I
First-‐Order Circuits •  Introduc)on •  The Source-‐Free RC Circuit •  The Source-‐Free RL Circuit •  Step Response of an RC Circuit •  Step Response of an RL Circuit EENG223: Circuit Theory I
First-‐Order Circuits: Introduc)on •  A ﬁrst-‐order circuit can only contain one energy storage element (a capacitor or an inductor). •  The circuit will also contain resistance. •  So there are two types of ﬁrst-‐order circuits: §  RC circuit §  RL circuit •  A ﬁrst-‐order circuit is characterized by a ﬁrst-‐
order diﬀeren)al equa)on. EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free Circuits •  A source-‐free circuit is one where all independent sources have been disconnected from the circuit a6er some switch ac7on. •  The voltages and currents in the circuit typically will have some transient response due to ini)al condi)ons (ini7al capacitor voltages and ini7al inductor currents). •  We will begin by analyzing source-‐free circuits as they are the simplest type. Later we will analyze circuits that also contain sources a6er the ini7al switch ac7on. EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RC Circuits • 
• 
A source-‐free RC circuit occurs when its dc source is suddenly disconnected. The energy already stored in the capacitor is released to the resistors. V0
•  Since the capacitor is ini)ally charged, we can assume that at 7me t=0, the ini)al voltage is: • 
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• 
• 
Then the energy stored: Applying KCL at the top node: By deﬁni7on, iC =C dv/dt and iR = v/R. Thus, EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RC Circuits V0
• 
• 
This is a ﬁrst-‐order diﬀeren-al equa-on, since only the ﬁrst deriva-ve of v is involved. Rearranging the terms: • 
Integra7ng both sides: • 
ln A is the integra7on constant. Thus • 
Taking powers of e produces: • 
From the ini-al condi-ons: v(0)=A=V0 • 
The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excita7on. EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RC Circuits • 
General form of the Diﬀeren7al Equa7ons (DE) and the response for a 1st-‐order source-‐free circuit: §  In general, a ﬁrst-‐order D.E. has the form: dx 1
+ x(t ) = 0 for t ≥ 0
dt τ
§  Solving this DE (as we did with the RC circuit) yields: x(t ) = x(0)e
−
t
τ
for t ≥ 0
§  here τ= (Greek leRer “Tau”) = )me constant(in seconds) EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RC Circuits • 
Notes concerning τ: 1) For the Source-‐Free RC circuit the DE is: §  So, for an RC circuit: τ
dv
1
+
v(t ) = 0
dt RC
for t ≥ 0
= RC
2) τ is related to the rate of exponen)al decay in a circuit as shown below. 3) It is typically easier to sketch a response in terms of mul)ples of τ than to be concerning with scaling of the graph. EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RC Circuits Ex. 7.1: In Fig. 7.5, let vC(0)= 15 V. Find vC , vx and ix for t>0. Solu)on • 
Equivalent Circuit for the above circuit can be generated: EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RC Circuits •  Equivalent Resistance seen by a Capacitor §  For the RC circuit in the previous example, it was determined that τ= RC. But what value of R should be used in circuits with mul)ple resistors? §  In general, a ﬁrst-‐order RC circuit has the following )me constant: τ = REQC
§  where REQ is the Thevenin resistance seen by the capacitor. §  More speciﬁcally, REQ = R (seen from the terminals of the capacitor for t>0 with independent sources killed.) EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RC Circuits Ex. : Refer to the circuit below. Let vC(0)= 45 V. DeterminevC , vx and io for t≥0. Solu)on • 
Consider Req seen from the capacitor. Req =
•  Time constant τ :
•  Then: 1
3
τ = Req C = 12 × = 4 s
vC (t ) = vC (0)e
v x (t ) =
12 × 6
+ 8 = 12 Ω
18
−t
4
= 45e −0.25t V
4
1
vC (t ) = 45e −0.25t = 15e −0.25t V
4+8
3
v x (t ) − vC (t ) 15e −0.25t − 45e −0.25t
io (t ) =
=
= −3.75e −0.25t V
8
8
EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RC Circuits Ex. 7.2: The switch in the circuit below has been closed for a long 7me, and it is opened at t= 0. Find v(t) for t≥0. Calculate the ini7al energy stored in the capacitor. Solu)on • 
For t<0 the switch is closed; the capacitor is an open circuit to dc, as represented in Fig. (a). •  For t>0 the switch is opened, and we have the RC circuit shown in Fig. (b). •  Time constant τ :
•  Then: •  The ini)al energy stored in the capacitor: EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RC Circuits Ex. : If the switch in Fig. below opens at t= 0, ﬁnd v(t) for t≥0 and wC(0). Solu)on • 
For t<0 the switch is closed; the capacitor is an open circuit to dc as shown in Fig. (a). 3
24 = 8 V
3+ 6
vC (0) = V0 = 8 V
vC (t ) =
for t < 0
(a) •  For t>0 the switch is opened, and we have the RC circuit shown in Fig. (b). Req =
•  Time constant τ :
•  Then: 12 × 4
=3 Ω
16
1
6
τ = Req C = 3 × = 0.5 s
v(t ) = vC (0)e
−t
0.5
(b) = 8e − 2t V
•  The ini)al energy stored in the capacitor: The image cannot be displayed. Your computer may not have enough memory to open the image, or the image
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EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RC Circuits Ex. : The switch in the circuit shown had been closed for a long 7me and then opened at 7me t = 0. a)  Determine an expression for v(t). b)  Graph v(t) versus t. c)  How long will it take for the capacitor to completely discharge? d)  Determine the capacitor voltage at 7me t=100ms.
