ENGN1218 Electronic Systems and Design
Topic 5 First-Order RC and RL Circuits
Passive Circuit Elements:
Resistance, Capacitance and Inductance
Overview • Element characteristics
• Resistance, capacitance and inductance
• Energy storage
ENGN1218 Electronic Systems and Design
Circuit Element Characteristics Circuit analysis is based on Ohm’s Law and Kirchhoff’s Laws
• The response of a circuit element • to a specific voltage or current
• Depends on the characteristics of that element • When current/voltage vary with time,
• Ohm’sLawcannotalwaysexplainthecircuitresponse • Inmanycases,thecircuitresponseisnon-linear.
• Because, resistance is NOT the only basic property of an electrical element.
• All electronic circuits have resistance plus two more basic properties
• CapacitanceandInductance
• Evident when current or voltage are changing
ENGN1218 Electronic Systems and Design
• Transient behaviour
Circuit Elements: Definitions
• Short-lived condition
• May occur when some form of electrical disturbance is applied e.g. sudden change in voltage
• Passive circuit elements
Take energy from the sources
• Either converts the energy to another form
• or stores the energy in an electric or magnetic field. • Can be represented by
• an equivalent network of resistors, capacitors and inductors
• called the reactance of the circuit element ENGN1218 Electronic Systems and Design
Circuit Elements: Definitions
• Linear circuit elements
• The current is proportional to the applied voltage • The resistor 𝑅,
• where 𝑅 is called resistance
• Capacitors C and inductors L , • called reactors
• Where for a given frequency, • the ratio is a constant
ENGN1218 Electronic Systems and Design
Circuit Elements: Inductance and Capacitance
• Inductance and capacitance
• Passive linear circuit elements • Storesenergyinacircuit.
• Deliversnon-instantaneousproperties
• Will change the time behaviour of a circuit
• Evident only when current and voltage are changing
• It takes time to store or dissipate energy • Transients
• CircuitTheorycanstillbeapplied • Equations will be time dependent
– involve integration and differentiation
ENGN1218 Electronic Systems and Design
Inductance and Capacitance: Stores Energy • Capacitance stores energy in an electrostatic field
• Due to a voltage across it
• An electric field is the repulsive force between like charges,
positive or negative
• Inductance stores energy in a magnetic field
• Due to a current flowing through it
• A magnetic field is produced by a current flowing through a wire (solenoidal) or between ends (poles) of domains in certain elements.
ENGN1218 Electronic Systems and Design
Circuit Elements: Practical Circuits
In practice no circuit element is an ideal resistance, capacitance or inductance.
• They are a combination of these concepts
Example of the Inductor (non-assessable):
1. An inductor can be represented by a coil of wire.
2. As wire has resistance, the element can be represented by inductance and resistance in series
3. A parallel capacitance takes into account the capacitance between the turns of wire of the coil and between the terminals
ENGN1218 Electronic Systems and Design
Circuit Element: Resistors
• Will not change the time behaviour of an electric signal
Resistors:
• Voltage/currentrelationship • does not involve time
• Any kind of waveform when applied to a resistor (sinewave, sawtooth, pulse etc.)
• will not change shape in the time dimension • No transient quantities
ENGN1218 Electronic Systems and Design
Resistors:
• Reduce current
• Dissipate energy
Circuit Element: Resistance
• provide an 𝐼𝑅 voltage drop
• Linear relationship between voltage and current
• Absorbs power (power always +ve)
• Don’tgenerateenergy
• Cannot store energy
• Used in heating elements
• Eg irons, toasters, heaters, electric kettles, electric
stoves, hair dryers, car defoggers.
