RC RL Circuits - ah-engr.com · RC Circuit: Resistor(s) and capacitor(s) RL Circuit: Resistor(s) and inductors(s) Called first-order circuits because they are modeled by first-order

First-Order Circuits (RC, RL) Ulaby Ch 5; Alexander Ch 7 e170, s20, DJD Page 1 of 22

Unless noted, images in this document are from:

Circuit Analysis and Design, by Ulaby, F; Maharbiz, M and Furse, C., 2018.

http://cad.eecs.umich.edu/

First-Order Circuits

Circuits with sources, resistors, and capacitors (or inductors), but not both

caps and inductors.

RC Circuit: Resistor(s) and capacitor(s)

RL Circuit: Resistor(s) and inductors(s)

Called first-order circuits because they are modeled by first-order

differential eqs.

General idea here:

Study what happens when we go from a DC state (everything is at steady state,

or not changing), then make an abrupt change, like throwing a switch, and

model the system as it approaches a new DC state (steady state).

DC Condition �� throw (open or close a switch) �� New DC Condition

…. Determine the transient response (time-varying response):

v(t) , i(t), p(t), w(t) etc.

Note: steady state does not mean current is not flowing. Steady state means

nothing is changing, i.e., all currents and voltages in system are constant.


Two types of circuits: RC and RL

Two basic set-ups:

Natural Response/Source-Free Response

Energy initially stored in capacitor (or inductor), switch thrown so that at

“end”, no energy is stored in capacitor (inductor)

Step Response/Force Response

No energy initially stored in cap (or inductor), switch thrown so that at

“end”, energy is stored in cap (inductor)


Natural Response of an RC Circuit (Source-Free Response)

Situation

• Switch has been as Position 1 for a long time

(i.e., the circuit has reached a steady

state. This is for time t ≤ 0.

• Since steady state, all currents and voltages

are constant.

�� = � �� = 0

• Cap acts like open circuit.

�� = 0�� = 0 • Voltage across cap just before t = 0 is:

�� = 0�� = �

Note: ��0�� is not necessarily Vs…. analyze

the circuit.

• Energy stored in the cap is:

�� = 0�� = 12 � ��

At t = 0, the switch is thrown to Position 2. What happens?



Adapted from: https://en.wikipedia.org/wiki/RC_circuit#/media/File:Series_RC_resistor_voltage.svg

Time, t ��

τ 0.368

2τ 0.135

3τ 0.050

4τ 0.018

5τ 0.007

��

� ��%



General Solution Method for RC Natural Response

1. Find Vo across cap. ��0�� = ��0�� = �

Solve DC problem just before switch is thrown.

2. Find time constant τ = RC (R = Req seen by cap, t > 0)

3. Apply equations

�� = �� ≥ 0 …. or �� = ��

"��

�� = − �$ ��%

& � ≥ 0

'�� = − (�

$ ��%& � ≥ 0

�� = 12 � (��

��%& � ≥ 0

�) = 12 � (� *1 − ��%

& + � ≥ 0

Notes:

• �� = � ,-.,% = � ��

/− 0&1 = � ��

/− 0)�1 = − 23

) �� ≥ 0

• Note: with one cap, the time response of all current and voltages in the

circuit vary as �^−�/6�.

• After 1 time constant τ, the capacitor voltage has gone 63% of the way to

its final value. Here, 0.37 or 37% of its initial value.


Example


Step Response of an RC Circuit (Forced Response)

Situation





are constant.

�� = � �� = 0

• Cap acts like open circuit.

�� = 0�� = 0 • Voltage across cap just before t = 0 is:

�� = 0�� = �

Note: ��0�� is not necessarily Vs1…. analyze

the circuit.

Note 2: A pure “step” has Vo = 0.


�� = 0�� = 12 � ��


For simplicity, let’s let Vo = 0 for now.




General Solution Method – Any RC Circuit

1. Find Vo across cap. ��0�� = ��0�� = �


2. Find Vf across cap at end of time. ��∞� = 8

Solve DC problem at t = ∞.

3. Find time constant τ = RC (R = Req seen by cap, t > 0)

4. Apply equations

�� = ��∞� + :��0�� − ��∞�;��%& � ≥ 0

or:

�� = 8 + < � − 8=��%& � ≥ 0

Notes:

• If � = 0, the cap voltage steps up from zero to 8.

