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Alternating current part 2
An AC circuit connected in series contains an inductor, capacitor, and ohmic resistor
Capacitor in AC circuit (C)
Pure inductor in AC circuit (L)
Ohmic resistor in AC circuit (R)
Capacitor voltage lags current by \[\frac{{{𝜋}}}{{{2}}}\]
Inductor voltage leads current by \[\frac{{{𝜋}}}{{{2}}}\]
Resistor voltage is in phase with current
In a circuit with inductor, capacitor and ohmic resistor
Connected in series
Current is constant, current equation is of the form:
\[𝑖 (t) = 𝐼_𝑚 sin (𝜔t ± ∅ ) \]
Current may lead or lag the voltage
Question: How to find total voltage difference, total impedance, and phase angle between current and voltage?
\[ 𝑉_m(tot) ≠𝑉_𝑅 + 𝑉_𝐶 + 𝑉_𝐿 \]
\[ Z ≠𝑅 + X_𝐶 + X_𝐿 \]
In this simulation, choose an RLC circuit
Connect an inductor, resistor and capacitor in series with an AC source
Click on each component and record the values of capacitor capacitance, inductor self-inductance, ohmic resistance, maximum source voltage and source frequency
Use the ammeter and current oscilloscope to determine the maximum source current and measure the maximum current in each part of the circuit
Use the voltmeter and voltage oscilloscope to determine the maximum source voltage and maximum voltage across each component
Calculate inductive reactance and capacitive reactance
Calculate maximum voltage across each component and compare with your measurements
Use vector representation to find total voltage
Measured values
Capacitance
Self-inductance
Ohmic resistance
Maximum source current
Maximum source voltage
Source frequency
C =....
L =....
R =....
Im=....
vm=....
f = ....
Values to calculate
Capacitive reactance
\[ XC=\frac{{{1}}}{{{2𝜋fC }}\]
Inductive reactance
\[ XL=2𝜋fL \]
Maximum resistor voltage
\[ V_ R=I_ m . R \]
Maximum capacitor voltage
\[ V_ C=I_ m . XC \]
Maximum inductor voltage
\[ V_ L=I_ m . XL \]
\[ XC=......\]
\[XL=......\]
\[ V_ R=......\]
\[ V_ C=......\]
\[ V_ L=......\]
Use the calculated values treating voltages as vectors and represent them on the following graph
First find
VL - VC= ...............
The subtraction result is in the direction of the larger value
(VR ) Transfer the subtraction vector to the head of the vector
Find the resultant of the two perpendicular vectors
\[(V_ m)^2=(V_ R)^2 + ( V_ L - V_ C)^2\]
\[(V_ m)= ............\]
Calculate impedance (total circuit impedance)
( Z ) represented by the symbol
\[(V_ m)^2=(V_ R)^2 + ( V_ L - V_ C)^2\]
\[(I_ m)^2 . Z^2=(I_ R)^2 .R^2+ ( I_ L.XL - I_ C.XC)^2\]
Current is equal because connected in series, so we can simplify
\[ Z^2= R^2+ (XL - XC)^2\]
Phase angle
The angle between current and voltage difference
From the drawn vectors
\[ ∅ = tan ^-1 \frac{{{( V_ L - V_ C)}}}{{{V_ R}}}\]
Or through impedances we can calculate phase angle
\[ ∅ = tan ^-1 \frac{{{( XL - XC)}}}{{{R}}}\]
If phase angle is negative, current leads voltage
If phase angle is positive, voltage leads current
Example 1
A circuit with inductor, resistor and capacitor
\[RLC\] connected in series with AC source where capacitor reactance is less than inductor reactance
XL > XC
Then one of the following is correct
Click here to show solution
Choose the correct answer
Example 2
(RLC )In the adjacent figure, a circuit
connected to AC source with equation
\[V(t) = 210 Sin (300 t ) \] Given that \[L= 0.5 \;H \;\;\;\;\;\;\;\; C= 5\; 𝜇F\;\;\;\;\;\;\;\;R=150 \;Ω\] Then the circuit impedance equals
Click here to show solution
Choose the correct answer
AC circuit (inductor - resistor)
AC circuit (resistor - capacitor)
Resonance circuit
Resonance circuits are used in radio and TV receivers
Frequency is changed or capacitor capacitance is changed or inductor self-inductance is changed until resonance is achieved
Usually frequency is changed until circuit impedance decreases allowing only one frequency signal from the antenna that matches the circuit frequency
( RLC) In this simulation choose an RLC circuit
Connect inductor, resistor and capacitor in series with AC source
Change source frequency until
XL = XC
Then we say resonance occurs
Then circuit impedance becomes minimum
Z = R
Thus current in circuit becomes maximum
\[I_ m =\frac{{{( V_ m)}}}{{{R}}}\]
At resonance \[V_L = V_C\]
\[ V_m = V_R \]
As if circuit contains only resistor and