e)  Determine the 7me at which the capacitor voltage is 10V. EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RL Circuits • 
• 
A source-‐free RL circuit occurs when its dc source is suddenly disconnected. The energy already stored in the inductor is released to the resistors. I0
• 
At 7me, t=0 , the intuctor has the ini)al current: t=0 • 
• 
• 
Then the energy stored: We can apply KVL around the loop above : By deﬁni7on, vL =L di/dt and vR = Ri. Thus, EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RL Circuits t=0 I0
• 
This is a ﬁrst-‐order diﬀeren-al equa-on, since only the ﬁrst deriva-ve of i is involved. Rearranging the terms and integra7ng: • 
Then: • 
• 
Taking powers of e produces: • 
Time constant for RL circuit becomes: The natural response of the RL circuit is an exponen7al decay of the ini7al current. EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RL Circuits • 
General form of the Diﬀeren7al Equa7ons (DE) and the response for a 1st-‐order source-‐free circuit: §  In general, a ﬁrst-‐order D.E. has the form: dx 1
+ x(t ) = 0 for t ≥ 0
dt τ
§  Solving this DE (as we did with the RL circuit) yields: x(t ) = x(0)e
§  Then: i (t ) = i (0)e
§  Where: τ =
L
R
−
t
τ
= I 0e
−
−
t
τ
for t ≥ 0
t
τ
for t ≥ 0
EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RL Circuits •  Equivalent Resistance seen by an Inductor §  For the RL circuit , it was determined that τ= L/R. As with the RC circuit, the value of R should actually be the equivalent (or Thevenin) resistance seen by the inductor. §  In general, a ﬁrst-‐order RL circuit has the following )me constant: L
τ=
REQ
§  where REQ is the Thevenin resistance seen by the inductor. §  More speciﬁcally, REQ = R (seen from the terminals of the capacitor for t>0 with independent sources killed.) EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RL Circuits Ex. 7.3: Assuming that i(0) =10 A, calculate i(t) and ix(t) in the circuit below. Solu)on • 
Thevenin resistance at the inductor terminals. we insert a voltage source with v0=1 V. Applying KVL to the two loops results (1) (2) •  Subs7tu7ng Eq. (2) into Eq. (1) gives. EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RL Circuits Ex. 7.3: Assuming that i(0) =10 A, calculate i(t) and ix(t) in the circuit below. Solu)on •  Hence, •  Time constant is: •  Thus, the current through the inductor is: EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RL Circuits Ex. 7.4: The switch in the circuit below has been closed for a long 7me. At t=0
the switch is opened. Calculate i(t) for t>0. Solu)on •  When t<0 the switch is closed, and the inductor acts as a short circuit to dc, •  Using current division: •  Current through an inductor cannot change instantaneously, EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RL Circuits Ex. 7.4: The switch in the circuit below has been closed for a long 7me. At t=0
the switch is opened. Calculate i(t) for t>0. Solu)on When t>0 the switch is open and the voltage source is disconnected. We now have the source-‐
free RL circuit in Fig. (b). •  The 7me constant is : •  Thus, EENG223: Circuit Theory I
First-‐Order Circuits: The Source-‐Free RL Circuits Ex. : Determine an expression for i(t). Sketch i(t) versus t. EENG223: Circuit Theory I
First-‐Order Circuits: Step Response of an RC Circuit •  Step Response (DC forcing func)ons) •  Consider circuits having DC forcing func)ons for t > 0 (i.e., circuits that have independent DC sources for t > 0). • 
• 
• 
• 