ENGN1218 Electronic Systems and Design
Circuit Element: Resistance
• Direct Current (DC)
• Constant • 𝐼
• where 𝑄𝐶 is charge
• and 𝑡𝑠 is the time charge flows
• 𝑃 𝑉. 𝐼 𝐼𝑅
• Alternating Current (AC)
• Time Varying
• 𝑖 , 𝑞 𝑖 𝜏 𝑑𝜏 𝑞𝑡)
•𝑝, 𝑤 𝑝 𝜏 𝑑𝜏
• where 𝑤𝐽 is work required to move charge
ENGN1218 Electronic Systems and Design
Circuit Symbol
Resistor R
Capacitor C
Inductor L
Units Voltage Current
1Ω 1𝑉 1𝐴
EnergyorPower
𝑃𝑉𝐼 𝑝 𝑡 𝑣 𝑡 𝑖 𝑡 𝑣 𝑡𝑖𝑡𝑅
Variable that cannot abruptly change
No restrictions
At DC Series Parallel
𝑅 𝑅 𝑅+…+𝑅
𝑣𝑡 ∝𝑖𝑡 𝑣𝑡 𝑅𝑖𝑡
𝐼𝑅 𝑖𝑡 𝑣𝑡 𝑅
𝑅1 𝑅1 ⋯𝑅1
ENGN1218 Electronic Systems and Design
In our next video…
We will look more closely at capacitance
ENGN1218 Electronic Systems and Design
Overview • Introduction to capacitors
• Charging of the capacitor • Capacitance
• Water tank analogy
Capacitor: Charging Current and Capacitance
ENGN1218 Electronic Systems and Design
Circuit Elements: Capacitors
Besides resistors, capacitors are the most common electrical components.
Our electronic age could not exist without them.
ENGN1218 Electronic Systems and Design
• For timing circuits
ENGN1218 ENGN2218
Capacitors: Some Applications
Some capacitor applications:
• Reduce voltage fluctuations (smoothing) in power supplies
– Provide electronic time delays
• Eg. 555 timer IC (controlling charging and discharging)
• Will pass AC signals, but block steady DC signals
– Can separate various sections of a circuit as far as DC signals are
concerned, but couple them for AC signals
– For coupling, eg. between stages of an audio system and to connect
a loudspeaker
• For filtering, eg. the tone control of an audio system • For tuning, eg. in a radio system
• To store energy, eg. a camera flash
ENGN2218 ENGN2218
ENGN1218 Electronic Systems and Design
The Capacitor
Circuit Elements: Capacitors
• Called a capacitor because it has the capacity to store energy in its electric field
• Consists of an insulator (dielectric) between two conductors
– The conductors are commonly made of aluminium foil
• Manufactured for specific values of capacitance
• Most common dielectrics are
– air, paper, mica, ceramic, polyester and electrolytic
• The type of capacitor is named after the dielectric
ENGN1218 Electronic Systems and Design
Capacitors: Types • There are three main types of capacitors
ENGN1218 Electronic Systems and Design
• Electrolytic capacitors:
• Reasonablecost
• Smaller size
Capacitors: Electrolytic Type
• Provide large values of capacitance
• Majorityarepolarised,
• must be connected correctly into a circuit
Negative electrode
• positive to positive terminal • Have a limited shelf life
Separated by electrolyte saturated gauze
Positive electrode (aluminium foil) • Rolledupstripsofaluminiumortantalumfoil
• Separated by electrolyte saturated gauze
• During manufacture chemical action creates a thin oxide film
which acts as a dielectric
• Typically used in power supplies to smooth rectified waveforms
Oxide film
ENGN1218 Electronic Systems and Design
• Capacitance value
Capacitor Values
Parameters that are used to specify capacitors:
• The farad is a very large unit.
• In practice generally between few pF to about 50mF
• Largervaluesareavailablebutnotdiscussedhere
• the larger the value of C, the more charge can be held for a given voltage • Tolerance
• Standardvaluesare5%,10%,20%ofthenominalcapacitance value
• Working voltage (voltage rating)
• Standardvaluestypicallybetween6.3Vand500V.
• Critical to keep the applied voltage below the breakdown point of the dielectric.
ENGN1218 Electronic Systems and Design
Standard Capacitor Values
ENGN1218 Electronic Systems and Design
Capacitance
Capacitance (C) is the ability of a dielectric to store electric charge • Measured in farads (F),
• A voltage has a field of electric lines of force electric field in dielectric
between opposite electric charges 𝑞 • An electric charge can be stored in the electric
field of the dielectric material
• The dielectric
• Insulator
• electrons cannot flow through the dielectric
• contacts the two conductors
• concentrates the electric field between the two
conductors.
number of electrons taken from conductor 𝐵 . ENGN1218 Electronic Systems and Design
Charging the capacitor
Charging the Capacitor
• The battery charges the capacitor
dielectric
• Electrons are lost from plate 𝐴
– which is connected to the positive side of battery
– and accumulate on plate 𝐵,
• Which is connected to the negative terminal of battery.
