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### Lecture 15 – Transient Response of 1st Order Circuits

### 1. DC Steady State

### 3. DC Steady-State + Transient Response

Good, Better, Best

Elder Oaks (October 2007):

As we consider various choices, we should remember that it is not enough that something is good. Other choices are better, and still others are best. Even though a particular choice is more costly, its far greater value may make it the best choice of all.

I have never known of a man who looked back on his working life and said, “I just didn't spend enough time with my job.”

Discussion #15 – 1st Order Transient Response

DC Steady-State

Transient Response

Discussion #15 – 1st Order Transient Response

R1

R2

vs

vs

+

–

+

–

L1

L2

1st Order CircuitsElectric circuit 1st order system: any circuit containing a single energy storing element (either a capacitor or an inductor) and any number of sources and resistors

R1

R2

C

1st order

2nd order

Discussion #15 – 1st Order Transient Response

Capacitor/Inductor Voltages/Currents

Review of capacitor/inductor currents and voltages

- Exponential growth/decay

NB: note the duality between inductors and capacitors

Capacitor current iC(t)

Inductor voltage vL(t)

NB: both can change instantaneously

Capacitor voltage vC(t)

Inductor current iL(t)

NB: neither can change instantaneously

Discussion #15 – 1st Order Transient Response

R1

+

–

t = 0

R2

C

Transient ResponseTransient response of a circuit consists of 3 parts:

- Steady-state response prior to the switching on/off of a DC source
- Transient response – the circuit adjusts to the DC source
- Steady-state response following the transient response

DC Source

Energy element

Switch

Discussion #15 – 1st Order Transient Response

1st and 3rd Step in Transient Response

Discussion #15 – 1st Order Transient Response

DC Steady-State

DC steady-state: the stable voltages and currents in a circuit connected to a DC source

Capacitors act like open circuits at DC steady-state

Inductors act like short circuits at DC steady-state

Discussion #15 – 1st Order Transient Response

R1

t → ∞

t = 0

vs

vs

+

–

+

–

R2

R2

R3

R3

C

C

DC Steady-StateInitial condition x(0): DC steady state before a switch is first activated

- x(0–): right before the switch is closed
- x(0+): right after the switch is closed

Final condition x(∞): DC steady state a long time after a switch is activated

Final condition

Initial condition

Discussion #15 – 1st Order Transient Response

t = 0

vs

+

–

R2

R3

C

DC Steady-State- Example 1: determine the final condition capacitor voltage
- vs= 12V, R1 = 100Ω, R2 = 75Ω, R3 = 250Ω, C = 1uF

Discussion #15 – 1st Order Transient Response

t → 0+

vs

+

–

R2

R3

C

DC Steady-State- Example 1: determine the final condition capacitor voltage
- vs= 12V, R1 = 100Ω, R2 = 75Ω, R3 = 250Ω, C = 1uF

- Close the switch and find initial conditions to the capacitor

NB: Initially (t = 0+) current across the capacitor changes instantly but voltage cannot change instantly, thus it acts as a short circuit

Discussion #15 – 1st Order Transient Response

t → ∞

vs

+

–

R2

R3

C

DC Steady-State- Example 1: determine the final condition capacitor voltage
- vs= 12V, R1 = 100Ω, R2 = 75Ω, R3 = 250Ω, C = 1uF

Close the switch and apply finial conditions to the capacitor

NB: since we have an open circuit no current flows through R2

Discussion #15 – 1st Order Transient Response

DC Steady-State

Remember – capacitor voltages and inductor currents cannot change instantaneously

- Capacitor voltages and inductor currents don’t change right before closing and right after closing a switch

Discussion #15 – 1st Order Transient Response

is

iL

L

R

DC Steady-State- Example 2: find the initial and final current conditions at the inductor

is = 10mA

Discussion #15 – 1st Order Transient Response

is

is

iL

L

R

iL

DC Steady-State- Example 2: find the initial and final current conditions at the inductor

is = 10mA

- Initial conditions – assume the current across the inductor is in steady-state.

NB: in DC steady state inductors act like short circuits, thus no current flows through R

Discussion #15 – 1st Order Transient Response

iL

DC Steady-State- Example 2: find the initial and final current conditions at the inductor

is = 10mA

- Initial conditions – assume the current across the inductor is in steady-state.