�� = 8 >1 − ��%& ?

• If 8 = 0, the cap voltage is a natural response:

�� = ��%&

• After 1 time constant τ, the voltage has gone 63% of the way to its final

value. It is gone 63% of @ 8 − �@

• Note the cap voltage can step up A 8 > �C, or step down A 8 < �C.

• If switch is thrown at � = �E:

�� = 8 + < � − 8=��%& � ≥ 0


Example

2

1


Natural Response of an RL Circuit (Source-Free Response)

Situation





are constant.

�F = G ��F�� = 0

• Inductor acts like short circuit.

�F� = 0�� = 0 • Current through inductor just before t = 0 is:

�F� = 0�� = H�

Note: ��0�� is not necessarily Is…. analyze

the circuit.


�F� = 0�� = 12 GH��



General Solution Method for RL Natural Response

1. Find Io across cap. �F0�� = �F0�� = H�


2. Find time constant 6 = F) (R = Req seen by inductor, t > 0)

3. Apply equations

�F�� = H�� ≥ 0 …. or �F�� = H��

"��

�F�� = −$H� ��%& � ≥ 0

'�� = −$H(� ��%& � ≥ 0

�F�� = 12 GH(��

��%& � ≥ 0

�) = 12 GH(� *1 − ��%

& + � ≥ 0

Notes:

• �F�� = G ,IJ,% = GH��

/− 0&1 = GH��

/− 0F/)1 = −$H��

� ≥ 0

• Note: with one cap, the time response of all current and voltages in the

circuit vary as �^−�/6�.

• After 1 time constant τ, the capacitor voltage has gone 63% of the way to

its final value. Here, 0.37 or 37% of its initial value.


Example


Example


Example Ans: i1(t) =2.88e−10t mA; i2(t) =0.72e−20t mA


Step Response of an RL Circuit (Force Response)

Situation

• Switches have been as Position 1 for a long

time (i.e., the circuit has reached a steady



are constant.

�F = G ��F�� = 0

• Inductor acts like short circuit.

�F� = 0�� = 0 • Current through inductor just before t = 0 is:

�F� = 0�� = H�

Note: ��0�� is not necessarily Is…. analyze

the circuit.


�F� = 0�� = 12 GH��



General Solution Method – Any RL Circuit

1. Find Io through inductor. �F0�� = �F0�� = H�


2. Find If through inductor at end of time. �F∞� = H8


3. Find time constant 6 = F) (R = Req seen by inductor, t > 0)

4. Apply equations

�F�� = �F∞� + :�F0�� − �F∞�;��%& � ≥ 0

or:

�F�� = H8 + <H� − H8=��%& � ≥ 0

Notes:

• If H� = 0, the inductor current steps up from zero to H8.

�F�� = H8 >1 − ��%& ?

• If H8 = 0, the cap voltage is a natural response:

�F�� = H��%&

• After 1 time constant τ, the current has gone 63% of the way to its final

value. It is gone 63% of @H8 − H�@

• Note the cap voltage can step up AH8 > H�C, or step down AH8 < H�C.


Example


General Response of voltages and currents through R and C (or L) elements.

K��: a voltage across any R, C or L, or current through any R, C or L.

1. Find Xo. K0�� = L� (see note below).

2. Find Xf. K∞� = K8


3. Find time constant 6 4. Apply equations

K�� = K∞� + :K0�� − K∞�;��%& � ≥ 0

or:

K�� = L8 + <L� − L8=��%& � ≥ 0

Note:

• Voltage across caps must be continuous ��0�� = ��0��

• Current through inductor must be continuous �F0�� = �F0��

• Voltage across resistors and inductors can change suddenly

Solve instantaneous DC problem at t = 0+ for �)0�� and �F0��

• Current through resistors and caps can change suddenly

Solve instantaneous DC problem at t = 0+ for �)0�� and ��0��

Documents

RC RL Circuits - ah-engr.com · RC Circuit: Resistor(s) and capacitor(s) RL Circuit: Resistor(s) and inductors(s) Called first-order circuits because they are modeled by first-order