For example:
• On conductor 𝐴, electron loss produces a positive charge
If 6.25 10 electrons accumulate on conductor B, the negative charge is 1C.
• On conductor 𝐵, electron gain produces a negative charge
• This redistribution of electrons produces an electric field in the dielectric
• Only need to consider the charge on one plate
– The number of electrons gained by conductor 𝐴 is equal to the
ENGN1218 Electronic Systems and Design
Capacitance: Charging Current
The charging current is the movement of electrons through the circuit from one conductor to the other
• Nocurrentpassesthroughthedielectric
• Istemporary(transient)
The capacitor voltage 𝑣 increases as charge is deposited onto the conductors • Asthecapacitorapproachesfullcharge
• The difference in voltage decreases and so the current decreases
ENGN1218 Electronic Systems and Design
Capacitance: Charging Current
• Current only flows until the capacitor is charged to the applied voltage
– There is no current when 𝑉 𝑣.
• Without any series resistance charging would be instantaneous
– However there is always some series resistance
• Charge is now stored in the electric field of the dielectric.
– Remains charged even after the voltage source is disconnected.
• The capacitance 𝐶 (in italics) measures how much charge is stored
– and is proportional to the voltage and the value of the charge.
𝑞 𝐶 𝑣 → 𝐶
ENGN1218 Electronic Systems and Design
Capacitance: The Water Tank Analogy
The water tank analogy
• By increasing the charging voltage 𝑣
– The electric field is stronger
– More charge 𝑞 is stored in the dielectric
Build up of electric potential 𝑣
• The amount of charge stored in the capacitorTotal amount of is proportional to the applied voltage. charge 𝑞 stored
• Capacitance 𝐶 is a physical constant
– indicates how much charge can be stored for a
given voltage – farads(F)
Bottom area of water tank (𝐶)
• A larger capacitance can store more charge
1F 1C 1𝑉
𝑞 𝐶 𝑣 → 𝐶
ENGN1218 Electronic Systems and Design
How much charge is stored in a 2μF capacitor with 50V across it?
A constant current of 2μA charges a capacitor for 20s. How much charge is stored?
Q 𝐶𝑉 210 50 100𝜇𝐶
• For a DC source
Examples: DC Sources
I orQI𝑡
QI𝑡 210 20
ENGN1218 Electronic Systems and Design
Capacitors: Historical Perspective • The unit of capacitance is named after
ENGN1218 Electronic Systems and Design
In our next video…
We will look at current, voltage and power.
ENGN1218 Electronic Systems and Design
Capacitors: Current, Voltage and Power
• Relationships between charge voltage, current and power
• Examples.
ENGN1218 Electronic Systems and Design
Capacitor: Sign Convention
• Passive sign convention for capacitor current and voltage
ENGN1218 Electronic Systems and Design
Capacitor: Current and Voltage
• 𝑞 𝑡 the charge on the plate
– proportional to voltage and the constant of capacitance,
• with respect to time
• To find relationship between current and capacitance – differentiate both sides of this equation
𝑞𝑡 C𝑣𝑡 where 𝑞𝑡 𝑖𝑡
𝑖𝑡 C 𝑑 𝑣 𝑡 𝑑𝑡
ENGN1218 Electronic Systems and Design
Capacitor: Voltage in terms of Current
𝑖𝑡 C 𝑑 𝑣 𝑡 𝑑𝑡
𝑑 𝑣 𝑡 𝑖 𝑡 𝑑 𝑡
• To determine voltage, integrate this expression over time ∞𝜏𝑡,assumingthat𝑣 ∞ 0
𝑣𝑡 𝑖𝜏𝑑𝜏
C 𝑖𝜏𝑑𝜏C 𝑖𝜏𝑑𝜏
𝑣 𝑡 1 𝑖𝜏𝑑𝜏𝑣 𝑡
is the voltage
Where 𝑣 𝑡 duetothechargethat accumulates on the capacitor between time ∞ 𝜏 𝑡
Capacitor: DC Voltage
What is the current response to DC voltage?