Current and voltage are in phase
Thus resonance frequency is
\[f_ 0 = \frac{{{ 1}}}{{{2𝜋√(𝑙.𝑐)}}}\]
Example 3
A circuit with inductor, resistor and capacitor
\[RLC\] connected to AC source with maximum voltage \[240\;V \] and maximum resistor voltage \[VR=120 V\] Source frequency was changed until maximum inductor voltage equals maximum capacitor voltage
\[VL = VC \] Then maximum current in circuit is
Given \[R=160 Ω\]
Click here to show solution
Choose the correct answer
Example 4
An RLC circuit connected to an antenna contains a coil with self-inductance
\[ L= 0.03\; H\] and a capacitor with capacitance
\[ C= 2\;𝜇𝐹\] can receive a wave with frequency
Click here to show solution
Choose the correct answer
Transformers
Transformers are used to step up or step down electrical voltage
A transformer consists of
A primary coil connected to an AC source
A secondary coil connected to a device
An iron core
It works based on the principle of mutual induction
Due to the changing current in the primary coil, the flux changes in the secondary coil inducing an emf and current
The primary and secondary coils have the same changing flux because of the iron core \[V_{emf}=\frac{d∅_B}{dt}\]
The induced emf in primary and secondary coils are
\[V_s= - N_s \frac{d∅_B}{dt}\]\[V_p= - N_p \frac{d∅_B}{dt}\] Dividing these equations gives
\[\frac {N_S}{N_P}=\frac {V_S}{V_P}\]
If the primary coil has more turns than the secondary, it's a step-down transformer
If the primary coil has fewer turns than the secondary, it's a step-up transformer
When a device (load resistance) is connected to the secondary, power is delivered \[P_S=I_S.V_S\]
This current creates a magnetic field in the secondary that induces an emf in the primary to maintain the original voltage
For ideal transformers
\[P_S=P_P \]\[I_S.V_S=I_P.V_P\]\[\frac{I_S}{I_P}=\frac{V_P}{V_S}=\frac{N_P}{N_S}\]
The inverse relationship between voltage and current means step-up transformers decrease current while step-down transformers increase current
Useful Information:
Isolation transformers neither step up nor step down
voltage and have equal turns in primary and secondary.
They're used to isolate circuits in sensitive equipment
to prevent current interference.
The primary current can be calculated using \[I_P=(\frac{N_S}{N_P})^2.\frac{V_P}{R}\]
The effective resistance in the primary can be calculated using
\[R_P=(\frac{N_P}{N_S})^2.R \]
Alternating current part 2 |
Capacitor in AC circuit (C) |
Pure inductor in AC circuit (L) |
Ohmic resistor in AC circuit (R) |
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Capacitance Self-inductance Ohmic resistance Maximum source current Maximum source voltage Source frequency C =.... L =.... R =.... Im=.... vm=.... f = .... Capacitive reactance Inductive reactance Maximum resistor voltage Maximum capacitor voltage Maximum inductor voltage \[ XC=......\] \[XL=......\] \[ V_ R=......\] \[ V_ C=......\] \[ V_ L=......\]
In a circuit with inductor, capacitor and ohmic resistor
Connected in series
Current is constant, current equation is of the form:
\[𝑖 (t) = 𝐼_𝑚 sin (𝜔t ± ∅ ) \]
Current may lead or lag the voltage
Question: How to find total voltage difference, total impedance, and phase angle between current and voltage?
\[ 𝑉_m(tot) ≠𝑉_𝑅 + 𝑉_𝐶 + 𝑉_𝐿 \]
\[ Z ≠𝑅 + X_𝐶 + X_𝐿 \]
In this simulation, choose an RLC circuit
Connect an inductor, resistor and capacitor in series with an AC source
Click on each component and record the values of capacitor capacitance, inductor self-inductance, ohmic resistance, maximum source voltage and source frequency
Use the ammeter and current oscilloscope to determine the maximum source current and measure the maximum current in each part of the circuit
Use the voltmeter and voltage oscilloscope to determine the maximum source voltage and maximum voltage across each component
Calculate inductive reactance and capacitive reactance
Calculate maximum voltage across each component and compare with your measurements
Use vector representation to find total voltage
Measured values
Values to calculate
\[ XC=\frac{{{1}}}{{{2𝜋fC }}\]
\[ XL=2𝜋fL \]
\[ V_ R=I_ m . R \]
\[ V_ C=I_ m . XC \]
\[ V_ L=I_ m . XL \]
Use the calculated values treating voltages as vectors and represent them on the following graph
First find
VL - VC= ...............