Discussion #15 – 1st Order Transient Response

is

iL

iL

L

R

L

R

DC Steady-State- Example 2: find the initial and final current conditions at the inductor

is = 10mA

- Initial conditions – assume the current across the inductor is in steady-state.
- Throw the switch

NB: inductor current cannot change instantaneously

Discussion #15 – 1st Order Transient Response

is

–

R

+

iL

iL

L

R

L

DC Steady-State- Example 2: find the initial and final current conditions at the inductor

is = 10mA

- Initial conditions – assume the current across the inductor is in steady-state.
- Throw the switch
- Find initial conditions again (non-steady state)

NB: polarity of R

NB: inductor current cannot change instantaneously

Discussion #15 – 1st Order Transient Response

DC Steady-State

- Example 2: find the initial and final current conditions at the inductor

is = 10mA

- Initial conditions – assume the current across the inductor is in steady-state.
- Throw the switch
- Find initial conditions again (non-steady state)
- Final conditions (steady-state)

t = 0

iL

L

R

NB: since there is no source attached to the inductor, its current is drained by the resistor R

Discussion #15 – 1st Order Transient Response

+

C

–

+

–

iC

vs

General Solution of 1st Order Circuits- Expressions for voltage and current of a 1st order circuit are a 1st order differential equation

NB: Review lecture 11 for derivation of these equations

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

- Expressions for voltage and current of a 1st order circuit are a 1st order differential equation

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

- Expressions for voltage and current of a 1st order circuit are a 1st order differential equation

NB: Constants

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

- Expressions for voltage and current of a 1st order circuit are a 1st order differential equation

NB: similarities

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

- Expressions for voltage and current of a 1st order circuit are a 1st order differential equation

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

- Expressions for voltage and current of a 1st order circuit are a 1st order differential equation

Capacitor/inductor voltage/current

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

- Expressions for voltage and current of a 1st order circuit are a 1st order differential equation

Forcing function

(F – for DC source)

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

- Expressions for voltage and current of a 1st order circuit are a 1st order differential equation

Combinations of circuit element parameters (Constants)

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

- Expressions for voltage and current of a 1st order circuit are a 1st order differential equation

Forcing Function F

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

- Expressions for voltage and current of a 1st order circuit are a 1st order differential equation

DC gain

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

- Expressions for voltage and current of a 1st order circuit are a 1st order differential equation

DC gain

Time constant

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

The solution to this equation (the complete response) consists of two parts:

- Natural response (homogeneous solution)
- Forcing function equal to zero
- Forced response (particular solution)

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

Natural response (homogeneous or natural solution)

- Forcing function equal to zero

Has known solution of the form:

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

Forced response (particular or forced solution)

F is constant for DC sources, thus derivative is zero

NB: This is the DC steady-state solution

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

Complete response (natural + forced)

Solve for α by solving x(t) at t = 0

Time constant

Final condition

Initial condition

Discussion #15 – 1st Order Transient Response

General Solution of 1st Order Circuits

Complete response (natural + forced)

Transient Response

Steady-State Response

Discussion #15 – 1st Order Transient Response

Full Transient Response

Discussion #15 – 1st Order Transient Response

R1

+

–

t = 0

R2

C

Transient ResponseTransient response of a circuit consists of 3 parts:

- Steady-state response prior to the switching on/off of a DC source
- Transient response – the circuit adjusts to the DC source
- Steady-state response following the transient response

DC Source

Energy element

Switch

Discussion #15 – 1st Order Transient Response

Transient Response

Solving 1st order transient response:

- Solve the DC steady-state circuit:
- Initial condition x(0–): before switching (on/off)
- Final condition x(∞): After any transients have died out (t → ∞)
- Identify x(0+): the circuit initial conditions
- Capacitors: vC(0+) = vC(0–)
- Inductors: iL(0+) = iL(0–)
- Write a differential equation for the circuit at time t = 0+
- Reduce the circuit to its Thévenin or Norton equivalent
- The energy storage element (capacitor or inductor) is the load
- The differential equation will be either in terms of vC(t) or iL(t)
- Reduce this equation to standard form
- Solve for the time constant
- Capacitive circuits: τ = RTC
- Inductive circuits: τ = L/RT
- Write the complete response in the form:
- x(t) = x(∞) + [x(0) - x(∞)]e-t/τ

Discussion #15 – 1st Order Transient Response

t = 0

+

vC(t)

–

C

vs

i(t)

+

–

Transient Response- Example 3: find vc(t) for all t
- vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF

Discussion #15 – 1st Order Transient Response

t = 0

+

vC(t)

–

C

vs

i(t)

+

–

Transient Response- Example 3: find vc(t) for all t

vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF

- DC steady-state
- Initial condition: vC(0)
- Final condition: vC(∞)