• Whenthevoltageacrossacapacitorisnotchangingwithtime 𝑣𝑡
DC voltage
𝑖𝑡 C𝑑𝑣𝑡 0𝐴 𝑑𝑡
The current through the capacitor is 0A
The capacitor blocks DC voltage and acts as an open circuit to DC
ENGN1218 Electronic Systems and Design
ENGN1218 Electronic Systems and Design
Capacitor: Step Change in Voltage
What is the current response to a step change in voltage?
• Thisoccurswhenthevoltagejumpsinstantaneouslyat time𝑡. 𝑣𝑡
Then𝑣 ∞as𝑑𝑡0
and𝑖𝑡 C𝑣𝑡 ∞𝐴 𝑣
• At this time the voltage across the capacitor is infinite which is impossible
Thus voltage across a capacitor cannot change instantaneously, the capacitor RESISTS an abrupt change in voltage
ENGN1218 Electronic Systems and Design
Capacitor: Switch Open and Closed • An electrical switch uses the extremes of resistance
𝑖𝑡 C→∞𝐴
𝑖𝑡 anyvalue determined by the circuit
• Consider when the switch has been closed for some time
𝑖𝑡 C0𝐴
capacitor is fully charge and no current flows
• Consider when the switch closes after being open
When switch closes there is an instantaneous change in the voltage across the capacitor which is impossible. The capacitor resists the change.
ENGN1218 Electronic Systems and Design
Capacitor: Power and Energy in terms of Voltage
The instantaneous power or rate of supply of energy (𝑤 𝑡 ) to a capacitance is 𝑝𝑡𝑣𝑡𝑖𝑡
𝑣 𝑡 . C 𝑑 𝑣 𝑡 𝑑 𝑤 𝑡 𝑑𝑡 𝑑𝑡
Integrate both sides to determine the energy stored in the electric field overtime∞𝜏𝑡,where𝑣 ∞ 0𝑉
𝑤 𝑡 C𝑣𝜏𝑑𝜏𝑣𝜏𝑑𝜏C 𝑣𝜏𝑑𝑣𝜏
1 1
2C𝑣 𝜏 | 2C𝑣 𝑡 0
Where 𝑥 𝑑𝑥 𝑥 and𝑞 𝑡 C𝑣𝑡
1 1𝑞𝑡 𝑤 𝑡 C𝑣 𝑡
Find the current waveform across a 5μF capacitor with the applied voltage as shown.
• For0𝑡6𝑚𝑠 •𝑖𝑡5μ 4000𝑡
60 10 60𝑚𝐴 𝑖𝑡
• 𝑖𝑡 C 𝑣 𝑡
𝑣𝑡 24
𝑡 4000𝑡, 0 𝑡 6𝑚𝑠 6𝑚
ENGN1218 Electronic Systems and Design
Capacitors: Example 1
𝑡 12000𝑡,6𝑚𝑠 𝑡 8𝑚𝑠 2𝑚
5𝜇. 4000
20 10 20𝑚𝐴 • For6𝑚𝑠𝑡8𝑚𝑠
• 𝑖𝑡 5μ 12000𝑡
5𝜇. 12000
ENGN1218 Electronic Systems and Design
Capacitors: Example 2 pg. 1/2
Find the voltage waveform across a 3μF capacitor, where the initial charge on the capacitor is 10V.
