1. Lecture 17: AC Circuits: Transformers
PHYS219 Fall semester 2014
Lecture 17: AC Circuits: Transformers
Dimitrios Giannios
Purdue University

2. Review: Resistor in AC circuit
Current I and potential V are in phase
Imax = Vmax/R
Pavg = I2rms R = (1/2) I2max R

3. Review: Inductor in AC circuit
current I and voltage V are out of phase by -90°
Imax = Vmax/XL, where XL=2πfL XL -reactance of inductor
Pavg = 0

4. Review: Capacitor in AC circuit
current I and voltage V are out of phase by +90°
Imax = Vmax/XC, where XC=1/(2πfC) XC -reactance of capacitor
Pavg = 0

5. Review: RL, RC circuit Imax = Vmax/Z, where Z- impedance
Pavg = I2rms R = (1/2) I2max R

6. What is a Transformer?

7. A Transformer: change Voltage by Trapping Flux with an Iron Core
Two coils closely coupled together by iron
Secondary (Coil 2)
Primary (Coil 1)
Increasing magnetic field

8. An Air-Core Transformer
Not very efficient

9. An Iron-Core Transformer
Np=N1
Ns=N2
Primary (Coil 1)
Secondary (Coil 2)
Input voltage must vary with time (for AC only)
Vin=ΔϕB/Δt=Δ(N1BA)/Δt=N1 AΔ(B)/Δt
Vout=ΔϕB/Δt=Δ(N2BA)/Δt=N2 AΔ(B)/Δt
Vin/Vout = N1/N2

10. Electric transformer: Primary and Secondary Coils
Resistive Load
Energy is conserved!
V2/V1 = N2/N1
Pin = Pout
I1V1 = I2V2
For an ideal transformer
I1/I2 = N2/N1

11. Adjusting the Output Voltage
8 turns
4 turns
Vmax
½Vmax

12. Adjusting the Output Voltage

13. EXAMPLE
A transformer has 10 turns on the primary and 100 turns on the secondary. If 110V rms is applied to the primary, what is the peak-to-peak voltage across the secondary?
Vmax
Vp to p V2

14. More Examples
If a 100 Ω load is connected to the secondary, what is the rms current in the primary?
110 V
100 Ω

15. Transformers in the US Power Grid
Why is power transmitted at such high voltages?
Vrms = 345, 000V
Irms = 1000 A
Typical values
PTransmitted=Vrms Irms=(345, 000V)(1000 A)=345 MW
SUPPOSE 10% of power is lost
PLost = 10% PTransmitted = 34.5MW
PLost = I2 rms Rline
Rline =34.5 Ω
Now suppose Vrms is half and Irms doubles, so the average power transmitted remains the same:
Vrms =0.5 x 345, 000V
Irms = 2 x 1000 A
Ptransmitted=Vrms Irms=345 MW (same as before)
Rline =34.5 Ω (as before)
PLost = I2 rms Rline = (2000A)234.5Ω=138MW
PLost = 40% Ptransmitted !!!

16. Minimizing Energy loss in the Grid
Ptransmitted = Vrms Irms  we need lots of power
Plost = I2rms Rline  we want low losses
For a given Irms = PTransmitted/Vrms
Plost=I2rms Rline = (PTransmitted/Vrms)2 Rline
To minimize losses we need to maximize voltage Vrms