Lab 4
Brief 5 AC RL and RC Circuits Electrical Circuits Lab I (ENGR 2105)
Dr. Kory Goldammer
Review of Complex Numbers and Transforms
Transforms
The Polar Coordinates / Rectangular Coordinates Transform
The Complex Plane
We can use complex numbers to solve for the phase shift in AC Circuits
Instead of (x,y) coordinates, we define a point in the Complex plane by (real, imaginary) coordinates
Real numbers are on the horizontal axis
Imaginary numbers are on the vertical axis
The Complex Plane (cont.)
Imaginary numbers are multiplied by j
By definition,
(Mathematicians use i instead of j, but that would confuse us since i stands for current in this class)
Imaginary Plane: Rectangular Coordinates
We can identify any point in the 2D plane using (real, imaginary) coordinates
Imaginary Plane: Transform to Polar Coordinates
We can identify any point in the complex plane using (r,) coordinates.
The arrow is called a Phasor.
r is the length of the Phasor, and is the angle between the Positive Real Axis and the Phasor
is the Phase Angle we want to calculate
Complex Plane Using Rectangular Coordinates
r
Imaginary Plane: Transform to Polar Coordinates
Complex Plane Using Rectangular Coordinates
(we will discuss the meaning of r later)
Use either the sin or cos term to find :
But we need in radians:
r
=7.07
Complex Math
For addition or subtraction, add or subtract the real and j terms separately.
(3 + j4) + (2 – j2) = 5 + j2
To multiply or divide a j term by a real number, multiply or divide the numbers. The answer is still a j term.
5 * j6 = j30
-2 * j3 = -j6
j10 / 2 = j5
Complex Math (cont. 1)
To divide a j term by a j term, divide the j coefficients to produce a real number; the j factors cancel.
j10 / j2 = 5
-j6 / j3 = -2
To multiply complex numbers, follow the rules of algebra, noting that j2 = -1
Complex Math (cont. 2)
To divide by a complex number: Can’t be done!
The denominator must first be converted to a Real number!
Complex Conjugation
Converting the denominator to a real number without any j term is called rationalization.
To rationalize the denominator, we need to multiply the numerator and denominator by the complex conjugate
Complex Number Complex Conjugate
5 + j3 5 – j3
–5 + j3 –5 – j3
5 – j3 5 + j3
–5 – j3 –5 + j3
Complex Math (cont. 2)
Multiply the original equation by the complex conjugate divided by itself (again, j2 = -1):
Phase Shift Time Domain - ω Domain Transforms
Transforming from the Time (Real World) Domain to the (Problem Solving Domain
Note that in the ω Domain, Resistance, Inductance and Capacitance all of units of Ohms!
| Element | Time Domain | ω Domain Transform |
| Applied Sinusoidal AC Voltage | (Volts) (ω=2πf) | Vp (Volts) |
| Series Current | (Amps) (ω=2πf) | (Amps) |
| Resistance | R (Ohms) | R (Ohms) |
| Inductance | L (Henry’s) | (Ohms) (ω=2πf) |
| Capacitance | C (Farads) | (Ohms) (ω=2πf) |
Solving For Current
Impedance - Ohms
So far, we’ve talked about resistance. Ohm’s law tells us:
Resitance limits, or “impedes”, the flow of current.
The more Ohms in the circuit, the less current.
Impedance (Z): The number of Ohms in a circuit.
Resistance has units of Ohms, but so do Capacitance and Inductance in the ω Domain!
Impedance – Ohms (cont. 1)
The impedance of the Capacitor:
When the frequency is small, ZC is large.
When the frequency is large, ZC is small
The impedance of the Inductor:
When the frequency is large, ZL is large.
When the frequency is small, ZL is small
Impedance – Ohms (cont. 2)
To find the total impedance of a series circuit:
| Element | Time Domain | ω Domain Transform |
| Applied Sinusoidal AC Voltage | (Volts) (ω=2πf) | Vp (Volts) |
| Series Current | (Amps) (ω=2πf) | (Amps) |
| Resistance | R (Ohms) | R (Ohms) |
| Inductance | L (Henry’s) | (Ohms) (ω=2πf) |
| Capacitance | C (Farads) | (Ohms) (ω=2πf) |
Ohm’s Law Still Works
Ohm’s law is well-named. It applies to anything that has units of Ohms!
Instead of , it becomes:
Or,
Ohms law works in both the time- and ω-domains!
Strategy
Solving in the time domain is difficult!
Transform v(t) to the ω domain
Calculate Z in the ω domain
Solve for Current in the ω domain:
Transform Current back to the time domain
Calculating current in an R (Resistor Only) Circuit – Nothing Changes!
Time Domain
ω Domain
Vp
Phase Angle for an R Circuit
Time Domain
0.1
Interpretation: The current in the circuit will be in phase with the applied voltage.
+j
+
I = 0.1 Amps
θ = 0o
Calculating current in an L (Inductor Only) Circuit
Time Domain
ω Domain
Vp
Phase Angle for an L Circuit
Time Domain
1
Interpretation: The current in the circuit will Lag the voltage by 90o (i.e. Current reaches it’s peak at a later time)
+j
+
I = -j1 Amps
Calculating current in a C (Capacitor Only) Circuit
Time Domain
ω Domain = 10 Volts
Vp
Phase Angle for a C Circuit
Time Domain
1
Interpretation: The current in the circuit will Lead the voltage by 90o
+j
+
I = j1 Amps
Calculating current in an RL Series Circuit
Time Domain
ω Domain =10Volts
Vp
Calculating Current for RL Circuit
Rectangular: Amps
Polar:
But we need in radians:
+j
+
θ
)
I =0.5 – j0.5 Amps
Calculating Current for RL Circuit – Time Domain
The Current is out of phase with the applied voltage by 45o
Transform Theta from Radians to Time
In the previous example, we said the two waveforms were out of phase by
How much is the phase shift in terms of time?
We know in 1 Period, the wave goes through , or 360o. So in this case, the shift is:
We know 1 period takes
3) The phase shift is of a period, so