Ans: 𝑣 𝑡 𝑖𝜏𝑑𝜏 𝑣 𝑡
100𝑚𝐴, 𝑖 𝑡 100𝑚𝐴,
0 𝑡 1𝑚𝑠 1𝑚𝑠𝑡2𝑚𝑠
0𝐴, • For0𝑡1𝑚𝑠
𝑡 2𝑚𝑠 • 𝑣𝑡 100𝑚𝑑𝑡𝑣0
3μ 𝑑𝑡10
1 0 0 𝑚 𝑡 0𝑡 1 0 3μ
33.33 10𝑡 0 10
𝑣 1𝑚𝑠 33.3310 110 10
ENGN1218 Electronic Systems and Design
For 𝑡 2𝑚𝑠
𝑣 2𝑚 33.3310 2𝑚𝑠 76.66
Capacitors: Example 2 pg. 2/2
For 1𝑚 𝑡 2𝑚𝑠
• 𝑣𝑡 100𝑚𝑑𝑡𝑣1𝑚
33.33 10𝑡 10 V , 0 𝑡 1𝑚𝑠
𝑣 𝑡 33.3310𝑡76.66𝑉, 1𝑚𝑠𝑡2𝑚𝑠
100𝑚𝑡 𝑡 43.33 3μ 1𝑚
33.33 10𝑡 1 10 43.33 33.33 10𝑡 76.66
For 𝑡 2𝑚𝑠
𝑣 2𝑚 10𝑉
𝑣 𝑡 3μ 0 𝑑𝑡𝑣 2𝑚 01010𝑉
Circuit Symbol Resistor R Capacitor C Inductor L Units 1𝑉 𝑞𝑡∝𝑣𝑡
𝑖𝑡 ∝𝑣𝑡 𝑣𝑡
𝑖𝑡 ∝𝑑𝑣𝑡 𝑑𝑥
EnergyorPower At DC
𝑝𝑡 𝑣𝑡𝑖𝑡 𝑣 𝑡 𝑖 𝑡𝑅 𝑤 𝑡 1C𝑣 𝑡 1𝑞𝑡 𝑅22C
Series Parallel
𝑅 𝑅 𝑅+…+𝑅
Open Circuit ?
ENGN1218 Electronic Systems and Design
1Ω
Voltage 𝑣𝑡∝𝑖𝑡 1
C, 1F 𝑣𝑡 C𝑖𝜏𝑑𝜏𝑣𝑡
𝑖𝑡𝑅 𝑖𝑡C𝑑𝑣𝑡 𝑑𝑡
𝑅1 𝑅1 ⋯ 𝑅1
ENGN1218 Electronic Systems and Design
In our next video…
We will determine how to calculate the capacitance of capacitors in series and parallel.
ENGN1218 Electronic Systems and Design
Derive the formulas for calculating the equivalent capacitance of capacitors in series and parallel
Capacitors: In Series and Parallel
Circuit Symbol Resistor R Capacitor C Inductor L Units 1𝑉 𝑞𝑡∝𝑣𝑡
𝑖𝑡 ∝𝑣𝑡 𝑣𝑡
𝑖𝑡 ∝𝑑𝑣𝑡 𝑑𝑥
EnergyorPower At DC
𝑝𝑡 𝑣𝑡𝑖𝑡 𝑣 𝑡 𝑖 𝑡𝑅 𝑤 𝑡 1C𝑣 𝑡 1𝑞𝑡 𝑅22C
Series Parallel
𝑅 𝑅 𝑅+…+𝑅
Open Circuit ?
ENGN1218 Electronic Systems and Design
1Ω
Voltage 𝑣𝑡∝𝑖𝑡 1
C, 1F 𝑣𝑡 C𝑖𝜏𝑑𝜏𝑣𝑡
𝑖𝑡𝑅 𝑖𝑡C𝑑𝑣𝑡 𝑑𝑡
𝑅1 𝑅1 ⋯ 𝑅1
ENGN1218 Electronic Systems and Design
Capacitors in Series
• N capacitors in series combine like resistors in parallel.
– Is equivalent to increasing the thickness of the dielectric
– The same current 𝑖 flows through all the capacitors, • All the capacitors have the same charge.
– Applying KVL
𝑣 𝑣 𝑣 ⋯ 𝑣 111
𝑣C 𝑖𝜏𝑑𝜏𝑣 𝑡 C 𝑖𝜏𝑑𝜏𝑣 𝑡 ⋯C 𝑖 𝜏 𝑑𝜏𝑣 𝑡
CC⋯C 𝑖𝜏𝑑𝜏𝑣𝑡𝑣𝑡⋯𝑣𝑡
C1 C1 C1 ⋯ C1
ENGN1218 Electronic Systems and Design
Capacitors in Parallel
• The total equivalent capacitance of N parallel capacitors
– Is the sum of the individual capacitances
– Connecting capacitors in parallel is equivalent to adding their plate areas together
• Capacitors in parallel
– have the same voltage across them.