The subtraction result is in the direction of the larger value
(VR ) Transfer the subtraction vector to the head of the vector
Find the resultant of the two perpendicular vectors
\[(V_ m)^2=(V_ R)^2 + ( V_ L - V_ C)^2\]
\[(V_ m)= ............\]
Calculate impedance (total circuit impedance)
( Z ) represented by the symbol
\[(V_ m)^2=(V_ R)^2 + ( V_ L - V_ C)^2\]
\[(I_ m)^2 . Z^2=(I_ R)^2 .R^2+ ( I_ L.XL - I_ C.XC)^2\]
Current is equal because connected in series, so we can simplify
\[ Z^2= R^2+ (XL - XC)^2\]
Phase angle
The angle between current and voltage difference
From the drawn vectors
\[ ∅ = tan ^-1 \frac{{{( V_ L - V_ C)}}}{{{V_ R}}}\]
Or through impedances we can calculate phase angle
\[ ∅ = tan ^-1 \frac{{{( XL - XC)}}}{{{R}}}\]
If phase angle is negative, current leads voltage
If phase angle is positive, voltage leads current
A circuit with inductor, resistor and capacitor
\[RLC\] connected in series with AC source where capacitor reactance is less than inductor reactance
XL > XC
Then one of the following is correct
Click here to show solution
Choose the correct answer
(RLC )In the adjacent figure, a circuit
connected to AC source with equation
\[V(t) = 210 Sin (300 t ) \] Given that \[L= 0.5 \;H \;\;\;\;\;\;\;\; C= 5\; 𝜇F\;\;\;\;\;\;\;\;R=150 \;Ω\] Then the circuit impedance equals
Click here to show solution
Choose the correct answer
AC circuit (inductor - resistor)
AC circuit (resistor - capacitor)
Resonance circuit
Resonance circuits are used in radio and TV receivers
Frequency is changed or capacitor capacitance is changed or inductor self-inductance is changed until resonance is achieved
Usually frequency is changed until circuit impedance decreases allowing only one frequency signal from the antenna that matches the circuit frequency
( RLC) In this simulation choose an RLC circuit
Connect inductor, resistor and capacitor in series with AC source
Change source frequency until
XL = XC
Then we say resonance occurs
Then circuit impedance becomes minimum Z = R
Thus current in circuit becomes maximum \[I_ m =\frac{{{( V_ m)}}}{{{R}}}\]
At resonance \[V_L = V_C\]
\[ V_m = V_R \]
As if circuit contains only resistor and
Current and voltage are in phase
Thus resonance frequency is
\[f_ 0 = \frac{{{ 1}}}{{{2𝜋√(𝑙.𝑐)}}}\]
A circuit with inductor, resistor and capacitor
\[RLC\] connected to AC source with maximum voltage \[240\;V \] and maximum resistor voltage \[VR=120 V\] Source frequency was changed until maximum inductor voltage equals maximum capacitor voltage
\[VL = VC \] Then maximum current in circuit is
Given \[R=160 Ω\]
Click here to show solution
Choose the correct answer
An RLC circuit connected to an antenna contains a coil with self-inductance
\[ L= 0.03\; H\] and a capacitor with capacitance
\[ C= 2\;𝜇𝐹\] can receive a wave with frequency
Click here to show solution
Choose the correct answer
Transformers
Transformers are used to step up or step down electrical voltage
A transformer consists of
A primary coil connected to an AC source
A secondary coil connected to a device
An iron core
It works based on the principle of mutual induction
Due to the changing current in the primary coil, the flux changes in the secondary coil inducing an emf and current
The primary and secondary coils have the same changing flux because of the iron core \[V_{emf}=\frac{d∅_B}{dt}\]
The induced emf in primary and secondary coils are
\[V_s= - N_s \frac{d∅_B}{dt}\]\[V_p= - N_p \frac{d∅_B}{dt}\] Dividing these equations gives
\[\frac {N_S}{N_P}=\frac {V_S}{V_P}\]
If the primary coil has more turns than the secondary, it's a step-down transformer
If the primary coil has fewer turns than the secondary, it's a step-up transformer
When a device (load resistance) is connected to the secondary, power is delivered \[P_S=I_S.V_S\]
This current creates a magnetic field in the secondary that induces an emf in the primary to maintain the original voltage
For ideal transformers
\[P_S=P_P \]\[I_S.V_S=I_P.V_P\]\[\frac{I_S}{I_P}=\frac{V_P}{V_S}=\frac{N_P}{N_S}\]
The inverse relationship between voltage and current means step-up transformers decrease current while step-down transformers increase current
Useful Information: Isolation transformers neither step up nor step down voltage and have equal turns in primary and secondary. They're used to isolate circuits in sensitive equipment to prevent current interference.
The primary current can be calculated using \[I_P=(\frac{N_S}{N_P})^2.\frac{V_P}{R}\]
The effective resistance in the primary can be calculated using
\[R_P=(\frac{N_P}{N_S})^2.R \]
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