NB: as t → ∞ the capacitor acts like an open circuit thus vC(∞) = vS

Discussion #15 – 1st Order Transient Response

t = 0

+

vC(t)

–

C

vs

i(t)

+

–

Transient Response- Example 3: find vc(t) for all t

vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF

- Circuit initial conditions: vC(0+)

Discussion #15 – 1st Order Transient Response

t = 0

+

vC(t)

–

C

vs

i(t)

+

–

Transient Response- Example 3: find vc(t) for all t

vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF

- Write differential equation (already in Thévenin equivalent) at t = 0

Recall:

Discussion #15 – 1st Order Transient Response

t = 0

+

vC(t)

–

C

vs

i(t)

+

–

Transient Response- Example 3: find vc(t) for all t

vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF

- Write differential equation (already in Thévenin equivalent) at t = 0

NB: solution is of the form:

Discussion #15 – 1st Order Transient Response

t = 0

+

vC(t)

–

C

vs

i(t)

+

–

Transient Response- Example 3: find vc(t) for all t

vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF

- Find the time constant τ

Discussion #15 – 1st Order Transient Response

t = 0

+

vC(t)

–

C

vs

i(t)

+

–

Transient Response- Example 3: find vc(t) for all t

vs= 12V, vC(0–) = 5V, R = 1000Ω, C = 470uF

- Write the complete response
- x(t) = x(∞) + [x(0) - x(∞)]e-t/τ

Discussion #15 – 1st Order Transient Response

R3

t = 0

+

vC(t)

–

+

–

R2

C

vb

va

+

–

Transient Response- Example 4: find vc(t) for all t

va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF

Discussion #15 – 1st Order Transient Response

R3

t = 0

+

vC(t)

–

+

–

R2

C

vb

va

+

–

Transient Response- Example 4: find vc(t) for all t

va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF

- DC steady-state
- Initial condition: vC(0)
- Final condition: vC(∞)

For t < 0 the capacitor has been charged by vb thus vC(0–) = vb

For t → ∞ vc(∞) is not so easily determined – it is equal to vT (the open circuited Thévenin equivalent)

Discussion #15 – 1st Order Transient Response

R3

t = 0

+

vC(t)

–

+

–

R2

C

vb

va

+

–

Transient Response- Example 4: find vc(t) for all t

va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF

- Circuit initial conditions: vC(0+)

Discussion #15 – 1st Order Transient Response

R3

t = 0

+

vC(t)

–

+

–

R2

C

vb

va

+

–

Transient Response- Example 4: find vc(t) for all t

va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF

- Write differential equation at t = 0
- Find Thévenin equivalent

Discussion #15 – 1st Order Transient Response

vC(t)

–

R1

R2

ia

C

R3

ib

Transient Response- Example 4: find vc(t) for all t

va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF

- Write differential equation at t = 0
- Find Thévenin equivalent

Discussion #15 – 1st Order Transient Response

Transient Response

- Example 4: find vc(t) for all t

va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF

- Write differential equation at t = 0
- Find Thévenin equivalent

+

vC(t)

–

R1

R2

R3

C

Discussion #15 – 1st Order Transient Response

Transient Response

- Example 4: find vc(t) for all t

va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF

- Write differential equation at t = 0
- Find Thévenin equivalent

+

vC(t)

–

RT

C

Discussion #15 – 1st Order Transient Response

+

vC(t)

–

+

–

vT

C

Transient Response- Example 4: find vc(t) for all t

va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF

- Write differential equation at t = 0
- Find Thévenin equivalent
- Reduce equation to standard form

Discussion #15 – 1st Order Transient Response

+

vC(t)

–

+

–

vT

C

Transient Response- Example 4: find vc(t) for all t

va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF

- Find the time constant τ

Discussion #15 – 1st Order Transient Response

+

vC(t)

–

+

–

vT

C

Transient Response- Example 4: find vc(t) for all t

va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF

- Write the complete response
- x(t) = x(∞) + [x(0) - x(∞)]e-t/τ

Discussion #15 – 1st Order Transient Response

R3

t = 0

+

vC(t)

–

+

–

R2

C

vb

va

+

–

Transient Response- Example 4: find vc(t) for all t

va= 12V, vb= 5V, R1 = 10Ω, R2 = 5Ω, R3 = 10Ω, C = 1uF

- Write the complete response
- x(t) = x(∞) + [x(0) - x(∞)]e-t/τ

Discussion #15 – 1st Order Transient Response

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