–ApplyingKCL 𝑖𝑖𝑖⋯𝑖 where𝑖 𝑡 C𝑣𝑡 𝑑 𝑑 𝑑
𝑖C𝑑𝑡𝑣 𝑡 C𝑑𝑡𝑣 𝑡 ⋯C𝑑𝑡𝑣 𝑡 𝑑𝑑
C 𝑑𝑡𝑣𝑡 C𝑑𝑡𝑣𝑡
C 𝐶 𝐶 ⋯ 𝐶
ENGN1218 Electronic Systems and Design
• Question 1: What is the equivalent capacitance of the circuit
45𝜇𝐹 15𝜇𝐹 60𝜇𝐹
12𝜇𝐹 8𝜇𝐹 20𝜇𝐹
C 1 111
60𝜇 60𝜇 20𝜇
ENGN1218 Electronic Systems and Design
• What is the equivalent capacitance of the circuit shown?
2𝜇𝐹 2𝜇𝐹2𝜇𝐹4𝜇𝐹
1.5𝜇𝐹 1.5𝜇𝐹
2𝜇𝐹 4𝜇𝐹 6𝜇𝐹
C 1 11
1 11 1
• Division of voltage across unequal capacitances
– Charge𝑄𝐼.𝑡600𝜇1600𝜇𝐶
𝑉 𝑄 600𝜇10𝑉 𝐶 60𝜇
• All capacitances have 𝑄 600𝜇𝐶 – Although the charge is the same
𝑉 𝑄 600𝜇 10𝑉 𝐶 60𝜇
– For capacitors in series, the voltage across each capacitor is inversely proportional to its capacitance.
• The smaller the capacitance the larger the proportion of the applied voltage.
• Example 3
• The charging current 𝐼 600𝜇𝐴 is flowing
• the voltage across each capacitor is different. ENGN1218 Electronic Systems and Design
𝑉 𝑄 600𝜇30𝑉 𝐶 20𝜇
Circuit Symbol Resistor R Capacitor C Inductor L Units 1𝑉 𝑞𝑡∝𝑣𝑡
𝑖𝑡 ∝𝑣𝑡 𝑣𝑡
𝑖𝑡 ∝𝑑𝑣𝑡 𝑑𝑥
EnergyorPower At DC
𝑝𝑡 𝑣𝑡𝑖𝑡 𝑣 𝑡 𝑖 𝑡𝑅 𝑤 𝑡 1C𝑣 𝑡 1𝑞𝑡 𝑅22C
Series Parallel
ENGN1218 Electronic Systems and Design
1Ω
Voltage 𝑣𝑡∝𝑖𝑡 1
C, 1F 𝑣𝑡 C𝑖𝜏𝑑𝜏𝑣𝑡
𝑖𝑡𝑅 𝑖𝑡C𝑑𝑣𝑡 𝑑𝑡
𝑅 𝑅𝑅+…+𝑅
Open Circuit
C1 C1 ⋯ C1
1 1 ⋯ 1
C C C+…+C
ENGN1218 Electronic Systems and Design
In our next video…
We will be introduced to inductance
ENGN1218 Electronic Systems and Design
Video 5 The Inductor
• Formal Introduction
• Applications • Types
ENGN1218 Electronic Systems and Design
Circuit Elements: Inductors
After resistors and capacitors, inductors are the most common electrical components.
They are extremely important in electronic circuits.
ENGN1218 Electronic Systems and Design
Inductors: Some Applications
Some inductor applications:
• Sometimes called ‘chokes’
– Commonly used to allow DC supply to flow whilst blocking AC supply
– Provide increased coupling resistance with increased frequency
– Can reduce the current of a specified frequency
– Can smooth out signal fluctuations in a signal
• Used in power supplies, transformers, radios, TVs, radar and electric motors.
ENGN1218 Electronic Systems and Design
Circuit Elements: Inductors
• Consists of man