FIRST-ORDER CIRCUITS
EEE130 ELECTRICAL CIRCUIT THEORY 1
Engr. Rogelio F. Bersano Jr.
Instructor
Outline
❖ Introduction
❖ The Source-Free RC Circuit
❖ The Source-Free RL Circuit
❖ Step Response of an RC Circuit
❖ Step Response of an RL Circuit
First-Order Circuits: Introduction
❑ 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 differential equation.
First-Order Circuits: The Source-Free Circuits
First-Order Circuits: The Source-Free RC Circuits
First-Order Circuits: The Source-Free RC Circuits
First-Order Circuits: The Source-Free RC Circuits
First-Order Circuits: The Source-Free RC Circuits
First-Order Circuits: The Source-Free RC Circuits
Example 1
First-Order Circuits: The Source-Free RC Circuits
Example 1: Solution
Solve for the time constant, 𝜏
𝜏 = 𝑅𝑒𝑞 𝐶
Solve for vx and ix
Now,
𝑅𝑒𝑞 = 5||(8 + 12)
𝑅𝑒𝑞
5 ∗ 20
=
=4Ω
5 + 20
Therefore,
𝜏 = 4 0.1 = 0.4 𝑠
Thus,
𝑣𝑐 = 𝑣0 𝑒
−𝑡ൗ
𝜏
= 15𝑒
−𝑡ൗ
0.4
𝑣𝑐 = 15𝑒 −2.5𝑡 𝑓𝑜𝑟 𝑡 ≥ 0
𝑣𝑥 =
12
𝑣𝑐 = 0.6 ∗ 15𝑒 −2.5𝑡
8 + 12
𝑣𝑥 = 9𝑒 −2.5𝑡
𝑖𝑥 =
𝑣𝑥
= 0.75𝑒 −2.5𝑡
12
for t > 0
First-Order Circuits: The Source-Free RC Circuits
Example 2
First-Order Circuits: The Source-Free RC Circuits
Example 2: Solution
Solve for the time constant, 𝜏
𝜏 = 𝑅𝑒𝑞 𝐶
Solve for vx and i0
Now,
𝑅𝑒𝑞 = 8 + (6||12)
𝑅𝑒𝑞 = 12 Ω
𝜏 = 12 1/3 = 4 𝑠
Thus,
𝑣𝑐 = 𝑣0
4
1
𝑣𝑐 =
∗ 45𝑒 −0.25𝑡
8+4
3
𝑣𝑥 = 15𝑒 −0.25𝑡
Therefore,
−𝑡
𝑒 ൗ𝜏
𝑣𝑥 =
=
−𝑡
45𝑒 ൗ4
𝑣𝑐 = 45𝑒 −0.25𝑡 𝑓𝑜𝑟 𝑡 ≥ 0
𝑖0 =
−𝑣𝑥
= −3.75𝑒 −.0.25𝑡
4
or
𝑑𝑣𝑐
1
𝑖0 = 𝐶
=
−0.25 ∗ 45𝑒 −0.25𝑡
𝑑𝑡
3
= −3.75𝑒 −0.25𝑡
for t > 0
First-Order Circuits: The Source-Free RC Circuits
Example 3
First-Order Circuits: The Source-Free RC Circuits
Example 3: Solution
At t < 0, the capacitor is fully charged
Solve for the time constant, 𝜏
𝑅𝑒𝑞 = 9 + 1 = 10 Ω
𝜏 = 𝑅𝑒𝑞 𝐶 = 10 ∗ 20𝑚 = 0.2 𝑠
Therefore,
Solve for vc,
𝑣𝑐 0− = 𝑣𝑐 0+ =
At t > 0,
9
∗ 20 = 15 𝑉
9+3
𝑣𝑐 = 𝑣0 𝑒
−𝑡ൗ
𝜏
= 15𝑒
−𝑡ൗ
0.2
𝑣𝑐 = 15𝑒 −5𝑡 𝑓𝑜𝑟 𝑡 ≥ 0
Solve for energy stored,
𝑊=
1 2
𝐶𝑉 = 0.5 20𝑚 15
2
𝑊 = 2.25 𝐽
2
First-Order Circuits: The Source-Free RL Circuits
First-Order Circuits: The Source-Free RL Circuits
First-Order Circuits: The Source-Free RL Circuits
First-Order Circuits: The Source-Free RL Circuits
First-Order Circuits: The Source-Free RL Circuits
Example 4
First-Order Circuits: The Source-Free RC Circuits
Example 4: Solution
Solve for Req at the inductor terminals
using Thevenin’s theorem
Substituting i2 to previous
equation gives us,
Hence,
Applying KVL at loop 1
Applying KVL at loop 2
Solve for the time constant,
Thus, the current through the
inductor is
First-Order Circuits: The Source-Free RC Circuits
Example 4: Solution using Differential Equations
Using Mesh Analysis,
Substituting i2 to previous
equation gives us,
Applying KVL at loop 1
Integrating both sides gives us,
Taking the powers of e, we
finally obtain
Applying KVL at loop 2
First-Order Circuits: The Source-Free RC Circuits
Example 4: Solution
The voltage across the inductor is
Since the inductor and the 2-Ω resistor
are in parallel,
First-Order Circuits: The Source-Free RL Circuits
Example 5
First-Order Circuits: The Source-Free RC Circuits
Example 5: Solution
At t < 0, the inductor is fully charged
Solve for Req and then the time constant
Solve for iL,
𝑖𝐿 0− = 𝑖𝐿 0+ =
At t > 0,
12
𝑖 =6𝐴
12 + 4 1
Thus,
First-Order Circuits: The Step Response of an RC/RL Circuit
First-Order Circuits: The Step Response of an RC/RL Circuit
First-Order Circuits: The Step Response of an RC/RL Circuit
First-Order Circuits: The Step Response of an RC/RL Circuit
First-Order Circuits: The Step Response of an RC/RL Circuit
Example 6
First-Order Circuits: The Source-Free RC Circuits
Example 6: Solution
At t < 0, the capacitor is fully charged.
Solve for the initial voltage,
𝑣𝑐 0− = 𝑣𝑐 0+ =
5
∗ 24 = 15 𝑉
5+3
For t > 0, the switch moved to position
B. Solve for the time constant
𝜏 = 𝑅𝑒𝑞 𝐶 = 4𝑘 ∗ 0.5𝑚 = 2 𝑠
Thus, from the general expression
of the complete response
𝑣𝑐 𝑡 = 𝑣 ∞ + 𝑣 0 − 𝑣 ∞ 𝑒
𝑣𝑐 𝑡 = 30 + 15 − 30
−𝑡ൗ
𝜏
𝑡ൗ
𝑒 2
𝑣𝑐 𝑡 = 15 2 − 𝑒 −0.5𝑡 𝑓𝑜𝑟 𝑡 > 0
At t = 0+ , 𝑣 0+ = 15 𝑉
At t =1 s,
At t = ∞, the capacitor will be opened
𝑣 ∞ = 30 𝑉
At t =4 s,
First-Order Circuits: The Step Response of an RC/RL Circuit
Seatwork
The circuit shown in the figure below is at steady state before
the switch closes. The response of the circuit is the voltage
v(t). Find v(t) for t > 0.