The general solu)on to a diﬀeren)al equa)on has two parts: x(t) = xh+ xp = homogeneous solu)on + par)cular solu)on or x(t) = xn+ xf = natural solu)on + forced solu)on •  xn is due to the ini)al condi)ons in the circuit •  and xf is due to the forcing func)ons (independent voltage and current sources for t > 0). •  xf in general take on the “form” of the forcing func)ons, •  So DC sources imply that the forced response func)on will be a constant(DC), •  Sinusoidal sources imply that the forced response will be sinusoidal, etc. EENG223: Circuit Theory I
First-‐Order Circuits: Step Response of an RC Circuit •  Step Response (DC forcing func)ons) •  Since we are only considering DC forcing func)ons in this chapter, we assume that : xf = B (constant). •  Recall that a 1st-‐order source-‐free circuit had the form Ae-t/τ. Note that there was a natural response only since there were no forcing func)ons (sources) for t > 0. So the natural response was xn = Ae − t /τ
for t > 0
•  The complete response for 1st-‐order circuit with DC forcing func)ons therefore will have the form: x(t) = xf + xn x(t ) = B + Ae − t /τ
•  The “Shortcut Method”: An easy way to ﬁnd the constants B and A is to evaluate x(t) at 2 points. Two convenient points at t = 0 and t = ∞ since the circuit is under dc condi)ons at these two points. This approach is some)mes called the “shortcut method.” EENG223: Circuit Theory I
First-‐Order Circuits: Step Response of an RC Circuit •  Step Response (DC forcing func)ons) •  The “Shortcut Method” : §  So, x(0) = B + Ae0= B + A §  And x(∞) = B + Ae-∞= B
•  Complete response yields the following expression: x(t ) = x(∞) + [ x(0) − x(∞)]e − t /τ
•  The Shortcut Method-‐ Procedure: The shortcut method will be the key method used to analyze 1st-‐order circuit with DC forcing func)ons: 1.  Analyze the circuit at t = 0-: Find x(0-) = x(0+)
2.  Analyze the circuit at t = ∞: Find x(∞) 3.  Find τ= REQC or τ= L/REQ
4.  Assume that x(t) has the form x(t) = x(∞)+[x(0) –x(∞)] e-t/τ using x(0) and x(∞) EENG223: Circuit Theory I
First-‐Order Circuits: Step Response of an RC Circuit •  Step Response (DC forcing func)ons) Notes: The “shortcut method” also works for source-‐free circuits, but x(∞) = B=0
since the circuit is dead at t = ∞. If variables other than vC or iL are needed, it is generally easiest to solve for vC or iL ﬁrst and then use the result to ﬁnd the desired variable. EENG223: Circuit Theory I
First-‐Order Circuits: Step Response of an RC Circuit Ex. 7.10: The switch in Fig. Below has been in posi7on A for a long 7me. At t=0 the switch moves to B. Determine v(t) for t>0 and calculate its value at t =1 s and 4 s. Solu)on •  Voltage across the capacitor just before t=0. Capacitor is open circuit under dc condi7ons: •  Capacitor voltage cannot change instantaneously: •  For t>0 (switch to B). Thevenin Resistance connected to the capacitor: •  Time constant: EENG223: Circuit Theory I
First-‐Order Circuits: Step Response of an RC Circuit Ex. 7.10: The switch in Fig. below has been in posi7on A for a long 7me. At t=0 the switch moves to B. Determine v(t) for t>0 and calculate its value at t 1 s and 4 s. Solu)on •  Since the capacitor acts like an open circuit to dc at steady state, v(∞) = 30 V. Thus, EENG223: Circuit Theory I
HW#3 Ex. 1 : Find v(t) and i(t) for t ≥0. Ex. 2 : Find v(t) and i(t) for t ≥0. EENG223: Circuit Theory I
First-‐Order Circuits: Step Response of an RL Circuit Ex. 3: Find v(t) and i(t) for t ≥0.

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